NEW INCOMPRESSIBLE SYMMETRIC TENSOR CATEGORIES IN POSITIVE CHARACTERISTIC

DAVE BENSON, PAVEL ETINGOF, AND VICTOR OSTRIK

Abstract. We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k) = p > 0, we use this method to construct generalizations Ver_{p^{n}}, Ver_{p^{n}}^{+} of the incompressible abelian symmetric tensor categories defined in [5] for p = 2 and in [25, 27] for n = 1. Namely, Ver_{p^{n}} is the abelian envelope of the quotient of the category of tilting modules for SL_{2}(k) by the n-th Steinberg module, and Ver_{p^{n}}^{+} is its subcategory generated by P GL_{2}(k)-modules. We show that Ver_{p^{n}} are reductions to characteristic p of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings ℤ[2 cos(2π/p^{n})], and that Ver_{p^{n}} embeds into Ver_{p^{n+1}}. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Ver_{p^{∞}} of the nested sequence Ver_{p} ⊂ Ver_{p^{2}} ⊂ · · ·. This would provide an analog of Deligne’s theorem in characteristic zero and a generalization of the result of [32], which shows that this conjecture holds for fusion categories, and moreover the fiber functor lands in Ver_{p}.

Contents

1. Introduction  2
2. Splitting ideals and abelian envelopes  5
2.1. Conventions  6
2.2. Splitting objects and splitting ideals  6
2.3. A finiteness property of splitting objects  9
2.4. The abelian category attached to an ideal and its derived category  11
2.5. The semigroup category structure on C  11
2.6. Properties of the tensor product on C  12
2.7. The unit object of C  13
2.8. The monoidal functor F  14
2.9. Properties of the tensor product on C in the case of a splitting ideal  16
2.10. The main theorems  19
2.11. Splitting objects in monoidal categories over discrete valuation rings  23
3. Properties of tilting modules for SL_{2}(k) and splitting objects of T_{n,p}  28
3.1. Splitting objects in T_{n,p}  28
3.2. The functor T_{<p^{n}−1} → T_{n,p} is fully faithful  29
3.3. Tensor ideals in T_{p}  29
3.4. Separatedness of T_{n,p} and its ideal of splitting objects  30
3.5. Donkin’s recursive formula and auxiliary properties of tilting modules  30
4. The Verlinde categories Ver_{p^{n}}  31
4.1. The definition of the Verlinde categories Ver_{p^{n}}, Ver_{p^{n}}^{+}  31

4.2. T_{n,p} is complete  32
4.3. Basic properties of Ver_{p^{n}}  33
4.4. The lift of Ver_{p^{n}} to characteristic zero  34
4.5. The Frobenius-Perron dimension and the Grothendieck ring of Ver_{p^{n}}  34
4.6. The Cartan matrix and the decomposition matrix of Ver_{p^{n}}  35
4.7. An explicit formula for the Cartan matrix of Ver_{p^{n}}  37
4.8. The objects 𝕋_{i} = F (T_{i})  39
4.9. Some properties of the Frobenius functor  40
4.10. The inclusion Ver_{p^{n−1}} ↪ Ver_{p^{n}}  41
4.11. The tensor product theorem for Ver_{p^{n}}  42
4.12. Tensor product of simple objects  44
4.13. The blocks of Ver_{p^{n}}  46
4.14. Ver_{p^{n}} is a Serre subcategory in Ver_{p^{n+k}}  46
4.15. Ext^{1} between simple objects  47
4.16. The Grothendieck ring of the stable category  49
4.17. Incompressibility  50
5. Examples  52
5.1. The category Ver_{p^{2}}  52
5.2. The category Ver_{p^{3}}^{+} in the case p = 3  54
References  56

1. Introduction

A tensor category C is said to have moderate growth if for any object X ∈ C the length of X^{⊗n} grows at most exponentially with n. In characteristic zero, a remarkable theorem of Deligne ([15]) states that every symmetric tensor category of moderate growth is super-Tannakian, i.e., fibers over the category of supervector spaces. In other words, it is the category of representations of an affine supergroup scheme. However, in any characteristic p > 0 this theorem fails. Namely, counterexamples in characteristic p ≥ 5 are given in [25, 27], in characteristic 2 in [34, 5], and in characteristic 3 for the first time in the present paper. In fact, in [5], the first two authors constructed a nested sequence of finite symmetric tensor categories in characteristic two,

Vec = C_{0} ⊂ C_{1} ⊂ C_{2} ⊂ ⋯

which are incompressible, i.e., do not fiber over anything smaller. The main purpose of this paper is to construct similar incompressible symmetric tensor categories in characteristic p > 2. Unfortunately, the approach of [5] does not work in odd characteristic, so we propose a completely different construction. This construction also works in characteristic two, and recovers the categories defined in [5], even though there are some differences in the behavior at the prime two.

Our categories are closely connected with the non-symmetric Verlinde tensor categories in characteristic zero which categorify the Verlinde fusion rules for SL_{2} ([22], 8.18), and because of this connection, we name them Ver_{p^{n}} (n ≥ 1). When p > 2, the category Ver_{p^{n}} decomposes as a tensor product Ver_{p^{n}}^{+} ⊠ sVec of the “even part” and the category of supervector spaces.

The underlying abelian category of Ver_{p^{n}} is easy to construct. We start with the natural two dimensional module V for the algebraic group SL_{2}(k), and take its tensor powers. Each V^{⊗m} has exactly one isomorphism class of indecomposable direct summand that does not appear in any previous tensor power, and this is the tilting module T_{m}. For m < p, T_{m} is isomorphic to the symmetric power S^{m}V, and for m = p^{k} − 1 it is the Steinberg module St_{k} = S^{p^{k}−1}V. We form the direct sum ⨁_{m=p^{n−1}−1}^{p^{n}−2}T_{m} and form its endomorphism ring A. Then Ver_{p^{n}} is the category A-mod of finite dimensional A-modules.

Constructing the symmetric tensor structure on Ver_{p^{n}} is more subtle. The essential point is to prove a splitting property of tilting modules.

Proposition 1.1. (Proposition 3.2) Every morphism between tilting modules T_{m} with m < p^{k} − 1 is split by tensoring with St_{k−1}.

In order to use this fact for putting an appropriate symmetric tensor structure on Ver_{p^{n}}, we set up a general categorical machinery of splitting ideals and abelian envelopes, which is of independent interest and likely to have other applications. Namely, given a Karoubian rigid monoidal category T , we say that Q ∈ T is a splitting object if tensoring with Q splits every morphism in T ; such objects form a thick ideal S ⊂ T . We say that T is separated if a morphism in T annihilated by tensoring with any splitting object is zero. Then we prove the following result.

Theorem 1.2. (Theorems 2.41,2.42) If T is separated and S is finitely generated as a left ideal, then there is a multitensor (in particular, abelian) category C(T ) together with a faithful monoidal functor F : T → C(T ) which identifies S with the ideal of projectives in C(T ). Moreover, if this functor is full then C(T ) is the abelian envelope of T (i.e., faithful monoidal functors from T to any multitensor category D correspond to tensor functors C(T ) → D).

We then apply this construction to the category T = T_{n,p} of tilting modules for SL_{2}(k) modulo the tensor ideal generated by St_{n}. This is enabled by Proposition 3.2, which implies that the category T_{n,p} is separated. This yields an abelian equivalence C(T_{n,p}) ≅ Ver_{p^{n}}, which equips Ver_{p^{n}} with a structure of a symmetric tensor category.

Our main theorem is as follows.

Theorem 1.3. Let k be an algebraically closed field of prime characteristic p. Then there are nested sequences of symmetric tensor categories Ver_{p^{n}}^{+} = Ver_{p^{n}}^{+}(k) ⊂ Ver_{p^{n}} = Ver_{p^{n}}(k) over k,

Ver_{p}^{+} ⊂ Ver_{p^{2}}^{+} ⊂ Ver_{p^{3}}^{+} ⊂ ⋯

and

Ver_{p} ⊂ Ver_{p^{2}} ⊂ Ver_{p^{3}} ⊂ ⋯

(Theorem 4.35) with the following properties.^{1}

(1) The category Ver_{p} is the semisimplification of the module category for ℤ/p. For n ≥ 2, the categories Ver_{p^{n}}, Ver_{p^{n}}^{+} are not semisimple (Theorem 4.5(i),(ii)).

(2) For p > 2, the category Ver_{p^{n}} decomposes as a tensor product Ver_{p^{n}}^{+} ⊠sVec (Corollary 4.12).

(3) We have Ver_{2^{n}} = C_{2n−2} and Ver_{2^{n}}^{+} = C_{2n−3}, the symmetric tensor categories contructed in [5] (Theorem 4.5(iii)).

1We do not define the category Ver_{2}^{+}.

(4) If p^{n} > 2, the Grothendieck ring of Ver_{p^{n}}^{+} is isomorphic to the ring of integers in the real part of the cyclotomic field of p^{n}-th roots of unity, namely ℤ[2 cos(2π/p^{n})]. In other words, Ver_{p^{n}}^{+} gives an abelian categorification of the ring ℤ[2 cos(2π/p^{n})]. The Grothendieck ring of Ver_{p^{n}} is isomorphic to the group ring of ℤ/2 over ℤ[2 cos(2π/p^{n})] if p > 2 and to ℤ[2 cos(2π/2^{n+1})] if p = 2 (Theorem 4.5(iv)).

(5) The Frobenius-Perron dimension of Ver_{p^{n}} is \frac{p^{n}}{2 sin^{2}(π/p^{n})} (Proposition 4.11).

(6) The categories Ver_{p^{n}}^{+}, p > 2 and Ver_{2^{n}} are incompressible, i.e., any (not necessarily braided) tensor functor out of them is a fully faithful embedding. In particular, these categories do not admit a tensor functor to a category of smaller Frobenius-Perron dimension. Likewise, the categories Ver_{p^{n}} and Ver_{2^{n}}^{+} are incompressible as symmetric tensor categories (Theorem 4.70).

(7) The category Ver_{p^{n}} admits a semisimple braided lift to characteristic zero, producing the semisimplification Ver_{p^{n}}(K) of the category of tilting modules for quantum SL_{2} at the root of unity of order p^{n} (Subsection 4.4).

(8) (The Steinberg tensor product theorem for Ver_{p^{n}}) There exist simple objects 𝕋_{j}^{[k]} of Ver_{p^{n}}, j = 0, ..., p − 1, k = 1, ..., n, and j ≠ p − 1 if k = 1, such that any simple object of Ver_{p^{n}} can be uniquely written in the form L_{i} = 𝕋_{i_{1}}^{[1]} ⊗ ... ⊗ 𝕋_{i_{n}}^{[n]}, where 0 ≤ i = \overline{i_{1}...i_{n}} ≤ p^{n−1}(p − 1) − 1 is an integer written in base p. The simple objects of Ver_{p^{n}}^{+} are the L_{i} with even i (Theorem 4.43).

(9) The category Ver_{p^{n}} has p − 1 blocks of sizes 1, p − 1, p(p − 1), ..., p^{n−2}(p − 1), so a total of n(p − 1) blocks. Moreover, all blocks of the same size are equivalent, even for different n. For instance, the blocks of size 1 are semisimple (i.e., generated by a simple projective object), and the blocks of size p−1 are equivalent to the unique nonsemisimple block of representations of the symmetric group S_{p} over k (Proposition 4.55, Proposition 5.1).

(10) The Grothendieck ring of the stable category of Ver_{p^{n}}^{+} is isomorphic to 𝔽_{p}[z]/z^{\frac{p^{n−1}−1}{2}} for p > 2 and 𝔽_{2}[z]/z^{2^{n−2}} for p = 2. The Grothendieck ring of the stable category of Ver_{2^{n}} is 𝔽_{p}[z]/z^{2^{n−1}−1} (Proposition 4.67).

(11) The determinant of the Cartan matrix of a block of size p^{r}(p − 1) in Ver_{p^{n}} is p^{p^{r}} for r ≥ 0, and its entries are 0 or powers of 2 (Proposition 4.69, Corollary 4.27).

(12) If p > 2 then Ver_{p^{n}} is a Serre subcategory in Ver_{p^{n+k}} (Proposition 4.58).

Theorem 1.3 implies that we can define tensor categories Ver_{p^{∞}} = ⋃_{n≥1}Ver_{p^{n}}. Motivated by the main result of [32], we make the following (perhaps imprudently) bold conjecture:

Conjecture 1.4. Any symmetric tensor category of moderate growth over an algebraically closed field k of characteristic p > 0 admits a fiber functor to Ver_{p^{∞}}.

Conjecture 1.4 is open even for finite tensor categories. In particular, this conjecture would imply that Deligne’s theorem holds for integral finite symmetric tensor categories for p > 2 (for example, for representation categories of triangular Hopf algebras); in other words, that such categories are representation categories of finite supergroup schemes. However, even this specialization of the conjecture is currently wide open.

In any case, the construction of the category Ver_{p^{∞}} opens the door for defining and studying many new interesting symmetric tensor categories, which may be constructed as representation categories of affine group schemes in Ver_{p^{∞}}. Examples of such group schemes are the general linear group GL(X) for X ∈ Ver_{p^{∞}}, as well as the orthogonal group O(X) and the symplectic group Sp(X) when X is equipped with a symmetric, respectively skewsymmetric isomorphism X → X^{∗}. In particular, we see that in characteristic p there are infinitely many versions of general linear groups of a given “rank” r, parametrized by various objects X of length r.

The results of this paper have applications to modular representation theory of elementary abelian p-groups which are discussed in [5], Subsection 4.4. Namely, let V be the tautological 2-dimensional k-representation of the additive group of the field 𝔽_{p^{n}} regarded as the unipotent subgroup of SL_{2}(𝔽_{p^{n}}), and let D_{n,p} be the quotient of the Karoubian tensor category generated by V (i.e., with indecomposables being direct summands of tensor powers of V) by the ideal of projectives, i.e., the corresponding stable category. Then it is shown in [5], Proposition 4.5 that D_{n,p} is equivalent to T_{n,p}. This implies that Ver_{p^{n}} is the abelian envelope of D_{n,p}, which greatly clarifies the structure of D_{n,p}.

The organization of the paper is as follows. In Section 2 we develop the technology of splitting ideals and abelian envelopes. In Section 3 we recall and prove a number of auxiliary results about tilting modules for SL_{2}(k). In Section 4 we apply the technology of Sections 2, 3 to constructing and studying the categories Ver_{p^{n}}. In particular, we prove the properties of Ver_{p^{n}} listed in Theorem 1.3, and also compute the Cartan and decomposition matrices of their blocks, the fusion rules, and Ext^{1} between simple objects. Finally, in Section 5 we compute explicitly the examples Ver_{p^{2}} and Ver_{3^{3}}. Remark 1.5. The categories Ver_{p^{n}} were also constructed simultaneously and independently by Kevin Coulembier in [13]. This is an application of his general theory of abelian envelopes alternative to the one we propose in Section 2.

Acknowledgements. We are grateful to Kevin Coulembier, Thorsten Heidersdorf and Raphaël Rouquier for useful discussions. This material is based on work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California (D. B. in Spring 2018, P. E. in Spring 2018 and Spring 2020, V. O. in Spring 2020). The work of P. E. was partially supported by the NSF grant DMS-1502244. The work of V. O. was partially supported by the NSF grant DMS-1702251 and by the Russian Academic Excellence Project ‘5-100’. The work of D. B. was partly supported by the Isaac Newton Institute for Mathematical Sciences during the Spring 2020 semester, under EPSRC grant EP/R014604/1. Many of the results in Section 5 were obtained by a computer calculation using MAGMA ([6]).

2. Splitting ideals and abelian envelopes

In this section we propose a general construction of abelian envelopes C(T ) of a certain class of Karoubian rigid monoidal categories T which we call separated and complete. Under a certain finiteness assumption on T , the resulting category C(T ) is a multitensor category with enough projectives. This construction is designed to build the categories Ver_{p^{n}} but is interesting in its own right, and we expect it to have applications beyond the main goals of this paper. Therefore we developed this theory in a greater generality than actually needed here. In particular, a number of statements in this section, specifically Lemma 2.4, Propositions 2.8, 2.9, Corollary 2.10, Propositions 2.31, 2.32, Corollary 2.39, Theorem 2.43, are not used in the subsequent sections of this paper.

2.1. Conventions. Throughout the paper, k will denote an algebraically closed field. We will freely use the theory of tensor categories and refer the reader to [22] for the basics of this theory. In particular, we will use the conventions of this book.

Namely, by an artinian category over k we mean a k-linear abelian category in which objects have finite length and morphism spaces are finite dimensional, and call such a category finite if it has finitely many (isomorphism classes of) simple objects and enough projectives. By a (multi)tensor category we will mean the notion defined in [22], Definition 4.1.1; in particular, such categories are artinian (hence abelian). We will denote the category of finite dimensional vector spaces by Vec_{k} or shortly by Vec and for char(k) ≠ 2 the category of finite dimensional supervector spaces by sVec_{k} or shortly sVec. If X is an object of a tensor category with finitely many simple objects, then we denote by FPdim(X) the Frobenius-Perron dimension of X ([22], Subsections 3.3, 4.5). We will also consider categories with tensor product which are not abelian but only Karoubian; in this case we will use the term Karoubian monoidal category. Similarly, by a tensor functor we will mean an exact monoidal functor between tensor categories, see [22], Definition 4.2.5 (note that such a functor is automatically faithful, see [22], Remark 4.3.10). In more general situations we will use the term monoidal functor; all monoidal functors we consider will be additive. Also, a symmetric tensor functor out of a symmetric tensor category into another, usually quite concrete one (such as Vec_{k}, sVec_{k} or Ver_{p^{n}} defined below) will be called a fiber functor.

Finally, we will need the notions of surjective and injective tensor functors, [22], Subsection 6.3. Namely, a tensor functor F : C → D is surjective if every object Y ∈ D is a subquotient of F (X) for some X ∈ C, and F is injective if it is fully faithful (i.e., embedding of a tensor subcategory).

2.2. Splitting objects and splitting ideals. Let k be an algebraically closed field. Throughout the paper, T will be a k-linear rigid monoidal Karoubian category with bilinear tensor product and finite dimensional Hom spaces (in particular, it is Krull-Schmidt).

Recall that a (thick) ideal P ⊂ T is a full Karoubian subcategory closed under tensoring on both sides with T (note that then it is also closed under left and right duals since Q is a direct summand in Q⊗Q^{∗}⊗Q and Q⊗^{∗}Q⊗Q). Thus, P is determined by its indecomposable objects, which form a subset of indecomposable objects of T . For a thick ideal P ⊂ T , the tensor ideal ⟨P⟩ generated by P is the collection of morphisms which factor through objects of P. We can form the quotient monoidal category T /⟨P⟩, which we will sometimes also denote by T /P.

Recall also that an ideal P is said to be finitely generated (as a left ideal) if there exists an object P ∈ P such that every indecomposable object Q ∈ P is a direct summand in X ⊗ P for some X ∈ T (note that this is a stronger condition than finite generation as a two-sided ideal, see Example 2.44(2)). In this case by the rigidity of T we may (and will) choose P so that every indecomposable object Q ∈ P is also a direct summand in P ⊗ Y for some Y ∈ T . Such an object P will be called a generator of P. For example, if P has finitely many isomorphism classes of indecomposables, then the direct sum P of representatives of these classes is a generator.

Recall that a morphism f : X → Y in T is called split if it is a direct sum of a zero morphism and an isomorphism. In other words, f is split if it is a composition ι ∘ π where π is a split epimorphism (projection to a direct summand) and ι is a split monomorphism (inclusion of a direct summand). It is easy to see that f is split if and only if there exists g : Y → X such that

(2.1)  f ∘ g ∘ f = f, g ∘ f ∘ g = g.

In fact, for this it suffices that there exist h : Y → X such that f ∘h∘f = f; then g := h∘f ∘h satisfies (2.1).

The following lemma is well known but we provide a proof for reader’s convenience.

Lemma 2.1. The direct sum of two morphisms is split if and only if each summand is split.

Proof. The “if” direction is obvious, so let us prove the “only if” direction. If f_{i} : X_{i} → Y_{i}, i = 1, 2, and f = f_{1} ⊕ f_{2} is split then there is h : Y_{1} ⊕ Y_{2} → X_{1} ⊕ X_{2} satisfying f ∘ h ∘ f = f. Let h_{ij} be the components of h. Then we have f_{1} ∘ h_{11} ∘ f_{1} = f_{1}, i.e., f_{1} is split. □

Definition 2.2. We will say that an ideal P ⊂ T is a splitting ideal if for any Q_{1}, Q_{2}, R ∈ P and any morphism f : Q_{1} → Q_{2} the morphism 1_{R} ⊗ f : R ⊗ Q_{1} → R ⊗ Q_{2} is split.

Note that in this case it follows by taking duals that the morphism f⊗1_{R} : Q_{1}⊗R → Q_{2}⊗R is also split.

Definition 2.3. We will say that an object R ∈ T is a splitting object if for any Q_{1}, Q_{2} ∈ T and a morphism f : Q_{1} → Q_{2} the morphisms 1_{R} ⊗ f : R ⊗ Q_{1} → R ⊗ Q_{2} and f ⊗ 1_{R} : Q_{1} ⊗ R → Q_{2} ⊗ R are split.

It is clear that splitting objects form a splitting ideal, which we denote by S = S(T ).

Lemma 2.4. Every splitting ideal consists of splitting objects.

Proof. Let P ⊂ T be a splitting ideal, R ∈ P and f : Q_{1} → Q_{2} be a morphism in T . Then g := 1_{R} ⊗ f : R ⊗ Q_{1} → R ⊗ Q_{2} is a morphism in P (as P is an ideal). Thus the morphism

1_{R⊗R^{∗}} ⊗ g : R ⊗ R^{∗} ⊗ R ⊗ Q_{1} → R ⊗ R^{∗} ⊗ R ⊗ Q_{2}

is split. But R is a direct summand in R ⊗ R^{∗} ⊗ R since R is rigid. This implies that 1_{R⊗R^{∗}} ⊗g = g ⊕h, so g itself is split by Lemma 2.1. Similarly, f ⊗1_{R} is split, as claimed. □

Lemma 2.4 implies that S is the unique maximal splitting ideal in T , containing all other splitting ideals.

A trivial example of an ideal is P = 0 (it is clearly splitting and finitely generated with P = 0). Here are some more interesting examples.

Example 2.5. Suppose that T is a multitensor category ([22], Definition 4.1.1) with enough projectives. Then the ideal P = Pr(T ) of projectives in T coincides with S(T ) and is a finitely generated splitting ideal. A generator P is the projective cover of 𝟙. Indeed, it is clear that every projective object is splitting. Conversely, if Q splits every morphism between projective objects then in particular it splits a morphism ξ : P_{1} → P_{0} whose cokernel is 𝟙, which implies that Q is projective, since Q is the cokernel of the split morphism ξ ⊗ 1_{Q}.

Example 2.6. Let T be the rigid monoidal category of rigid finite dimensional bimodules over a finite dimensional non-semisimple Frobenius algebra B (i.e., those projective as a left and as a right module); note that the functor of double dual in this category is given by the Nakayama automorphism. Then the subcategory P of projective bimodules is a finitely generated ideal (with generator P = B^{e} := B ⊗ B^{op}), but is not a splitting ideal. Indeed, there are non-split endomorphisms of the bimodule B^{e} (for example, nonzero elements of the radical of B^{e}), and they are not split by tensoring with any rigid bimodule. In fact, in this case there are no nonzero splitting ideals at all, since for any bimodule Q the bimodule B^{e} ⊗_{B} Q ⊗_{B} B^{e} = B ⊗_{k} Q ⊗_{k} B is free.

Let us now describe all splitting ideals in T .

Lemma 2.7. (a) Let R ∈ T be an indecomposable splitting object. Then the following conditions on an object Q ∈ T are equivalent:

(i) there exists X ∈ T such that Hom(X ⊗ Q, R) ≠ 0;

(ii) there exists X ∈ T such that Hom(R, X ⊗ Q) ≠ 0;

(iii) there exists X ∈ T such that R is a direct summand of X ⊗ Q.

(b) Let Q, R ∈ T be indecomposable splitting objects. Then there exists X ∈ T such that R is a direct summand in X ⊗ Q if and only if there exists X ∈ T such that Q is a direct summand of X ⊗ R.

Proof. (a) It is clear that (iii) implies (i) and (ii).

Suppose Hom(X ⊗ Q, R) ≠ 0. Then Hom(X, R ⊗ Q^{∗}) ≠ 0. Thus R ⊗ Q^{∗} ≠ 0. Hence the natural map 1 ⊗ ev : R ⊗ Q^{∗} ⊗ Q → R is not zero (as it corresponds to the identity morphism of R ⊗ Q^{∗}). Also this morphism is split since R is a splitting object. Since R is indecomposable, 1 ⊗ ev is a split epimorphism. Hence R is a direct summand of Y ⊗ Q, where Y = R ⊗ Q^{∗}. Thus (i) implies (iii).

Suppose Hom(R, X ⊗ Q) ≠ 0. Then Hom(R ⊗ ^{∗}Q, X) ≠ 0. Thus R ⊗ ^{∗}Q ≠ 0. Hence the natural map 1 ⊗ coev : R → R ⊗^{∗}Q ⊗ Q is not zero (as it corresponds to the identity morphism of R ⊗^{∗}Q). Also this morphism is split since R is a splitting object. Since R is indecomposable, 1 ⊗ coev is a split monomorphism. Thus R is a direct summand of Y ⊗ Q, where Y = R ⊗^{∗}Q. Thus (ii) implies (iii).

(b) Suppose R is a direct summand of X ⊗ Q. Then Hom(R, X ⊗ Q) ≠ 0. It follows that Hom(X^{∗} ⊗ R, Q) ≠ 0. So by (a) there is Y ∈ T such that Q is a direct summand of Y ⊗ R. □

Now consider the set S = S(T ) of isomorphism classes of indecomposable splitting objects in T . Then all splitting objects are direct sums of objects from S. Define a relation on S by the condition that Q ∼ R if there exist X, Y ∈ T such that R is a direct summand in X ⊗ Q ⊗ Y. Clearly, this relation is reflexive and transitive, and it is also symmetric by Lemma 2.7(b), so it is an equivalence relation. Note that by Lemma 2.7(a) Q ∼ R iff there exist X, Y ∈ T such that Hom(Q, X ⊗ R ⊗ Y) ≠ 0 iff there exist X, Y ∈ T such that Hom(X ⊗R ⊗Y, Q) ≠ 0. Note also that Q^{∗} ∼ Q since Q is a direct summand in Q⊗Q^{∗} ⊗Q.

Let \overline{S} := S/ ∼. Every class c ∈ \overline{S} generates a minimal splitting ideal P(c). It is easy to see that if c_{1} ≠ c_{2} then P(c_{1}) ⊗ P(c_{2}) = 0 (as these ideals are disjoint).

Thus we have the following proposition.

Proposition 2.8. Splitting ideals of T are in 1-1 correspondence with subsets K ⊂ \overline{S} defined by the formula P_{K} := ⨁_{c∈K}P(c). Among them, finitely generated splitting ideals correspond to such sums that are finite with P(c) finitely generated for each c ∈ K. In particular, S = P\overline{S}.

Proof. This follows from Lemma 2.4. □

We can also define the equivalence relations ∼_{ℓ}, ∼_{r} as follows: Q ∼_{ℓ} R if there is X ∈ T such that Q is a direct summand in X ⊗ R, and Q ∼_{r} R if there is X ∈ T such that Q is a direct summand in R ⊗ X (the symmetry of these relations follows from Lemma 2.7(b)). Note that by Lemma 2.7(a) Q ∼_{ℓ} R iff there is X ∈ T such that Hom(Q, X ⊗ R) ≠ 0 and Q ∼_{r} R iff there is X ∈ T such that Hom(Q, R ⊗ X) ≠ 0. The relation ∼_{ℓ} splits an equivalence class c ∈ \overline{S} into the disjoint union of classes c_{•j}, j ∈ I_{c}, and the relation ∼_{ℓ} splits c into the disjoint union of classes c_{i•}, i ∈ I_{c}. We can define c_{ij} := c_{i•} ∩ c_{•j}, with c_{ij}^{∗} = c_{ji}, then c is a disjoint union of c_{ij}. Let P(c)_{i•}, P(c)_{•j}, P(c)_{ij} be the subcategories of P additively generated by c_{i•}, c_{•j}, c_{ij}, respectively; clearly, P_{•j} is a left ideal and P_{i•} is a right ideal. Thus we have decompositions

P(c) = ⨁_{i∈I_{c}}P_{i•} = ⨁_{j∈I_{c}}P_{•j} = ⨁_{i,j∈I_{c}}P(c)_{ij}.

2.3. A finiteness property of splitting objects.

Proposition 2.9. Let Q, R be indecomposable splitting objects in T . If Hom(R, Q) ≠ 0 then R is a direct summand in Q ⊗ Q^{∗} ⊗ Q.

Proof. Step 1. Let Y ∈ T be indecomposable. Let 𝔪_{Y} be the radical of the local algebra End(Y). For each indecomposable object X ∈ T set U(X, Y) := X ⊗ Hom(X, Y)^{∗} if X is not isomorphic to Y, and U(Y, Y) := Y ⊗ 𝔪_{Y}^{∗}, and extend this definition by additivity to all objects X ∈ T . Then U(X, Y) is equipped with a canonical morphism ξ : U(X, Y) → Y which is not a split epimorphism, and any morphism g : X → Y which is not a split epimorphism factors through ξ.

Step 2. Now let R, Q be indecomposable splitting objects of T . Let

U := U(Q ⊗ Q^{∗} ⊗ Q ⊕ R ⊗ Q^{∗} ⊗ Q, R).

Then the map ξ ⊗ 1_{Q^{∗}} : U ⊗ Q^{∗} → R ⊗ Q^{∗} is split. Thus we can write U ⊗ Q^{∗} ≅ N ⊕ K, R ⊗ Q^{∗} ≅ N ⊕ L, so that ξ ⊗ 1 = 1_{N} ⊕ 0_{K→L}.

Step 3. Now assume that f : R → Q is a morphism. Then the map

f ⊗ 1_{Q^{∗}} : R ⊗ Q^{∗} → Q ⊗ Q^{∗}

is split. So we can write Q ⊗ Q^{∗} ≅ M ⊕ C, and we have a split epimorphism

g : R ⊗ Q^{∗} = N ⊕ L → M.

Step 4. Also, the map

f ∘ ξ ⊗ 1_{Q^{∗}} : U ⊗ Q^{∗} ≅ N ⊕ K → Q ⊗ Q^{∗} ≅ M ⊕ C

is split. It is a direct sum of a map η : N → M and 0_{K→C}, so the map η is split. So we can write N ≅ W ⊕ V, M ≅ W ⊕ Y, so that η = 1_{W} ⊕ 0_{V →Y}. Thus the map g : N ⊕ L → M can be written as g : W ⊕ V ⊕ L → W ⊕ Y. Since g is a split epimorphism and g|_{V} = 0, we obtain a split epimorphism ḡ : W ⊕ L → W ⊕ Y, such that ḡ|_{W} = 1_{W}. Let h : L → W be

the corresponding component of ḡ. Then composing ḡ with the automorphism (\begin{matrix}1 & −h \\ 0 & 1 \end{matrix}) of

W ⊕ L, we obtain a split epimorphism ḡ′ = 1_{W} ⊕ g̃, where g̃: L → Y is a split epimorphism.

Step 5. Now assume additionally that f ≠ 0. Then we claim that the map ξ ⊗ 1_{Q^{∗}} is not a split epimorphism, i.e., L ≠ 0. Indeed, the morphism

∘ξ : Hom(1, U ⊗ R^{∗}) ≅ Hom(R, U) → Hom(1, R ⊗ R^{∗}) ≅ Hom(R, R)

is not surjective, as ξ is not split (so 1_{R} can’t be in the image). Hence the morphism ξ ⊗ 1_{R^{∗}} : U ⊗ R^{∗} → R ⊗ R^{∗} is not a split epimorphism. But R, Q belong to the same P(c)_{ij}, so there is X ∈ T such that R^{∗} is a direct summand of Q^{∗}⊗X. Thus ξ⊗1_{Q^{∗}⊗X} = ξ⊗1_{Q^{∗}} ⊗1_{X} is not a split epimorphism either (indeed, the direct sum of two morphisms can only be a split epimorphism if both summands are). Hence ξ ⊗ 1_{Q^{∗}} is not a split epimorphism, as claimed.

Step 6. Moreover, we claim that Y ≠ 0. Assume that Y = 0. Consider the map θ : L → R ⊗ Q^{∗} = W ⊕ V ⊕ L given by θ = (θ_{1}, θ_{2}, θ_{3}), where θ_{1} = −h, θ_{2} = 0, θ_{3} = 1_{L}. It is easy to see that the map (f ⊗ 1_{Q^{∗}}) ∘ θ : L → Q ⊗ Q^{∗} is zero. Let ζ be the element of Hom(L ⊗ Q, R) corresponding to θ. The composition f ∘ ζ ∈ Hom(L ⊗ Q, Q) is zero. Thus the map ζ is not a split epimorphism, as f ≠ 0. Also L ⊗ Q is a direct summand in R ⊗ Q^{∗} ⊗ Q, since L is a direct summand in R ⊗ Q^{∗}. Thus by the definition of U, ξ, there exists an element ζ̃ ∈ Hom(L ⊗ Q, U) which maps to ζ. Let θ̃ ∈ Hom(L, U ⊗ Q^{∗}) correspond to ζ̃. Then θ̃ maps to θ. However, the image of any element of Hom(L, U ⊗ Q^{∗}) factors through N, so cannot equal θ (as L ≠ 0), which contradicts the assumption that Y = 0. Thus, Y ≠ 0.

Step 7. Since Y is a direct summand in both L and Q ⊗ Q^{∗}, this implies that

Hom(Q ⊗ Q^{∗}, L) ≠ 0.

Thus the map

∘(ξ ⊗ 1_{Q^{∗}}) : Hom(Q ⊗ Q^{∗}, U ⊗ Q^{∗}) → Hom(Q ⊗ Q^{∗}, R ⊗ Q^{∗})

is not surjective. Hence the map

∘ξ : Hom(Q ⊗ Q^{∗} ⊗ Q, U) → Hom(Q ⊗ Q^{∗} ⊗ Q, R)

is not surjective either. So there exists a morphism π : Q ⊗ Q^{∗} ⊗ Q → R that does not factor through ξ. By the definition of U, ξ, this implies that π is a split epimorphism, hence R is a direct summand in Q ⊗ Q^{∗} ⊗ Q, as claimed. □

Corollary 2.10. For every indecomposable splitting object Q there are finitely many indecomposable splitting objects R such that Hom(R, Q) ≠ 0; namely, at most the number of distinct indecomposable direct summands in Q ⊗ Q^{∗} ⊗ Q.

Proof. This follows immediately from Proposition 2.9. □

Remark 2.11. Suppose that in Proposition 2.9, there is a monoidal functor F : T → C from T to a multitensor category C with enough projectives which is fully faithful on projectives, and the objects Q, R are projective in C (we will show below that this is so under mild assumptions, but the proof uses Proposition 2.9). Then there is a much more direct proof of Proposition 2.9. Namely, suppose that R is the projective cover of a simple object X ∈ C. Then, Hom(R, Q) ≠ 0 implies that X is a composition factor in Q. Thus X ⊗ Q^{∗} ≠ 0 (as X ⊗ X^{∗} is a subquotient of X ⊗ Q^{∗}), and X ⊗ Q^{∗} ⊗ Q is a direct summand in Q ⊗ Q^{∗} ⊗ Q. But we have a morphism 1 ⊗ ev : X ⊗ Q^{∗} ⊗ Q → X, which is nonzero since it corresponds to the identity endomorphism of the nonzero object X ⊗ Q^{∗}. Since X ⊗ Q^{∗} ⊗ Q is projective, it follows that it contains R as a direct summand.

In fact, the proof of Proposition 2.9 is just an adaptation of this argument which does not refer to the category C.

2.4. The abelian category attached to an ideal and its derived category. Let P ⊂ T be an ideal. Let I be the set of isomorphism classes of indecomposables in P with representatives P_{i}, i ∈ I. Let us say that P has the finiteness property if for any Q ∈ P there are finitely many i ∈ I such that Hom(P_{i}, Q) ≠ 0. For example, by Corollary 2.10, the ideal S (and, in fact, any splitting ideal) automatically has the finiteness property.

From now on we will assume that P has the finiteness property. Let C := ⨁_{i,j∈I}Hom(P_{i}, P_{j})^{∗}. This space is naturally a coalgebra whose dual is the algebra A := C^{∗} = ∏_{i,j∈I} Hom(P_{i}, P_{j}) with operation ab = b ∘ a (note that this product is well defined and does not involve infinite summations because of the finiteness property of P). For example, if I is finite then A = End(P)^{op}, where P := ⨁_{i∈I}P_{i}.

Let C = C(T , P) := C-comod be the category of finite dimensional C-comodules (or, equivalently, C^{∗}-modules when I is finite). Equivalently, C is the category of additive functors P^{op} → Vec which are of finite length, i.e., vanish on almost all P_{i}. Then C is an artinian category with enough projectives and simple objects L_{i}, i ∈ I. Namely, we have a natural inclusion ι : P ↪ C as a full subcategory given by ι(Q) := ⨁_{i∈I}Hom(P_{i}, Q), whose image is the ideal Pr(C) of projective objects in C (note that dim ι(Q) < ∞ since Hom(P_{i}, Q) = 0 for almost all i). We will identify P with Pr(C) using ι and, abusing notation, will often denote the image of Q ∈ P under ι also by Q. Then P_{i} are projective covers of L_{i}.

Consider now the bounded above derived category D^{−}(C). By a standard theorem of homological algebra ([26], III.5.21), we can realize D^{−}(C) as the bounded above homotopy category of projectives K^{−}(Pr(C)) ≅ K^{−}(P). This allows us to realize the objects of D^{−}(C) explicitly as complexes

... → P_{n+2} → P_{n+1} → P_{n},

where P_{j} ∈ Pr(C). Moreover, the subcategory C ⊂ D^{−}(C) (the heart of the tautological t-structure) is realized as the subcategory of such complexes which are exact in nonzero degrees, i.e., the subcategory of resolutions

... → P_{2} → P_{1} → P_{0}.

2.5. The semigroup category structure on C. Note that P is a semigroup category ([22], p.25; i.e., it has a tensor product with associativity isomorphism satisfying the pentagon axiom, but has no unit in general). This implies

Proposition 2.12. The category K^{−}(P) ≅ D^{−}(C) has a natural structure of a semigroup category, given by tensoring bounded above complexes of objects from P. Therefore, the category C is equipped with the induced semigroup category structure (given by taking the zeroth cohomology of the tensor product), which extends the tensor product on P.

Proof. The proof is standard. Namely, given X, Y ∈ C, fix projective resolutions P_{•} of X and Q_{•} of Y. We define the derived tensor product X ⊗^{L} Y to be the object of D^{−}(C) represented by the complex P_{•} ⊗ Q_{•}. In particular, we set X ⊗ Y := H^{0}(X ⊗^{L} Y) ∈ C.

Let us see why X, Y ↦ X ⊗^{L} Y is a well defined bifunctor. Let P′_{•}, Q′_{•} be other resolutions of X, Y. Then there exist morphisms of resolutions f : P_{•} → P′_{•}, g : Q_{•} → Q′_{•} which extend the identity morphisms 1_{X}, 1_{Y}, and they are defined uniquely up to homotopy. Similarly, we have f′ : P′_{•} → P_{•}, g′ : Q′_{•} → Q_{•}, and f ∘ f′, f′ ∘ f, g ∘ g′, g′ ∘ g are homotopic to the identity morphisms. Now consider the morphism f ⊗ g : P_{•} ⊗ Q_{•} → P′_{•} ⊗ Q′_{•}. This is well defined up to homotopy, and the inverse is f′ ⊗ g′ up to homotopy. Thus f ⊗ g defines an isomorphism X ⊗^{L} Y → (X ⊗^{L} Y)′ of the derived tensor products defined using these two pairs of resolutions, and it is easy to show that this isomorphism is independent of the choice of f, g. Moreover, if we have three resolutions for X and for Y then it is shown in a standard way that the composition around the corresponding triangle is the identity. Thus X ⊗^{L} Y and hence X ⊗ Y is well defined. □

Remark 2.13. In fact, to define X ⊗ Y, one does not need the full resolutions of X and Y by objects of P. All one needs is presentations of X, Y, i.e., their representations as cokernels of morphisms f : P_{1} → P_{0} and g : Q_{1} → Q_{0} between objects from P. In this case we have a morphism

f ⊗ 1 + 1 ⊗ g : P_{1} ⊗ Q_{0} ⊕ P_{0} ⊗ Q_{1} → P_{0} ⊗ Q_{0},

and X ⊗ Y is just the cokernel of this morphism. We refer the reader to [30, Section 3] for more details.

Example 2.14. 1. If P = 0 then C = 0.

2. In the setting of Example 2.5 we have C = T , and the semigroup structure on C constructed above recovers the original monoidal structure on T .

3. Similarly, in Example 2.6, C ⊃ T is the category of all finite dimensional B-bimodules with tensor product over B, and X ⊗^{L} Y is the usual derived tensor product over B.

2.6. Properties of the tensor product on C. Let us now study the properties of the tensor product on C defined in Proposition 2.12.

Lemma 2.15. (i) Let Q_{•} be a projective resolution of an object Y ∈ C. Then for each T ∈ T the complexes T ⊗ Q_{•} and Q_{•} ⊗ T in K^{−}(P) are acyclic in strictly negative degrees.

(ii) Let X ∈ P. Then for any Y ∈ C we have H^{−i}(X ⊗^{L} Y) = H^{−i}(Y ⊗^{L} X) = 0 for all i > 0.

Proof. (i) Let P_{1} → P_{2} → P_{3} be an exact sequence of projective objects of C. This means that the complex

Hom(Q, P_{1}) → Hom(Q, P_{2}) → Hom(Q, P_{3})

is acyclic for any Q ∈ P. Now consider the complex

(2.2)  Hom(Q, T ⊗ P_{1}) → Hom(Q, T ⊗ P_{2}) → Hom(Q, T ⊗ P_{3}).

It can be written as

Hom(T^{∗} ⊗ Q, P_{1}) → Hom(T^{∗} ⊗ Q, P_{2}) → Hom(T^{∗} ⊗ Q, P_{3}).

But T^{∗} ⊗ Q ∈ P. Thus the complex (2.2) is acyclic. This implies that the complex

T ⊗ P_{1} → T ⊗ P_{2} → T ⊗ P_{3}

is acyclic, which implies the claim for T ⊗ Q_{•}. The claim for Q_{•} ⊗ T is proved similarly. (ii) follows from (i). □

Corollary 2.16. The bifunctor (X, Y) ↦ X ⊗ Y on C is right exact in both arguments, and

X ⊗^{L_{i}} Y = H^{−i}(X ⊗^{L} Y),

where ⊗^{L_{i}} is the i-th left derived functor of the tensor product. Moreover, if X is projective then the functors X⊗?,? ⊗ X on C are exact.

Proof. This follows immediately from Lemma 2.15(ii). □

Corollary 2.17. The monoidal category T acts on C on the left by exact functors T_{l}(X) := T ⊗ X, T ∈ T , X ∈ C, commuting with the action of C on itself by right multiplication. There is also a similar action on the right given by T_{r}(X) := X ⊗ T which commutes with the action of C on itself by left multiplication. Also, these actions map projective objects to projective ones.

Proof. This follows from Lemma 2.15(i): the left action is given by T_{l}(X) = H^{0}(T ⊗ Q_{•}), where Q_{•} is a projective resolution of X. It is clear that the functors T_{l} are well defined and right exact, and L_{j}T_{l}(X) = H^{−j}(T ⊗ Q_{•}) = 0 for j > 0. Thus the functors T_{l} are exact. The same applies to the functors T_{r}. □

2.7. The unit object of C. Now assume that P is a finitely generated ideal, and P ∈ P a generator. Let 1 be the unit object of T . Since P is rigid, we have the evaluation morphism ev : P^{∗} ⊗ P → 1. Consider the morphism τ : P^{∗} ⊗ P ⊗ P^{∗} ⊗ P → P^{∗} ⊗ P given by the formula

τ = ev ⊗ 1 ⊗ 1 − 1 ⊗ 1 ⊗ ev.

We define the object 𝟙 in the category C to be the cokernel of τ (note that we use a different font to distinguish it from 1).

Proposition 2.18. For Q ∈ P ⊂ T there are isomorphisms

Q ≅ Q ⊗ 𝟙, Q ≅ 𝟙 ⊗ Q

which are functorial in Q.

Proof. By Corollary 2.16, Q ⊗ 𝟙 is the cokernel of the morphism

d_{2} := 1_{Q} ⊗ τ : Q ⊗ P^{∗} ⊗ P ⊗ P^{∗} ⊗ P → Q ⊗ P^{∗} ⊗ P.

Let us extend this morphism to the “bar complex”

(2.3)  Q ⊗ P^{∗} ⊗ P ⊗ P^{∗} ⊗ P → Q ⊗ P^{∗} ⊗ P → Q → 0,

where the last map is d_{1} := 1_{Q} ⊗ev (it is easy to see that d_{1}d_{2} = 0). To construct a functorial isomorphism Q⊗𝟙 ≅ Q, it suffices to show that this complex is exact. Since P is a generator, every Q is a direct summand in X ⊗ P for some X ∈ C, so it suffices to show that (2.3) is exact for Q = X ⊗ P. Hence, it suffices to show that (2.3) is split for Q = P.

Since P is rigid, (2.3) for Q = P is split in the third term, with splitting h_{1} : P → P ⊗P^{∗}⊗P given by h_{1} := coev ⊗ 1_{P}, where coev : 1 → P ⊗ P^{∗} is the coevaluation morphism. So it remains to show that (2.3) is split in the second term as well. To this end, we define a homotopy

h_{2} : P ⊗ P^{∗} ⊗ P → P ⊗ P^{∗} ⊗ P ⊗ P^{∗} ⊗ P

by

h_{2} := coev ⊗ 1^{⊗3} = h_{1} ⊗ 1 ⊗ 1.

It is easy to check that h_{1}d_{1} + d_{2}h_{2} = 1, which implies the statement.

This proves that there is a canonical isomorphism Q → Q⊗𝟙. The isomorphism Q → 𝟙⊗Q is constructed similarly. □

Corollary 2.19. The semigroup category C is actually a monoidal category with unit 𝟙. Moreover, if T is braided or symmetric then so is C.

Proof. Since C is the category of resolutions in P, the functorial isomorphism 𝟙 ⊗ Q ≅ Q for Q ∈ P from Proposition 2.18 gives rise to an isomorphism 𝟙 ⊗ ≅ Id_{C}. Similarly, we have an isomorphism ⊗ 𝟙 ≅ Id_{C}. Thus, 𝟙 is a unit in C, hence C is monoidal. The second statement is clear. □

Example 2.20. The following example shows that the assumption that P is finitely generated cannot be dropped. Let char(k) = 2 and Q = kℤ/2 be the regular representation of ℤ/2 over k. For a set M let (ℤ/2)^{M} be the group of maps M → ℤ/2, and for m ∈ M let Q_{m} be the representation of Fun(M, ℤ/2) on the vector space Q given by ρ_{Q_{m}}(g) = ρ_{Q}(g(m)), g ∈ (ℤ/2)^{M}, where ρ_{Q} defines the action of ℤ/2 on Q. Let T′ = T′(M) be the full subcategory of Rep_{k}((ℤ/2)^{M}) spanned by the tensor products Q_{N} := ⨂_{m∈N}Q_{m} for finite subsets N ⊂ M. Let T = T (M) be the quotient of T′ by the tensor ideal generated by Q_{ij} = Q_{i} ⊗ Q_{j}, where i, j ∈ M are distinct (i.e., we set all morphisms factoring through a direct sum of Q_{ij} to zero). Then T has indecomposables 1 and Q_{m}, m ∈ M, with Hom(1, 1) = Hom(1, Q_{i}) = Hom(Q_{i}, 1) = k, Hom(Q_{i}, Q_{i}) = k^{2}, but Hom(Q_{i}, Q_{j}) = 0 for i ≠ j (since any morphism of representations Q_{i} → Q_{j} factors through Q_{ij}). It is easy to show that for any subset N ⊂ M the ideal P_{N} = ⟨Q_{i}, i ∈ N⟩ is a splitting ideal in T , which is finitely generated if and only if N is finite. The abelian semigroup category C(T , P_{N}) is C_{N} := ⨁_{m∈N}Rep_{k}(ℤ/2), which is monoidal (i.e., has a unit object) if and only if N is finite.

2.8. The monoidal functor F. We continue to assume that P ⊂ T is a finitely generated ideal, and C = C(T , P).

Corollary 2.21. There is a monoidal functor F : T → C such that F (T) = T_{l}(𝟙) = T_{r}(𝟙) (so F (1) = 𝟙). Moreover, if T is braided or symmetric then so is F.

Proof. This follows from Corollary 2.19 and Corollary 2.17. Namely, we have a monoidal functor F_{l} : T → End(C_{C}) ≅ C given by F_{l}(T) := T_{l}(𝟙) and a monoidal functor F_{r} : T → End(_{C}C)^{op} ≅ C given by F_{r}(T) := T_{r}(𝟙). Moreover, if Q ∈ P then F_{l}(Q) and F_{r}(Q) are canonically isomorphic to Q. However, for any T ∈ T we have that F_{l}(T) is the cokernel of the natural morphism

F_{l}(1 ⊗ τ) : F_{l}(T ⊗ P^{∗} ⊗ P ⊗ P^{∗} ⊗ P) → F_{l}(T ⊗ P^{∗} ⊗ P),

while F_{r}(T) is the cokernel of the natural morphism

F_{r}(1 ⊗ τ) : F_{r}(T ⊗ P^{∗} ⊗ P ⊗ P^{∗} ⊗ P) → F_{r}(T ⊗ P^{∗} ⊗ P).

This implies that F_{l} ≅ F_{r}, which gives a construction of the functor F ≅ F_{l} ≅ F_{r} with the claimed properties. □

Corollary 2.22. Every projective object in C is rigid.

Proof. Indeed, by Corollary 2.21, every projective object in C is of the form F (Q), where Q ∈ P, hence rigid. But a monoidal functor maps rigid objects to rigid ones. □

Corollary 2.23. The functors P ⊗? and ? ⊗ P on C are faithful.

Proof. Let f : X → Y be a morphism in C. Then the image of 1_{P} ⊗ f is P ⊗ Im(f). Now, Hom(X, P ⊗ Im(f)) = Hom(P^{∗} ⊗ X, Im(f)). But P^{∗} is a generator of P. Thus if P ⊗ Im(f) = 0 then Hom(Q, Im(f)) = 0 for all projectives Q, hence Im(f) = 0 and f = 0, as claimed. □

Proposition 2.24. The map F_{X,Y} : Hom(X, Y) → Hom(F (X), F (Y)) is surjective if either X or Y is in P. Thus, the tensor ideal ⟨Pr⟩ in C of morphisms factoring through projectives is contained in the image of F.

Proof. By definition for Q, R ∈ P the map F_{Q,R} : Hom(Q, R) → Hom(F (Q), F (R)) is an isomorphism. Now let X ∈ T and f : F (Q) → F (X) be a morphism. We would like to show that f = F (g) for some morphism g : Q → X.

Recall that F (X) is a quotient of F (X ⊗ P^{∗} ⊗ P). Since F (Q) is projective, the morphism f admits a lift f′ : F (Q) → F (X ⊗ P^{∗} ⊗ P). Since F (X ⊗ P^{∗} ⊗ P) also is projective, there exists g′ : Q → X ⊗ P^{∗} ⊗ P such that F (g′) = f′. Let g be the composition of g′ with the morphism 1_{X} ⊗ ev : X ⊗ P^{∗} ⊗ P → X. Then F (g) = f, as claimed.

Thus, the map Hom(Q, X) → Hom(F (Q), F (X)) is surjective. Now, to show that the map Hom(X, Q) → Hom(F (X), F (Q)) is also surjective, write it as the map Hom(Q^{∗} ⊗ X, 1) → Hom(F (Q^{∗} ⊗ X), F (1)), which is surjective since Q^{∗} ⊗ X ∈ P. This proves the proposition. □

Let J_{P} be the kernel of F, i.e., the collection of morphisms f in T such that F (f) = 0. Clearly, J_{P} is a tensor ideal. The following proposition describes J_{P} explicitly.

Proposition 2.25. Let f : X → Y be a morphism in T . The following conditions are equivalent:

(a) f ∈ J_{P};

(b1) f ⊗ 1_{P} = 0 in T ;

(b2) 1_{P} ⊗ f = 0 in T ;

(c1) For any object Q ∈ P we have f ⊗ 1_{Q} = 0 in T ;

(c2) For any object Q ∈ P we have 1_{Q} ⊗ f = 0 in T .

Proof. If f ∈ J_{P} then F (f) = 0, so F (f) ⊗ 1_{P} = F (f ⊗ 1_{P}) = 0. But X ⊗ P, Y ⊗ P are projective and F is fully faithful on projectives. Thus f ⊗ 1_{P} = 0. Similarly 1_{P} ⊗ f = 0. Thus, (a) implies (b1) and (b2).

If f ⊗ 1_{P} = 0 then f ⊗ 1_{P ⊗X} = f ⊗ 1_{P} ⊗ 1_{X} = 0 for all X ∈ C, so for any object Q ∈ P we have f ⊗ 1_{Q} = 0. Thus, (b1) implies (c1). Similarly, (b2) implies (c2).

If f ⊗ 1_{Q} = 0 then f ⊗ 1_{Z} = 0 for any Z ∈ C, i.e., F (f) = 0, hence f ∈ J_{P}. Thus (c1) implies (a). Similarly, (c2) implies (a). □

Definition 2.26. If J_{P} = 0, i.e., F is faithful, we will say that the ideal P is faithful.

Example 2.27. The ideals P in Examples 2.5 and Example 2.6 are faithful. On the other hand, if T = T_{1} ⊕ T_{2} with T_{1} ≠ 0 and P ⊂ T_{2} then P is not faithful, as ⟨T_{1}⟩ ⊂ J_{P}. Also, in Example 2.20 with M finite, P_{N} is faithful if and only if N = M, i.e., P_{N} = S is the ideal of all splitting objects. In general, if P = P_{N} then J_{P} = ⟨P_{M\N}⟩.

Definition 2.28. Let us say that T is of finite type if the ideal S = S(T ) is finitely generated. If so, denote the ideal J_{S} by J = J(T ). Let us say that T is separated if J = 0, i.e., S is faithful.

In other words, T is separated if it has “enough splitting objects”, i.e., splitting objects separate morphisms (Proposition 2.25(c1,c2)), which motivates the terminology.

Remark 2.29. It is clear that T is of finite type if and only if the sets \overline{S} and I_{c}, c ∈ \overline{S} are finite.

Example 2.30. A multitensor category with enough projectives is separated. On the other hand, the category of projective bimodules from Example 2.6 is not separated, since it has no nonzero splitting objects (so J is the collection of all morphisms in T ). Similarly, if G is a semisimple algebraic group in characteristic p then Rep(G) is not separated, for the same reason.

Note that if X ∈ S and 1_{X} ∈ J then X ⊗X^{∗} = 0, hence X = 0 (as X is a direct summand in X ⊗ X^{∗} ⊗ X). Thus, faithfulness is not a very restrictive condition: if T is of finite type then the category T /J is separated (with the same indecomposable objects as T ), and C(T , S) = C(T /J, S). Also we obtain the following proposition (which is not used below).

Proposition 2.31. Let T be of finite type. Let P = P_{K} for a subset K ⊂ \overline{S} (as in Proposition 2.8). Then J_{P} contains ⟨P_{S\K}⟩. In particular, P is faithful if and only if P = S and T is separated.

Proof. This follows immediately since P_{K} ⊗ P_{S\K} = 0. □

Let us say that S is indecomposable if S ≠ 0 and Q ∼_{ℓ} R for all indecomposable Q, R ∈ S. In this case, we get

Proposition 2.32. If T is separated and S is indecomposable then it is the unique minimal (nonzero) thick ideal in T .

Proof. Lemma 2.7 implies that S is minimal. Let I ≠ 0 be any thick ideal in T and 0 ≠ X ∈ I. Since T is separated, there is Q ∈ S such that Q ⊗ X ≠ 0. Then Q ⊗ X generates S, so S ⊂ I. □

Let P ⊂ T be a finitely generated ideal and T̂_{P} be the full Karoubian completion of F (T ) = T /J inside C = C(T , P) (i.e., the Karoubian completion of the full subcategory with the same objects as F (T )). Then T̂_{P} is a full Karoubian monoidal subcategory of C, and the monoidal abelian category C(T̂_{P}, P) coincides with C. We will call T̂_{P} the completion of T with respect to P.

Definition 2.33. We will say that T is complete with respect to P if the natural inclusion T /J ↪ T̂_{P} is an equivalence.

It is clear that T is complete with respect to P if and only if for any T ∈ T , the natural map Hom(1, T) → Hom(𝟙, F (T)) is surjective.

If T is of finite type and P = S, we will denote T̂_{P} simply by T̂, and say that T is complete if it is complete with respect to S. In this case we will also write C(T ) for C(T , S). Thus, T is separated iff F : T → C(T ) is faithful and complete iff F is full. In particular, the category T̂ is separated and complete, and T is separated and complete iff the natural functor T → T̂ is an equivalence. In this case the functor F is a fully faithful embedding.

2.9. Properties of the tensor product on C in the case of a splitting ideal. Now assume that P ⊂ T is a finitely generated splitting ideal, and as before let C = C(T , P).

Proposition 2.34. (i) For any Y ∈ C, Q ∈ P the tensor products Q ⊗ Y and Y ⊗ Q are projective.

(ii) Tensoring with any object Q ∈ P on either side splits every morphism in C.

Proof. (i) Let f : Q_{1} → Q_{0} be a presentation of Y. By Corollary 2.16, the cokernel of the morphism 1_{Q} ⊗f : Q⊗Q_{1} → Q⊗Q_{0} is Q⊗Y. But since P is a splitting ideal, this morphism is split. Hence Q ⊗ Y is a direct summand in Q ⊗ Q_{0}, hence projective.

(ii) Let f : X → Y be a morphism in C, and K, K′ be the kernel and cokernel of f, respectively. Then by Corollary 2.16, Q ⊗ K, Q ⊗ K′ are the kernel and cokernel of 1_{Q} ⊗ f, respectively. But by (i) these objects are projective. This implies that the morphism 1_{Q} ⊗ f is split. Similarly, the morphism f ⊗ 1_{Q} is split, as claimed. □

Recall that a multiring category over a field k is a k-linear artinian monoidal category with biexact tensor product, [22], Definition 4.2.3.

Proposition 2.35. The bifunctor (X, Y) ↦ X ⊗ Y on C is exact in both arguments. Thus C is a multiring category.

Proof. Let Q_{•} be a projective resolution of Y ∈ C. Then by Corollary 2.16 H^{−i}(X ⊗^{L} Y) is the −i-th cohomology of the complex X ⊗ Q_{•}. So it suffices to show that if P_{1}, P_{2}, P_{3} are projective and the complex P_{1} → P_{2} → P_{3} is acyclic then the complex

X ⊗ P_{1} → X ⊗ P_{2} → X ⊗ P_{3}

is also acyclic. That is, we need to show that the complex

Hom(R, X ⊗ P_{1}) → Hom(R, X ⊗ P_{2}) → Hom(R, X ⊗ P_{3})

is acyclic for every projective R.

Let Q be projective. Then by Proposition 2.34, Q ⊗ X is projective, so by Corollary 2.16 the complex

Hom(R, Q ⊗ X ⊗ P_{1}) → Hom(R, Q ⊗ X ⊗ P_{2}) → Hom(R, Q ⊗ X ⊗ P_{3})

is acyclic. Since X ⊗ P_{i} are projective (again by Proposition 2.34), this can be written as

Hom(Q^{∗} ⊗ R, X ⊗ P_{1}) → Hom(Q^{∗} ⊗ R, X ⊗ P_{2}) → Hom(Q^{∗} ⊗ R, X ⊗ P_{3}).

Taking Q = R ⊗^{∗}R, we have Q^{∗} ⊗ R = R ⊗ R^{∗} ⊗ R. Thus, using that R is a direct summand in R ⊗ R^{∗} ⊗ R (as R is rigid), we get that the complex

Hom(R, X ⊗ P_{1}) → Hom(R, X ⊗ P_{2}) → Hom(R, X ⊗ P_{3})

is acyclic, implying that the functor X⊗? is exact. Similarly, the functor ? ⊗ X is exact, as claimed. □

Proposition 2.36. Let D be a multiring category, and X ∈ D be the cokernel of a morphism f : Q_{1} → Q_{0} such that the objects Q_{0}, Q_{1} are rigid. Then X is also rigid. Namely, X^{∗} is the kernel of f^{∗} : Q_{0}^{∗} → Q_{1}^{∗} and ^{∗}X is the kernel of ^{∗}f : ^{∗}Q_{0} → ^{∗}Q_{1}.

Proof. Define X^{∗} to be the kernel of the map f^{∗} : Q_{0}^{∗} → Q_{1}^{∗}. Let i : X^{∗} → Q_{0}^{∗}, j : f(Q_{1}) → Q_{0} be the natural inclusions, and f̄ : Q_{1} → f(Q_{1}) the natural projection. We claim that the map ev_{Q_{0}} ∘ (i ⊗ j) : X^{∗} ⊗ f(Q_{1}) → 𝟙 is zero. To show this, it suffices to show that the map ev_{Q_{0}} ∘ (i ⊗ f) : X^{∗} ⊗ Q_{1} → 𝟙 is zero (indeed, tensoring with X^{∗} is right exact, so the map 1 ⊗ f̄ : X^{∗} ⊗ Q_{1} → X^{∗} ⊗ f(Q_{1}) is surjective, and f = j ∘ f̄). But we have ev_{Q_{0}} ∘ (i ⊗ f) = ev_{Q_{1}} ∘ (f^{∗} ∘ i ⊗ 1), and by definition f^{∗} ∘ i = 0, which implies the claim.

This means that the map ev_{Q_{0}}∘(i⊗1) : X^{∗}⊗Q_{0} → 𝟙 gives rise to a map ev_{X} : X^{∗}⊗X → 𝟙.

Similarly, let p : Q_{0} → X be the natural projection and consider the coevaluation map coev_{Q_{0}} : 𝟙 → Q_{0} ⊗ Q_{0}^{∗}. It gives rise to a map (p ⊗ 1) ∘ coev_{Q_{0}} : 𝟙 → X ⊗ Q_{0}^{∗}. Note that since the tensor product by X is left exact, X ⊗ X^{∗} ⊂ X ⊗ Q_{0}^{∗}.

We claim that (p ⊗ 1) ∘ coev_{Q_{0}} lands in X ⊗ X^{∗}, i.e., that the composition

(p ⊗ f^{∗}) ∘ coev_{Q_{0}} : 𝟙 → X ⊗ Q_{1}^{∗}

is zero. We have = (p ⊗ f^{∗}) ∘ coev_{Q_{0}} = (p ∘ f ⊗ 1) ∘ coev_{Q_{1}}. But by definition p ∘ f = 0, as claimed.

Thus, the map (p ⊗ 1) ∘ coev_{Q_{0}} gives rise to a map coev_{X} : 𝟙 → X ⊗ X^{∗}.

Now the axioms of the left dual for X^{∗} ([22], Subsection 2.10) are easily deduced from those for Q_{0}^{∗}. For example, let us check that the composition X → X ⊗ X^{∗} ⊗ X → X is the identity (the other axiom is proved similarly). Indeed, let x ∈ X and let y ∈ X be the image of x under this composition. Fix a lift x_{1} ∈ Q_{0} of x and let x_{2} be the image of x_{1} in Q_{0} ⊗ Q_{0}^{∗} ⊗ Q_{0}, and x_{3} the projection of x_{2} to X ⊗ Q_{0}^{∗} ⊗ Q_{0}. Then x_{3} has a unique preimage x_{4} in X ⊗ X^{∗} ⊗ Q_{0}. Let x_{5} be the image of x_{4} in X ⊗ X^{∗} ⊗ X. By the definition of coev_{X} we have (1 ⊗ ev_{X})(x_{5}) = y. By the definition of ev_{X}, it follows that (1 ⊗ ev_{Q_{0}})(x_{3}) = y. Thus it follows from the rigidity of Q_{0} that y is the image of x_{1} in X, i.e., y = x, as desired.

Similarly, X admits a right dual ^{∗}X, which is the kernel of the map ^{∗}f : ^{∗}Q_{0} → ^{∗}Q_{1}. Thus, X is rigid and the proposition is proved. □

Proposition 2.37. (i) The monoidal category C = C(T , P) is rigid, i.e., it is a multitensor category.

(ii) Let P = S, so that C = C(T ). Then we have C = ⨁_{c∈\overline{S}}C(c), where C(c) are indecomposable multitensor subcategories. Namely, C(c) is the subcategory of objects generated by c. In particular, 𝟙 = ⨁_{c∈\overline{S}}𝟙_{c}.

(iii) We have C(c) = ⨁_{i,j∈I_{c}}C(c)_{ij}, where C(c)_{ij} are component categories of C(c) ([22], Definition 4.3.5). In particular, 𝟙_{c} = ⨁_{i∈I_{c}}𝟙_{c,i} where 𝟙_{c,i} are simple, and C(c)_{ij} = 𝟙_{c,i}C(c)𝟙_{c,j}.

(iv) S is indecomposable if and only if 𝟙 ∈ C is indecomposable, i.e., C is a tensor category.

Proof. (i) Any X ∈ C can be represented as a cokernel of a morphism f : Q_{1} → Q_{0} between projective objects. By Corollary 2.22, the objects Q_{0} and Q_{1} are rigid. Thus by Proposition 2.36 and Proposition 2.35, X is rigid, as claimed.

The proof of (ii), (iii) and (iv) is straightforward. □

Remark 2.38. Example 2.6 shows that the splitting assumption for P cannot be dropped, as the category of finite dimensional bimodules over a finite dimensional non-semisimple Frobenius algebra is not rigid.

As a by-product we obtain the following corollary (which is not used below).

Corollary 2.39. If T is separated then the algebra End(1) is semisimple.

Proof. This follows since End(𝟙) is semisimple and commutative and F_{1,1} : End(1) → End(𝟙) is injective (for P = S). □

Example 2.40. Let char(k) = 2 and T_{0} := Rep_{k}(ℤ/2), with the usual monoidal structure. Then T_{0} has two indecomposable objects, 1 and P (the regular representation). Let T be the following modification of T_{0}: we set End(1) := k[ε]/ε^{2} instead of k. The monoidal structure is modified in an obvious way (namely, the tensor product of ε with any morphism X → Y where X = P or Y = P is zero). Then S is generated by P, so J = kε, and T /J = C(T /J) = Rep_{k}(ℤ/2) (i.e., T is not separated but complete). This example shows that J does not have to be a thick ideal, and that the separatedness assumption in Corollary 2.39 cannot be dropped.

2.10. The main theorems.

Theorem 2.41. T admits a faithful monoidal functor E into a multitensor category D with enough projectives which is full on projectives (i.e., any morphism between projectives is in the image) if and only if T is separated. Moreover, if such a functor exists, it is unique up to an isomorphism, with D ≅ C(T ) and E ≅ F. The same holds for braided and symmetric categories.

Proof. Suppose T is separated. Then by Proposition 2.37, there is a monoidal functor F : T → C(T ) which satisfies the conditions of the theorem.

Conversely, let E : T → D be a faithful monoidal functor into a multitensor category with enough projectives which is full on projectives. Then, since E is full on projectives, there exists Q ∈ T such that E(Q) = U. Moreover, since U splits every morphism in D and E is faithful, we have Q ∈ S.

Now let f : X → Y be a morphism in J. Then 1_{Q} ⊗ f = 0, hence 1_{E(Q)} ⊗ E(f) = 0, i.e., 1_{U} ⊗ E(f) = 0. Thus U ⊗ Im(E(f)) = Im(1_{U} ⊗ E(f)) = 0 in D. This holds for every projective U, which implies Im(E(f)) = 0, hence E(f) = 0. Thus f = 0 by the faithfulness of E. Thus, T is separated (the ideal of projectives is generated as a left ideal by the projective cover of 𝟙, hence is finitely generated).

Also, in this case S coincides with the ideal of projectives Pr ⊂ D, so we get that D ≅ C(T ) and E ≅ F. □

Recall ([20], Section 9) that a multitensor category C ⊃ T is called the abelian envelope of T if it has the following universal property: for any multitensor category D there is an equivalence between the category of tensor functors E : C → D and the category of faithful monoidal functors \overline{E} : T → D, defined by the formula \overline{E} = E|_{T}. The same definition is made for braided and symmetric categories (in which case the functors have to be braided).

Theorem 2.42. (i) T admits a fully faithful monoidal functor E : T → D into a multitensor category D with enough projectives if and only if it is separated and complete. Moreover, in this case there exists a tensor embedding Ẽ : C(T ) ↪ D such that E ≅ Ẽ ∘ F. The same applies to braided and symmetric categories.

(ii) In this case C(T ) is the abelian envelope of T .

Proof. (i) If T is separated and complete then by Proposition 2.37, we have a fully faithful monoidal functor F : T → C(T ) satisfying the conditions in the theorem.

Conversely, let E : T → D be a fully faithful monoidal functor into a multitensor category D with enough projectives. Let E ⊂ D be the tensor subcategory generated by E(T ), [22], Definition 4.11.1 (full and closed under subquotients). Then E is a multitensor category with enough projectives, and we have a fully faithful monoidal functor E : T → E. Thus we may assume without loss of generality that E = D (i.e., E is surjective). In this case, let U be a projective object of D. Since E is surjective, there exists T ∈ T such that U is a subquotient of E(T). Since U is both projective and injective, it is in fact a direct summand of E(T). Since F is full, this means that U = E(Q) for some direct summand Q of T. Thus, F is full on projectives. By Theorem 2.41 this implies that T is separated and D ≅ C(T ), E ≅ F. Finally, since F is full, T is complete.

(ii) This follows from [20], Theorem 9.2.1. More precisely, this theorem applies to symmetric categories, but the proof extends without significant changes to the case of braided categories and categories without a fixed braided structure. □

Let D be a multitensor category with enough projectives. Recall that the stable category Stab(D) is the quotient D/Pr, where Pr is the ideal of projectives. The indecomposables of Stab(D) are the non-projective indecomposables of D, so an object of Stab(D) may be realized as an object in D with no nonzero projective direct summands. There is a natural monoidal functor Stab : D → Stab(D).

Suppose E is a rigid monoidal Karoubian category and E : E → Stab(D) a faithful monoidal functor. Consider the fiber product A := E ×_{Stab(D)} D, whose objects are triples (X, Y, f), where X ∈ E, Y ∈ D, and f : E(X) → Stab(Y) is an isomorphism, and morphisms (X_{1}, Y_{1}, f_{1}) → (X_{2}, Y_{2}, f_{2}) are pairs (g, h), g : X_{1} → X_{2}, h : Y_{1} → Y_{2} such that f ∘ E(g) = Stab(h)∘f. This is naturally a rigid monoidal Karoubian category. If (X, Y, f) and (X, Y, f′) are objects of A then there exists an automorphism g : Y → Y such that Stab(g) = f′ ∘ f^{−1} and it defines an isomorphism (X, Y, f) ≅ (X, Y, f′). Thus isomorphism classes of objects in A are isomorphism classes of pairs (X, Y) such that E(X) ≅ Stab(Y), i.e., isomorphism classes of pairs (X, Q) where Q is a projective object of D; namely, (X, Q) = X ⊕ Q. Thus the indecomposables of A are the indecomposables of E and the indecomposable projectives of D. Moreover, for X, X′ ∈ E and projective Q, Q′ ∈ D we have

Hom_{A}(X, X′) = Hom_{E}(X, X′), Hom_{A}(Q, Q′) = Hom_{D}(Q, Q′),
Hom_{A}(Q, X) = Hom_{D}(Q, E(X)), Hom_{A}(X, Q) = Hom_{D}(E(X), Q).

The category A is equipped with a natural faithful monoidal functor Ẽ : A → D.

Note that if E is pseudomonic, i.e., full on isomorphisms (in particular, if it is fully faithful) then A is naturally equivalent to a rigid monoidal isomorphism-closed subcategory of D containing the tensor ideal generated by projective objects.

Theorem 2.43. There is a 1-1 correspondence between (equivalence classes of): (1) Separated categories T such that C(T ) ≅ D;

(2) Rigid monoidal Karoubian categories \overline{T} together with a faithful monoidal functor \overline{F} : \overline{T} → Stab(D); which sends complete categories T to full subcategories \overline{T} ⊂ Stab(D), and vice versa. Namely, this correspondence is defined by the formulas

\overline{T} = T /Pr, T = \overline{T} ×_{Stab(D)} D.

Proof. Given T ⊂ D, define \overline{T} := T /Pr. Then \overline{T} is a rigid monoidal Karoubian category with a faithful monoidal functor \overline{F} : \overline{T} → Stab(D). Conversely, suppose E is a rigid monoidal Karoubian category with a faithful monoidal functor E : E → Stab(D) and set Ẽ:= E ×_{Stab(D)} D. Then Ẽ is a separated category, and it is clear that C(Ẽ) ≅ D.

Now, it is clear that ˜ \overline{E} = E, and the equality \overline{T̃} = T follows from Proposition 2.24. □

Example 2.44. 1. Suppose D is semisimple. Then Stab(D) = 0, so Theorem 2.43 says that a separable category T such that C(T ) = D must be complete and coincide with D (which is easy).

2. Let C be a finite multitensor category and T = Stab^{−1}(Vec) the preimage of Vec under the projection Stab : C → Stab(C). The indecomposable objects of T are 𝟙 and the indecomposable projectives of C (note that 𝟙 may be decomposable in C but not in T ). Then the ideal P ⊂ T of projectives coincides with S, T is separated and C(T ) = C. But if 𝟙 is decomposable and has length m ≥ 2 then the map k = Hom_{T} (1, 1) → Hom_{C}(𝟙, 𝟙) = k^{m} is not surjective, hence F is not full and T is not complete.

A simple example of this is the case when C = Mat_{N}(Rep_{k}(ℤ/2)), char(k) = 2, where N is a finite set (i.e., the objects of C are matrices of objects of Rep_{k}(ℤ/2) with rows and columns labeled by N and tensor product given by matrix multiplication using the usual tensor product of Rep_{k}(ℤ/2)). In this case the indecomposable objects of T are 1 and also R_{ij}, i, j ∈ N, where R_{ij} is the regular representation of ℤ/2 standing in the (i, j)-th position. Note that this example also makes sense when N is an infinite set, in which case the ideal S (generated by all the R_{ij}) is not finitely generated as a left ideal, but is finitely generated as a two-sided ideal (in fact, by any one of the R_{ij}).

3. Let C = Rep_{k}(ℤ/p) where char(k) = p > 2. We have a faithful symmetric monoidal functor E : sVec ↪ Stab(C) sending the non-trivial simple object to the indecomposable module L = Ω(1) of size p − 1 (here Ω is the Heller shift). Let T = Stab^{−1}(Im(E)). Then T has three indecomposables: 1, L, P, with L ⊗ P = P^{p−1}, P ⊗ P = P^{p} and L ⊗ L = P^{p−2} ⊕ 1, and Hom(1, P) = Hom(P, 1) = k, Hom(P, P) = k^{p}, Hom(P, L) = Hom(L, P) = k^{p−1}, Hom(L, L) = k[z]/z^{p−1}, Hom(1, L) = Hom(L, 1) = 0. In this case S := ⟨P ⟩ is a finitely generated ideal and T is separated, but the functor F is not full, since the morphism 0 = Hom_{T} (1, L) → Hom(𝟙, F (L)) = k is not injective. Thus T is not complete.

On the other hand, in this example we may replace sVec by the full monoidal subcategory of Stab(C) generated by 1 and L (i.e., Hom(1, L) = Hom(L, 1) = k rather than zero), and the functor E with the tautological embedding. In this case T = Stab^{−1}(Im(E)) is both separated and complete; it is the full monoidal subcategory of C spanned by 1, L, P. We have T = C for p = 3, but for p > 3 it is smaller than C (and not abelian). But for any p > 2 the category C is the abelian envelope of T .

4. Let char(k) = 2 and C = Rep_{k}(ℤ/2). Then Stab(C) = Vec is the category of finite dimensional vector spaces. Let H be a finite dimensional Hopf algebra over k and E := Rep(H). Let E : E → Vec = Stab(C) be the forgetful functor. Let T = E ×_{Stab(C)} C. Then T is separated and C(T ) = C. Note that E is not full on isomorphisms in general, so T is not equivalent to an isomorphism-closed subcategory of C, but it admits a faithful monoidal functor F : T → C.

Example 2.45. Here is a prototypical example of a separated and complete category T . Let C be a multitensor category with enough projectives which is tensor generated by an object X ∈ C. This means that any object of C is a subquotient of a direct sum Y of tensor products of objects X^{(n)}, where X^{(n)} is the n-th dual of X. Let T be the full rigid monoidal Karoubian subcategory of C generated by X, i.e., its objects are direct summands (rather than subquotients) of direct sums Y as above. Then by Theorem 2.42, S ⊂ T is the ideal of projective objects in C, and C = C(T ).

This example becomes interesting when the category T has an independent (often diagrammatic) description not involving C, and the corresponding universal property. This happens in a number of concrete examples, notably the following.

1. Let C = Rep^{ab}(S_{t}) is the abelian envelope of the Deligne category Rep(S_{t}) for t ∈ ℤ_{+}, char(k) = 0 ([16, 9]). Let X be the “tautological” object (the interpolation of the permutation representation of S_{n}). Then T is the Karoubian Deligne category Rep(S_{t}) ([16]). The indecomposable objects of T are X_{λ} labeled by partitions λ, with X_{0} = 1. The blocks in this category are described in [9]; there are semisimple blocks corresponding to a single isolated partition and also semi-infinite chains of partitions, whose structure depends on t. The ideal S is then spanned by X_{λ} for the isolated λ and also all other λ except the first element of each chain; these exceptions are exactly the partitions for which t ≥ λ_{1} + |λ| (they are in bijection with partitions of t defined by λ ↦ (t − λ_{1}, λ_{1}, ..., λ_{n})). Indeed, it is precisely these X_{λ} that become projective in C ([19], 4.4).

2. Let C = Rep(GL(m|n)), char(k) = 0, and assume for simplicity that m ≥ n. Let X = k^{m|n} be the defining representation. Then the category T is the category of modules that occur as direct summands in X^{⊗i} ⊗ X^{∗⊗j}. Let t = m − n and Rep(GL_{t}) be the Karoubian Deligne category for parameter t. It is shown in [11, 8] that the only nontrivial tensor ideals in Rep(GL_{t}) are J_{0} ⊃ J_{1} ⊃ J_{2}..., such that J_{i} is the kernel of the natural functor Rep(GL_{t}) → Rep(GL(t + i|i)). Then T = Rep(GL_{t})/J_{n} (see [8, 10, 20]).

3. Let G be a simply connected simple algebraic group and q a root of unity of sufficiently large order. Let C = Rep(G_{q}) be the category of finite dimensional representations of the corresponding quantum group (char(k) = 0), and T = Tilt(G_{q}) ⊂ Rep(G_{q}) be the category of tilting modules; it is obtained by the above construction from the direct sum of the fundamental representations X = ⨁_{i}L_{ω_{i}} (note that all projectives in C are tiltings, so T is separated and complete and C = C(T )). If G = SL_{n}, the category T is the category of modules occurring as direct summands in tensor powers of the tautological representation V, and it is expected to have a nice diagrammatic description as a braided category using webs (a root of unity analog of [7]). In particular, we see that C is the abelian envelope of T , which is a quantum analog of [14], Theorem 3.3.1.

4. Let G be as above, p a prime, k a field of characteristic p, C = Rep(G) be the category of finite dimensional representations of G over k, and T ⊂ C be the category of tilting modules. Then C is the abelian envelope of T ([14], Theorem 3.3.1). Since the category C has no nonzero projective objects, this is not quite an instance of the above theory. However, it is a limit of such instances. More precisely, let G_{n} be the n-th Frobenius kernel of G. Then we have the restriction functor Rep(G) → Rep(G_{n}). Denote by T (n) the image of T in Rep(G_{n}). Then Rep(G_{n}) = C(T (n)). This follows from the fact that the n-th Steinberg representation St_{n} of G is projective when restricted to G_{n}.

5. Let T_{p} be the category of tilting modules for the group SL_{2}(k) (char(k) = p), with indecomposable tilting objects T_{i}, i ≥ 0, with highest weight i. Then T_{p} has a thick ideal I_{n} generated by the n-th Steinberg representation St_{n} := T_{p^{n}−1}, which is spanned by the objects T_{i} with i ≥ p^{n} − 1. Let T_{n,p} := T_{p}/I_{n} ([5]). It is shown in [5], Theorem 4.14 that for n ≥ 0 the category T_{n+1,2} has an abelian envelope C_{2n}, a finite symmetric tensor category in which it sits as a full monoidal generating subcategory. Thus, T_{n+1,2} is separated and complete, and C_{2n} = C(T_{n+1,2}).

However, while these are nice examples, the true power of the above theory is seen when the category C(T ) is not known a priori, and this theory provides its first construction. This is what happens for T_{n,p} in the case p > 2, and the main goal of this paper is to define and study the corresponding categories C(T_{n,p}).

2.11. Splitting objects in monoidal categories over discrete valuation rings. For applications, we will need to generalize the above theory of splitting ideals and associated abelian envelopes to the deformation-theoretic setting, when the ground ring is not a field but rather a complete discrete valuation ring (DVR), which in applications will be of mixed characteristic.

Let R be a complete DVR with maximal ideal 𝔭 and uniformizer ε. Let k := R/𝔭 be its residue field, and K be the fraction field of R.

Definition 2.46. By a flat monoidal Karoubian category over R we will mean an R-linear monoidal Karoubian category T for which Hom(X, Y) is a free finite rank R-module for any X, Y ∈ T .

If T is a flat monoidal Karoubian category over R then we have the monoidal Karoubian category T_{k} := T /𝔭 over the residue field k. We also have the category T_{K}, the Karoubian closure of K ⊗_{R} T (which may not be Karoubian by itself), a monoidal Karoubian category over K.^{2} Moreover, it is easy to show that if T_{k} is rigid then so are T , T_{K}.

We have a natural bijection between objects of T and T_{k} given by X ∈ T ↦ X_{k} ∈ T_{k}, Y ∈ T_{k} ↦ Ỹ ∈ T , such that Hom(X_{k}, Z_{k}) = k ⊗_{R} Hom(X, Z). Also, any object X ∈ T defines an object X_{K} ∈ T_{K}, with Hom(X_{K}, Z_{K}) = K ⊗_{R} Hom(X, Z). Thus any object Y ∈ T_{k} uniquely lifts to an object of Ỹ_{K} ∈ T_{K} (although the lift of an indecomposable in T may be decomposable in T_{K}).

Note that the assignments X ↦ X_{k} and X ↦ X_{K} are symmetric monoidal functors T → T_{k} and T → T_{K}. Moreover, if f : X → Y is a split surjection or injection in T then so are f_{k} : X_{k} → Y_{k} and f_{K} : X_{K} → Y_{K}.

A flat monoidal category T over R cannot have nonzero splitting objects in the sense of Definition 2.3, since any element x ∈ 𝔭, x ≠ 0 defines a morphism x_{Q^{∗}} : Q^{∗} → Q^{∗} which is not split by tensoring with Q. However we can make a less restrictive definition.

Definition 2.47. 1. A morphism f : Q_{1} → Q_{0} in T is R-split if for any X ∈ T the induced morphism of free R-modules f_{X} : Hom(X, Q_{1}) → Hom(X, Q_{0}) is split (i.e., its cokernel is a free R-module).

2. An object Q ∈ T is a splitting object if tensoring with Q on either side splits any R-split morphism.

Example 2.48. Any split morphism in T is automatically R-split. Indeed, let f : X → Y be a split morphism in T . Then f = 0 ⊕ g, where g is an isomorphism. Since 0 and g are R-split, so is f.

Lemma 2.49. Let f : X → Y be an R-split morphism in T .

(i) If f is nonzero then the corresponding morphism f_{k} : X_{k} → Y_{k} is nonzero.

(ii) If f_{k} is split then f is split.

(iii) If Q ∈ T then the morphisms 1_{Q} ⊗ f and f ⊗ 1_{Q} are R-split.

(iv) If Q_{k} is a splitting object of T_{k} then Q is a splitting object of T .

Proof. (i) Suppose f_{k} = 0. For an object Z ∈ T consider the map f ∘ : Hom(Z, X) → Hom(Z, Y). The map f_{k} ∘ : Hom(Z_{k}, X_{k}) → Hom(Z_{k}, Y_{k}) is zero. Thus f ∘ = εg for some map g : Hom(Z, X) → Hom(Z, Y). But f∘ is R-split, which implies that g = 0, i.e., f∘ = 0. Thus f = 0.

2Note that the field K is not algebraically closed, but this is not important for our considerations unless specified otherwise. For background on multitensor categories over arbitrary fields see [22], Subsection 4.16.

(ii) Since f_{k} is split, up to composing with isomorphisms on both sides it can be written as a direct sum of the zero morphism X′ → Y′ and the identity morphism Z → Z, i.e., as a block matrix

f_{k} ∼ (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) .

This means that f can be similarly written as a block matrix

f ∼ (\begin{matrix}εa & εb \\ εc & 1 + εd \end{matrix}) .

Thus up to composing with isomorphisms

f ∼ (\begin{matrix}ε(a − εb(1 + εd)^{−1}c) & 0 \\ 0 & 1 + εd \end{matrix}) .

Since f is R-split, ε(a − εb(1 + εd)^{−1}c) must be R-split, hence it is zero by (i). Thus

f ∼ (\begin{matrix}0 & 0 \\ 0 & 1 + εd \end{matrix}) ,

i.e., f is a direct sum of zero and an isomorphism. Thus, f is split.

(iii) The morphism 1_{Q} ⊗ f is R-split since the corresponding map

(1_{Q} ⊗ f)∘ : Hom(Z, Q ⊗ X) → Hom(Z, Q ⊗ Y)

coincides with the map f∘ : Hom(Q^{∗} ⊗Z, X) → Hom(Q^{∗} ⊗Z, Y), which is split by definition. Similarly, f ⊗ 1_{Q} is R-split.

(iv) Assume that Q_{k} is splitting. Let f : X → Y be a morphism in T . Then the morphism 1_{Q_{k}} ⊗ f_{k} = (1_{Q} ⊗ f)_{k} is split. Also the morphism 1_{Q} ⊗ f is R-split by (iii). Thus by (ii) the morphism 1_{Q} ⊗ f is split. Similarly, the morphism f ⊗ 1_{Q} is split. Thus, Q is a splitting object. □

It follows from Lemma 2.49(iii) that splitting objects form a thick ideal S ⊂ T . Thus, if Q is a splitting object then so is Q^{∗}. Hence, if f is an R-split morphism and Q is a splitting object then Q splits the morphism f^{∗} (as Q^{∗} splits f).

From now on assume that the category T_{k} is separated and its ideal of splitting objects S(T_{k}) is finitely generated as a left ideal. Thus we can define the multitensor category C(T_{k}) with enough projectives and a faithful monoidal functor F : T_{k} → C(T_{k}).

Lemma 2.50. (i) Let P be a generator of S(T_{k}). Then the morphism coevP̃ : 1 → P̃ ⊗ P̃^{∗} is R-split.

(ii) If Q is a splitting object of T then Q_{k} is a splitting object of T_{k}. In other words, S(T_{k}) = S_{k}.

Proof. (i) By applying the faithful monoidal functor F : T_{k} → C(T_{k}) and using Proposition 2.24 we get that for any X ∈ T_{k} the map coev_{P} ∘ : Hom(X, 1_{k}) → Hom(X, P ⊗ P^{∗}) is injective. This implies that the map

(coevP̃∘)_{K} : K ⊗_{R} Hom(X̃, 1) → K ⊗_{R} Hom(X̃, P̃ ⊗ P̃^{∗})

is also injective, and thus has the same rank as coev_{P} (the dimension of the source). Thus coevP̃ is split as a morphism of R-modules, as claimed.

(ii) Let Q be an indecomposable splitting object of T , and P a generator of S(T_{k}). Then Q_{k} ⊗ P ≠ 0 as T_{k} is separated. Consider the morphism f = 1_{Q} ⊗ coevP̃ : Q → Q ⊗ P̃ ⊗ P̃^{∗}. Since Q_{k} ⊗ P ≠ 0, we have f_{k} ≠ 0, hence f ≠ 0. On the other hand, the morphism coevP̃ is R-split by (i), so the morphism f is split. Since Q is indecomposable, this implies that f is a split injection. Thus f_{k} : Q_{k} → Q_{k} ⊗ P ⊗ P^{∗} is also a split injection, i.e., Q_{k} is a direct summand in Q_{k} ⊗ P ⊗ P^{∗}, hence split, as claimed. □

For simplicity assume that T has finitely many indecomposables. Then we can define the category C(T ) in the same way as we did over a field. Namely, let P be the direct sum of all indecomposable splitting objects of T_{k}, P̃ the corresponding object of T , A = End(P̃)^{op}, and C(T ) = Rep(A) be the category of finitely generated A-modules. Then C(T ) can be realized as a category of resolutions

... → Q_{2} → Q_{1} → Q_{0},

where Q_{i} ∈ T are splitting objects (i.e., complexes Q^{•} of splitting objects such that the complex Hom(P̃, Q^{•}) is exact). As before, this is an R-linear abelian monoidal category with right exact tensor product. Moreover, we have a faithful monoidal functor F : T → C(T ) given by F (X) = Hom(P̃, X).

Of course, we cannot hope that the tensor product in C(T ) is exact, since for example C(T ) contains the category of finitely generated R-modules with tensor product over R (quotients of a multiple of 𝟙_{i} for some i). However, it turns out that it is exact on a certain class of objects we call flat.

Definition 2.51. An object X ∈ C(T ) is flat if it is presented by an R-split morphism f : Q_{1} → Q_{0}, Q_{1}, Q_{0} ∈ S; i.e., X = Coker(F (f)).

For instance, every splitting object is automatically flat.

Lemma 2.52. The following conditions on X ∈ C(T ) are equivalent:

(i) X is flat;

(ii) X is a free R-module;

(iii) Any morphism f : Q_{0} → Q_{1} with Q_{0}, Q_{1} ∈ S presenting X is R-split.

Proof. Clearly (iii) implies (i).

Suppose X is flat, and presented by an R-split morphism f : Q_{1} → Q_{0}. Then X is the cokernel of the map f∘ : Hom(P̃, Q_{1}) → Hom(P̃, Q_{0}). By definition, this map is R-split, hence X is a free R-module. Thus (i) implies (ii).

Suppose that X is a free R-module, and presented by a morphism f : Q_{1} → Q_{0}. Then by Proposition 2.24 for Z ∈ T the natural map Hom(Z, Q_{i}) → Hom(F (Z), F (Q_{i})) is an isomorphism. Thus we have an exact sequence

Hom(Z, Q_{1}) → Hom(Z, Q_{0}) → Hom(F (Z), X).

Thus the cokernel of the map f∘ : Hom(Z, Q_{1}) → Hom(Z, Q_{0}) is an R-submodule of Hom(F (Z), X) which is, in turn, contained in Hom_{R}(F (Z), X). But Hom_{R}(F (Z), X) is a free R-module since X is free as an R-module. Thus, Coker(f∘) is a free R-module and f is R-split. □

Corollary 2.53. Let Q_{0}, Q_{1} ∈ S. A morphism f : Q_{1} → Q_{0} is R-split if and only if F (f) : F (Q_{1}) → F (Q_{0}) is split as a morphism of R-modules.

Proof. Suppose f is R-split. Then F (f) = f∘ : Hom(P̃, Q_{1}) → Hom(P̃, Q_{0}) is split as a morphism of R-modules. Conversely, if F (f) is split then f presents the object Coker(F (f)), which is free as an R-module, so by Lemma 2.52 f is R-split. □

Corollary 2.54. For every Z ∈ T the object F (Z) is flat.

Proof. This follows from Lemma 2.52 since Hom(P̃, Z) = F (Z) is a free R-module. □

Proposition 2.55. X ∈ C(T ) is flat if and only if it is rigid, and in this case the functors X⊗? and ? ⊗ X are exact.

Proof. We start with the following lemma.

Lemma 2.56. For any flat Y ∈ C(T ) and projective Q ∈ C(T ) the tensor products Q ⊗ Y and Y ⊗ Q are projective.

Proof. Let f : Q_{1} → Q_{0} be a presentation of Y by an R-split morphism. By a generalization of Corollary 2.16 to R-linear categories, the cokernel of the morphism

1_{Q} ⊗ F (f) : Q ⊗ F (Q_{1}) → Q ⊗ F (Q_{0})

is Q ⊗ Y. Since f is R-split and Q = F (Q′) for some splitting object Q′, the morphism 1_{Q} ⊗ F (f) = F (1_{Q′} ⊗ f) is split (as so is 1_{Q′} ⊗ f). Hence Q ⊗ Y is a direct summand in Q ⊗ F (Q_{0}) = F (Q′ ⊗ Q_{0}), hence projective. □

Lemma 2.57. The following conditions on X ∈ C(T ) are equivalent:

(i) X is flat;

(ii) the functor X⊗ is exact;

(iii) the functor ⊗X is exact.

Proof. The proof that (i) implies (ii), (iii) is the same as that of Proposition 2.35, using Lemma 2.56 instead of Proposition 2.34.

To prove that (ii) implies (i), we may assume that X ∈ C(T )_{ij} for some i, j. If the functor X⊗ is exact then in particular it is exact on the subcategory of finitely generated R-modules, R-mod ⊂ C(T )_{jj} (quotients of a multiple of 𝟙_{j}). If M ∈ R-mod then X ⊗M is just the usual tensor product X ⊗_{R} M. Since this functor (of M) is exact, X is flat as an R-module, i.e., free (as it is finitely generated). Hence it is flat as an object of C(T ) by Lemma 2.52. □

Lemma 2.58. Let X ∈ C(T ) be a flat object which is the cokernel of an R-split morphism f : Q_{1} → Q_{0}. Then X is rigid. Namely, X^{∗} is the kernel of f^{∗} : Q_{0}^{∗} → Q_{1}^{∗} and ^{∗}X is the kernel of ^{∗}f : ^{∗}Q_{0} → ^{∗}Q_{1}.

Proof. The proof is the same as that of Proposition 2.36, using Lemma 2.57 instead of Proposition 2.35. □

Lemma 2.58 shows that any flat object is rigid. Also, for every rigid X the functor X⊗ is exact, which implies that X is flat by Lemma 2.57. This completes the proof of Proposition 2.55. □

Thus, in terms of the realization of C(T ) as the category of finitely generated left Amodules, the tensor product looks as follows: we have an (A, A ⊗_{R} A)-bimodule T, and

X ⊗ Y = T ⊗_{A⊗_{R}A} (X ⊗_{R} Y).

The exactness of this functor on flat objects proved in Proposition 2.55 is equivalent to the condition that T is projective as a right module (in particular, it is free over R).

We see that the algebra A has a structure of a pseudo-Hopf algebra over R ([24], 4.2). Thus A_{k} is a pseudo-Hopf algebra over k and A_{K} := K ⊗_{R} A is one over K. We have C(T_{k}) = Rep(A_{k}). Also it follows that the category T_{K} is separated (namely, for any splitting object Q ∈ T the object Q_{K} is splitting in T_{K}) and C(T_{K}) = Rep(A_{K}).

Remark 2.59. If X ∈ C(T ) is a flat object then X^{∗}, ^{∗}X are both the dual R-module of X with the action of a ∈ A given by a_{X^{∗}} = S(a)_{X}^{∗}, a_{^{∗}X} = S^{−1}(a)_{X}^{∗} where S : A → A is the antipode.

If there exists a pseudo-Hopf algebra A over R which is free of finite rank as an R-module and C_{k} = Rep(A_{k}), C_{K} = Rep(A_{K}), then we will say that C_{K} is a lift of C_{k} over K. Note that a specific object of C_{k} may admit more than one lift over K, or no such lifts at all.

For example, let H be a (quasi-)Hopf algebra over R which is free of finite rank as an Rmodule (for example, the group algebra RG of a finite group G). Then the tensor category Rep(H_{K}) is a lift of Rep(H_{k}) over K.

We see that in the above setting C(T_{K}) is a lift of C(T_{k}) over K.

From now on assume that the algebra A_{K} is split over K, or, equivalently, the category C_{K} is split (note that this can always be achieved by a finite extension of R). For brevity denote C(T ), C(T )_{k}, C(T )_{K} by C, C_{k}, C_{K}. Given a simple object Y ∈ C_{K}, it is well known that one can choose (non-uniquely) a flat object \overline{Y} ∈ C such that \overline{Y}_{K} = Y. For a simple object X ∈ C_{k} let [Y : X] be the multiplicity of X in the Jordan-Hölder series of the reduction \overline{Y}_{k} ∈ C_{k}. This notation is justified because this multiplicity is, in fact, independent on the choice of \overline{Y} (see [21]). The decomposition matrix D of C is the matrix D with entries D_{XY} = [Y : X].

Proposition 2.60. (i) (CDE triangle) Let C_{k}, C_{K} be the Cartan matrices of C_{k}, C_{K}, and D be the decomposition matrix of C. Then the decomposition matrix for lifts of projectives in C_{k} into projectives in C_{K} is D^{T}, and we have C_{k} = DC_{K}D^{T}.

(ii) FPdim(C_{k}) = FPdim(C_{K});

(iii) If T_{K} is complete then so is T_{k}.

Proof. (i) This holds for any flat deformation of finite dimensional algebras over a DVR, [21].

(ii) Let d_{k}, d_{K} be the vectors of Frobenius-Perron dimensions in C_{k}, C_{K} ([22], Subsections 3.3, 4.5). By (i) we have

FPdim(C_{k}) = d_{k}^{T}C_{k}d_{k} = d_{k}^{T}DC_{K}D^{T} d_{k} = d_{K}^{T}C_{K}d_{K} = FPdim(C_{K}).

(iii) We need to show that the functor F_{k} : T_{k} → C_{k} is full. Since it is faithful, it suffices to show that dim Hom_{T_{k}}(T_{i}, T_{j}) ≥ dim Hom_{C_{k}}(F_{k}(T_{i}), F_{k}(T_{j})). But

dim Hom_{T_{k}}(T_{i}, T_{j}) = dim Hom_{T_{K}}(T_{i}, T_{j}) = dim Hom(F_{K}(T_{i}), F_{K}(T_{j})),

so it suffices to show that dim Hom(F_{k}(T_{i}), F_{k}(T_{j})) ≥ dim Hom(F_{K}(T_{i}), F_{K}(T_{j})). But this follows since F_{K}(T_{i}) are lifts of F_{k}(T_{i}) over K (as A-modules). □

Corollary 2.61. If T_{K} is semisimple then T_{k} is complete and the Cartan matrix of C_{k} is given by C = DD^{T}.

Proof. This follows immediately from Proposition 2.60, since the Cartan matrix of C_{K} = T_{K} is the identity matrix. □

3. Properties of tilting modules for SL_{2}(k) and splitting objects of T_{n,p}

3.1. Splitting objects in T_{n,p}. Let T = T_{p} be the category of tilting SL_{2}(k)−modules, char(k) = p > 0. Its indecomposable object of highest weight m ∈ ℤ_{≥0} will be denoted T_{m}. For r ∈ ℤ_{≥0} let St_{r} = T_{p^{r}−1} be the r-th Steinberg module; we will also denote St_{1} as St. For k ∈ ℤ_{≥1} let T_{<p^{k}−1} be the additive subcategory of T generated by T_{m} with m < p^{k} − 1.

Let L_{m} be the simple SL_{2}(k)−module of highest weight m ∈ ℤ_{≥0}. Let M ↦ M^{(1)} be the Frobenius twist. We first recall the following standard facts.

Proposition 3.1. (i) ([29, E.1]) L_{a} = T_{a} for 0 ≤ a ≤ p − 1.

(ii) (Steinberg tensor product theorem, [29, 3.17]) L_{a+bp} = L_{a}⊗L_{b}^{(1)}, 0 ≤ a ≤ p−1, b ∈ ℤ_{≥0}. (iii) ([29, E.9]) One has

(3.1)  T_{a+pb} = T_{a} ⊗ T_{b}^{(1)} for p − 1 ≤ a ≤ 2p − 2, b ∈ ℤ_{≥0}.

In particular St_{r} = St ⊗ St_{r−1}^{(1)} for r ∈ ℤ_{≥1}.

An important role below will be played by the following proposition.

Proposition 3.2. Let k ∈ ℤ_{≥1} and X, Y ∈ T_{<p^{k}−1}. Then for any morphism f : X → Y the morphism f ⊗ 1 : X ⊗ St_{k−1} → Y ⊗ St_{k−1} is split.

Proof. We will use the following lemma.

Lemma 3.3. Assume that k ∈ ℤ_{≥1} and m ≤ p^{k} − 1. Then L_{m} ⊗ St_{k−1} is tilting.

Proof. We use induction in k. In the base case k = 1, St_{k−1} = St_{0} = 1 and the result follows from Proposition 3.1(i).

Now assume that Lemma 3.3 holds for some k. Let m ≤ p^{k+1} − 1. We write m = a + bp where 0 ≤ a ≤ p − 1 and 0 ≤ b ≤ p^{k} − 1, so L_{m} = L_{a+bp} = L_{a} ⊗ L_{b}^{(1)} by Proposition 3.1(ii). We have

L_{m} ⊗ St_{k} = L_{a} ⊗ L_{b}^{(1)} ⊗ St ⊗ St_{k−1}^{(1)} = T_{a} ⊗ St ⊗ (L_{b} ⊗ St_{k−1})^{(1)}.

By the induction assumption L_{b} ⊗ St_{k−1} is tilting, so St ⊗ (L_{b} ⊗ St_{k−1})^{(1)} is also tilting by Proposition 3.1(iii). Thus T_{a} ⊗ St ⊗ (L_{b} ⊗ St_{k−1})^{(1)} is also tilting as a product of two tilting modules. We have thus proved the induction step and the Lemma follows. □

Now consider the exact sequences

0 → Ker (f) → X → Im(f) → 0,
0 → Im(f) → Y → Coker(f) → 0.

Let M be one of Ker (f), Im(f), Coker(f). The Jordan-Holder factors of M are of the form L_{m}, 0 ≤ m < p^{k} − 1. Thus M ⊗ St_{k−1} has a filtration with subquotients of the form L_{m} ⊗ St_{k−1}, 0 ≤ m < p^{k} − 1 which are tilting by Lemma 3.3. Since all Exts between tilting modules vanish (see [29, Corollary E.2 (b)]) this filtration splits and M ⊗ St_{k−1} is tilting. Thus the sequences

0 → Ker (f) ⊗ St_{k−1} → X ⊗ St_{k−1} → Im(f) ⊗ St_{k−1} → 0,
0 → Im(f) ⊗ St_{k−1} → Y ⊗ St_{k−1} → Coker(f) ⊗ St_{k−1} → 0

also split and the result follows. □

Remark 3.4. The inequality for m in Lemma 3.3 is the best possible: the module L_{p^{k}} ⊗ St_{k−1} is not tilting. Namely this module has highest weight p^{k} + p^{k−1} − 1 and dimension 2p^{k−1} (since L_{p^{k}} = L_{1}^{(k)}); this dimension is less than the dimension of the Weyl module W_{p^{k}+p^{k−1}−1}, which equals p^{k} + p^{k−1}.

3.2. The functor T_{<p^{n}−1} → T_{n,p} is fully faithful.

Proposition 3.5. The composite functor T_{<p^{n}−1} ↪ T_{p} → T_{n,p} is fully faithful.

Proof. This functor is full by definition. To prove its faithfulness we need to show that no nonzero morphism in T_{<p^{n}−1} factors through I_{n} = T_{≥p^{n}−1}. Assume that this is not the case and let f̃ ∈ Hom(T_{m}, T_{k}) be a morphism in ⟨I_{n}⟩ with 0 ≤ m, k < p^{n} − 1. Then the morphism f ∈ Hom(1, T_{m}^{∗} ⊗ T_{k}) which is the image of f̃ under the canonical isomorphism Hom(T_{m}, T_{k}) ≃ Hom(1, T_{m}^{∗}⊗T_{k}) is also in ⟨I_{n}⟩. Thus there exists an object T_{s} with s ≥ p^{n}−1 and morphisms 1 → T_{s} and T_{s} → T_{m}^{∗} ⊗ T_{k} such that their composition is not zero.

Lemma 3.6. (see for example [11], Lemma 5.3.3 and references therein) Assume Hom(1, T_{s}) ≠ 0 in T_{p}. Then s = 2p^{l} − 2 for some l ∈ ℤ_{≥0} and the socle of T_{s} is one-dimensional.

Proof. For 0 ≤ s ≤ 2p − 2 the lemma follows from the description of tilting modules in [29, E.1]. Now assume that s > 2p−2 and write s = a+bp where p−1 ≤ a ≤ 2p−2 and b ∈ ℤ_{≥0}. Then by [29, Lemma E.9] we have T_{s} = T_{a} ⊗ T_{b}^{(1)}. Thus Hom(1, T_{s}) = Hom(T_{a}^{∗}, T_{b}^{(1)}). Thus T_{a}^{∗} must have a quotient on which the first Frobenius kernel of SL_{2}(k) acts trivially. Again by the description in [29, E.1] we see that a = 2p − 2 and the quotient is the object 1 = L_{0}. Thus Hom(1, T_{b}^{(1)}) ≠ 0, so Hom(1, T_{b}) ≠ 0. By induction we get that b = 2p^{l−1} − 2, so s = 2p^{l} − 2 as required.

The module T_{2p^{l}−2} is a lift to SL_{2}(k) of the injective hull of the trivial module over the l-th Frobenius kernel, see [29, E.9]; this implies the statement about the socle. □

Lemma 3.6 implies that the map T_{s} → T_{m}^{∗} ⊗ T_{k} is injective and s = 2p^{l} − 2 with l ≥ n. This is impossible as all weights of T_{m}^{∗} ⊗ T_{k} have absolute value ≤ m + k < 2(p^{n} − 1). □

3.3. Tensor ideals in T_{p}. Lemma 3.6 can be used to establish the classification of tensor ideals in T .

Proposition 3.7. ([11], Theorem 5.3.1) The only nonzero tensor ideals in T_{p} are the ideals ⟨I_{k}⟩, k ≥ 0.

Proof. Let I be a nonzero tensor ideal in T_{p}. Let f ∈ I be a nonzero morphism, f : X → Y. Then f defines a nonzero element f̃ ∈ Hom(1, X ⊗Y^{∗}) which belongs to I. Thus there exists a nonzero morphism g : 1 → T_{m} for some m ≥ 0 which belongs to I. Let m be the smallest such number. By Lemma 3.6, m = 2p^{k} − 2 for some k ≥ 0, and I contains Hom(1, T_{2p^{k}−2}). Since the map Hom(1, T_{2p^{l}−2}) → Hom(1, St_{l} ⊗ St_{l}^{∗}) induced by the inclusion of the direct summand T_{2p^{l}−2} ↪ St_{l} ⊗ St_{l}^{∗} is an isomorphism, we see that 1_{St_{k}} is contained in I, i.e., ⟨I_{k}⟩ ⊂ I. This implies that 1_{St_{l}} is contained in I for all l ≥ k. Thus, Hom(1, T_{2p^{l}−2}) is contained in I if and only if l ≥ k, i.e., if and only if it is contained in ⟨I_{k}⟩. Hence I = ⟨I_{k}⟩, since a tensor ideal is uniquely determined by its components in Hom(1, T) where T runs over indecomposable objects. □

3.4. Separatedness of T_{n,p} and its ideal of splitting objects. Now for n ≥ 0 let I_{n} = I_{n,p} be the thick ideal in T = T_{p} generated by St_{n}. We have I_{n} = T_{≥p^{n}−1}, the additive subcategory of T_{p} spanned by the T_{m} with m ≥ p^{n} − 1. Thus the tensor ideal ⟨I_{n}⟩ consists of morphisms which factor through a direct sum of T_{m}, m ≥ p^{n} − 1.

For n ≥ 1 consider the category T_{n,p} := T_{p}/I_{n}. This is a rigid monoidal Karoubian category with indecomposables T_{i}, 0 ≤ i ≤ p^{n} − 2.

Corollary 3.8. The category T_{n,p} is separated.

Proof. By Proposition 3.2, St_{n−1} = T_{p^{n−1}−1} is a splitting object in T_{n,p}. Also it is clear that if St_{n−1} ⊗ X = 0 for an object X then X = 0, which implies the statement by Proposition 3.7. □

Let S_{n,p} be the ideal of splitting objects of T_{n,p}.

Proposition 3.9. We have S_{n,p} = I_{n−1}/I_{n}.

Proof. By Proposition 3.7 and Corollary 3.8, S_{n,p} = I_{m}/I_{n} for some m < n. Thus S_{n,p} is indecomposable. Hence the claim follows from Corollary 3.8 and Proposition 2.32. □

3.5. Donkin’s recursive formula and auxiliary properties of tilting modules. We recall Donkin’s recursive formula for the tilting SL_{2}(k)−modules, see [17, Example 2, p. 47].

Let W_{a} denote the standard (i.e., Weyl) SL_{2}(k)–module with highest weight a. Recall that the modules T_{i} = W_{i} = L_{i} are simple for 0 ≤ i ≤ p − 1, and T_{0} = 1. Let V := L_{1}. Recall the following result.

Lemma 3.10. ([29, E.1]) (i) One has

(3.2)  V ⊗ T_{i} = T_{i−1} ⊕ T_{i+1} if 1 ≤ i ≤ p − 2.

(ii) For p ≤ a ≤ 2p − 2 the module T_{a} has a submodule W_{a} such that T_{a}/W_{a} = W_{2p−2−a}; thus its composition series is T_{a} = [L_{2p−2−a}, V^{(1)} ⊗ L_{a−p}, L_{2p−2−a}].

Using Proposition 3.1(iii), this implies

Proposition 3.11. (i) for p > 2 we have

(3.3)  V ⊗ T_{a+pb} = \{ \begin{matrix}T_{p} ⊗ T_{b}^{(1)} = T_{p+pb} & if & a = p − 1, \\ 2T_{p−1+pb} ⊕ T_{p+1+pb} & if & a = p, \\ T_{a−1+pb} ⊕ T_{a+1+pb} & if & p < a < 2p − 2, \\ T_{2p−3+pb} ⊕ T_{p−1} ⊗ (V ⊗ T_{b})^{(1)} & if & a = 2p − 2. \end{matrix}

(ii) For p = 2 we have

(3.4)  V ⊗ T_{a+2b} = \{ \begin{matrix}T_{2} ⊗ T_{b}^{(1)} = T_{2+2b}, a = 1 \\ 2T_{1+2b} ⊕ V ⊗ (V ⊗ T_{b})^{(1)}, a = 2. \end{matrix}

We also have the following result describing tensor products of tilting modules with small highest weights.

Proposition 3.12. The tensor product T_{m} ⊗ T_{r} for 1 ≤ m, r ≤ p − 1 is given by the formula

T_{m} ⊗ T_{r} = ⨁_{|m−r|≤k≤2(p−2)−m−r}T_{k} ⊕ ⨁_{p−1≤k≤m+r}T_{k},

if m + r ≥ p, and

T_{m} ⊗ T_{r} = ⨁_{|m−r|≤k≤m+r}T_{k}

if m + r < p, where the summation is over integers k such that m + r − k is even.

Proof. This follows immediately from Lemma 3.10 and Equations (3.3),(3.4) □

Proposition 3.13. Assume that 0 ≤ i ≤ p − 1 and m = pr + p − 2 for some r ∈ ℤ_{≥0}. Then T_{i} ⊗ T_{k} contains T_{m} as a summand if and only if k = m − i, in which case T_{m} appears with multiplicity 1.

Proof. For p = 2 the statement is clear from (3.4), so we may assume that p > 2. It is clear that T_{i} ⊗ T_{m−i} contains T_{m} with multiplicity 1, so we only need to prove the “only if” statement. The case i = 0 is clear as T_{0} = 1. For i ≥ 1 we proceed by induction in i. The base case i = 1 is clear by inspection of (3.3) and (3.2). Suppose 1 < i ≤ p − 1 and the statement is known for i−1, and let us prove it for i. So by (3.2), T_{i−1} ⊗V ⊗T_{k} contains T_{m}. By the induction assumption, V ⊗ T_{k} must contain T_{m−i+1}. But m − i + 1 ≡ −i − 1 ≢ −1 (mod p). Thus by (3.3) we have either k = m − i or k = m − i + 2 and V ⊗ T_{k} contains T_{m−i+1} with multiplicity 1. For the sake of contradiction assume that k = m − i + 2. Then T_{i−1} ⊗ V ⊗ T_{k} = (T_{i−2} ⊕ T_{i}) ⊗ T_{k} contains T_{m} with multiplicity 1. But T_{i−2} ⊗ T_{k} already contains T_{i−2+k} = T_{m}, so T_{i} ⊗ T_{k} cannot contain T_{m} (as the multiplicity is 1). Thus this case is impossible and k = m − i, as desired. □

4. The Verlinde categories Ver_{p^{n}}

4.1. The definition of the Verlinde categories Ver_{p^{n}}, Ver_{p^{n}}^{+}. Let p be a prime. Recall that Ver_{p} denotes the Verlinde category, which is the semisimplification of the category of modular representations Rep_{k}(ℤ/p) ([25, 27, 32]).

Let n ≥ 1 be an integer. By Corollary 3.8, the category T_{n,p} is separated, and by Proposition 3.9 its ideal of splitting objects S_{n,p} = I_{n−1}/I_{n} is spanned by the objects T_{i}, p^{n−1} − 1 ≤ i ≤ p^{n} − 2 and is indecomposable. Thus, using the main construction of Section 2, we can define finite symmetric tensor categories

Ver_{p^{n}} = Ver_{p^{n}}(k) := C(T_{n,p}).

If p^{n} > 2, we can also consider subcategories T_{n,p}^{+} ⊂ T_{n,p} spanned by T_{i} with even i, and the corresponding subcategories Ver_{p^{n}}^{+} := C(T_{n,p}^{+}) ⊂ Ver_{p^{n}}. This makes Ver_{p^{n}} into faithfully ℤ/2-graded tensor categories with trivial components Ver_{p^{n}}^{+} ([22], Subsection 4.14).

Definition 4.1. We will call the categories Ver_{p^{n}}, Ver_{p^{n}}^{+} the Verlinde categories.

The motivation for this terminology will be clarified in Subsection 4.4.

4.2. T_{n,p} is complete.

Theorem 4.2. (i) The category T_{n,p} is complete, so it is a full rigid monoidal Karoubian subcategory of Ver_{p^{n}} = C(T_{n,p}).

(ii) Ver_{p^{n}} is the abelian envelope of T_{n,p}.

Proof. (i) For p = 2 the result follows from [5], so we may assume that p ≥ 3.

We need to show that for any X, Y ∈ T_{n,p}, the map Hom(X, Y) → Hom(F (X), F (Y)) is an isomorphism, or, equivalently, that the map Hom(1, Y ⊗X^{∗}) → Hom(𝟙, F (Y)⊗F (X)^{∗}) = Hom(𝟙, F (Y ⊗ X^{∗})) is an isomorphism. Thus, it suffices to show that for any i ∈ [0, p^{n} − 2] the map Hom(1, T_{i}) → Hom(𝟙, F (T_{i})) is an isomorphism. For i ≥ p^{n−1} − 1 this follows from Proposition 2.24, so it suffices to treat the case i ≤ p^{n−1} − 2.

Let S = T_{p^{n−1}−1}. To compute Hom(𝟙, F (T_{i})), we will use the presentation of 𝟙 as the cokernel of the map

τ : S^{∗} ⊗ S ⊗ S^{∗} ⊗ S → S^{∗} ⊗ S.

Note that S ≅ S^{∗}. Hence we get that

Hom(𝟙, F (T_{i})) = Ker(Hom(S^{⊗2}, T_{i}) → Hom(S^{⊗4}, T_{i})).

Thus it suffices to check that the sequence

Hom_{T_{n,p}}(1, T_{i}) → Hom_{T_{n,p}}(S^{⊗2}, T_{i}) → Hom_{T_{n,p}}(S^{⊗4}, T_{i})

is exact. This sequence can be rewritten as

(4.1)  Hom_{T_{n,p}}(1, T_{i}) → Hom_{T_{n,p}}(S^{⊗2}, T_{i}) → Hom_{T_{n,p}}(S^{⊗2}, T_{i} ⊗ S^{⊗2}).

Let us now consider the same sequence in T_{p}:

(4.2)  Hom_{T_{p}}(1, T_{i}) → Hom_{T_{p}}(S^{⊗2}, T_{i}) → Hom_{T_{p}}(S^{⊗2}, T_{i} ⊗ S^{⊗2}),

which is clearly exact. But since i ≤ p^{n−1} − 2 and p ≥ 3, the sequence (4.2) actually belongs to T_{<p^{n}−1}, hence is isomorphic to (4.1) by Proposition 3.5. Thus (4.1) is exact, as required.^{3}

(ii) now follows from Theorem 2.42. □

Another proof of Theorem 4.2(i) is given in Remark 4.10 below.

Theorem 4.2 allows one to easily compute the dimension of the space of invariants in V^{⊗r}, where V := F (T_{1}) ∈ Ver_{p^{n}}. It is clear that if r is odd then this space is zero, so it suffices to compute the invariants in V^{⊗2m}. Let d_{mn} = dim Hom(𝟙, V^{⊗2m}).

Proposition 4.3. Let f_{n}(z) = ∑_{m≥0} d_{mn}z^{m}. Then

(4.3)  f_{n}(z) = \frac{(t + t^{−1})(t^{p^{n}−1} − t^{−p^{n}+1})}{t^{p^{n}} − t^{−p^{n}}},

where z = (t + t^{−1})^{−2}.

Proof. By Theorem 4.2, d_{mn} = dim Hom(1, T_{1}^{⊗2m}). The rest of the argument is the same as in [5], proof of Proposition 3.9. □

Remark 4.4. Proposition 4.3 generalizes [5], Proposition 3.3. Note that there is a misprint in [5], Proposition 3.3 and the proof of Proposition 3.9: 2^{n} should be replaced by 2^{n+1}.

3This method of proof was suggested by K. Coulembier.

4.3. Basic properties of Ver_{p^{n}}.

Theorem 4.5. (i) If n = 1 then Ver_{p^{n}} = Ver_{p}, Ver_{p^{n}}^{+} = Ver_{p}^{+}.

(ii) We have a fully faithful monoidal functor F : T_{n,p} → Ver_{p^{n}} which defines an equivalence between S_{n,p} and the category of projectives in Ver_{p^{n}}. In particular, Ver_{p^{n}} has p^{n−1}(p − 1) simple objects whose projective covers are F (T_{i}), p^{n−1} −1 ≤ i ≤ p^{n} −2, and all simple objects in Ver_{p^{n}} are self-dual.

(iii) We have Ver_{2^{n}} = C_{2n−2} and if n ≥ 2 then Ver_{2^{n}}^{+} = C_{2n−3}, where C_{i} are the categories defined in [5].

(iv) For p^{n} > 2 let O_{n,p} be the ring of integers in the field ℚ(2 cos(2π/p^{n})) = ℚ(q^{2} + q^{−2}), where q = q_{n,p} := exp(πi/p^{n}); that is, O_{n,p} = ℤ[2 cos(2π/p^{n})] ([35], p.16, Proposition 2.16). Then the Frobenius-Perron dimension FPdim defines an isomorphism of the Grothendieck ring Gr(Ver_{p^{n}}^{+}) onto O_{n,p}.

(v) There are no tensor functors Ver_{p^{n}}^{+} → Ver_{p^{m}}^{+} for m < n.

Proof. Parts (i), (ii) are immediate from the previous results. Part (iii) follows from (ii) and [5], Theorem 4.14.

Let us prove (iv). Note that the split Grothendieck ring of T_{n,p} is a ℤ_{+}-ring of finite rank ([22], Section 3), so for T ∈ T_{n,p} we can define its Frobenius-Perron dimension FPdim(T). Let

(4.4)  Q_{n,p}(2 cos x) := \frac{sin(p^{n}x)}{sin x}

be the Chebyshev polynomial of the second kind. By [5], Corollary 3.5,

Q_{n,p}(FPdim(T_{1})) = 0,

which implies that FPdim(T_{1}) = FPdim(F (T_{1})) = 2 cos(π/p^{n}). Thus we have FPdim(T_{2}) = 2 cos(2π/p^{n}) + 1 for p > 2 and FPdim(T_{2}) = 2 cos(2π/p^{n}) + 2 for p = 2. Hence the image of FPdim contains O_{n,p}. On the other hand, since the rank of Ver_{p^{n}}^{+} is p^{n−1}(p − 1)/2, which equals the degree of ℚ(2 cos(2π/p^{n})) over ℚ, we see that FPdim lands in ℚ(2 cos(2π/p^{n})), hence in O_{n,p}, since the Frobenius-Perron dimension of any object is an algebraic integer ([22], Proposition 3.3.4). Thus FPdim : Gr(Ver_{p^{n}}^{+}) → O_{n,p} is an isomorphism.

(v) follows from (iv). □

Remark 4.6. For p = 2 Theorem 4.5 is proved in [5], Corollary 2.2 (the ring O_{n,2} is denoted there by O_{n−2}).

Let Q_{n,p} = Q_{n,p}^{+} − Q_{n,p}^{−} be the splitting of the Chebyshev polynomial of the second kind Q_{n,p} (see (4.4)) into the parts with positive and negative coefficients.

Corollary 4.7. (The universal property of Ver_{p^{n}}) Let D be a symmetric tensor category over k. Then the category of k-linear symmetric monoidal functors E: Ver_{p^{n}} → D is equivalent to the category of objects X ∈ D such that ∧^{3}X = 0 and Q_{n,p}^{+}(X) ≅ Q_{n,p}^{−}(X), equipped with an isomorphism ∧^{2}X ≅ 𝟙. This equivalence is given by E ↦ E(V), where V := F (T_{1}). In particular, the group of tensor automorphisms of the identity functor of Ver_{p^{n}} is ℤ/2 for p > 2 and trivial for p = 2.

Proof. This follows from Theorem 4.2 and [5], Corollary 3.5. The last statement follows since the group of automorphisms of V preserving the identification ∧^{2}V ≅ 𝟙 is ℤ/2 for p > 2 and trivial for p = 2. □

Remark 4.8. The condition that dim X = 2 in [5], Corollary 3.5 is redundant since this follows in any characteristic from the conditions ∧^{2}X ≅ 𝟙, ∧^{3}X = 0. Indeed, let d = dim X. Then for p > 3 we have \frac{d(d−1)}{2} = 1, \frac{d(d−1)(d−2)}{6} = 0, so d = 2. For p = 3 we have \frac{d(d−1)}{2} = 1, so d^{2} − d − 2 = 0, which in characteristic 3 can be written as (d − 2)^{2} = 0, hence again d = 2. Finally, for p = 2 we have dim(∧^{3}X) = dim(∧^{2}X)d (see for example [23]), which implies that d = 0 = 2.

4.4. The lift of Ver_{p^{n}} to characteristic zero. Let W (k) be the ring of Witt vectors of k, K_{0} its fraction field, and \overline{K}_{0} the algebraic closure of K_{0}. Fix n and let ζ = ζ_{n,p} ∈ \overline{K}_{0} be a primitive p^{n}-th root of unity. Consider the discrete valuation ring R_{n,p} := W (k)[ζ] with uniformizing element ζ − 1. Let K be the fraction field of R_{n,p} (a ramified extension of K_{0} of degree p^{n−1}(p − 1)). In this subsection we will show that the category Ver_{p^{n}}(k) admits a flat deformation to a braided category over R_{n,p}, whose generic fiber is the semisimple Verlinde category Ver_{p^{n}}(K) associated to the quantum group SL_{2} at a p^{n}-th root of unity. To this end, we will use the theory of splitting objects for categories defined over a complete DVR developed in Subsection 2.11.

Let T L(ζ) be the lopsided version of the Temperley-Lieb category with parameter δ = ζ^{1/2} + ζ^{−1/2}, see [31], Subsection 2.5; it is defined over ℤ[ζ], hence over R_{n,p}, and we’ll use the R_{n,p}-linear version. Then T L(ζ) is a flat braided monoidal Karoubian category over R_{n,p} with indecomposables T̃_{i}, i ≥ 0. Also, T L(ζ)_{k} = T_{p} is the symmetric category of tilting modules for SL_{2}(k), while T L(ζ)_{K} is (the lopsided version of) the braided category of tilting modules for the quantum group SL_{2}^{ζ^{1/2}} over K. Moreover, T L(ζ) has a tensor ideal I(ζ) generated by the Steinberg module St = T̃_{p^{n}−1} over the quantum group, which specializes to I_{n} over k, and localizes to I(ζ)_{K} := K ⊗_{R_{n,p}} I(ζ) over K (note that other tensor ideals I_{k}, k ≠ n, do not lift to tensor ideals in T L(ζ)). Thus we can define the flat braided monoidal category T (ζ) := T L(ζ)/I(ζ) over R_{n,p} with indecomposables T̃_{i}, 0 ≤ i ≤ p^{n} − 2, whose reduction T (ζ)_{k} is the symmetric category T_{n,p} with indecomposables T_{i}, 0 ≤ i ≤ p^{n} − 2, while the localization T_{K} is the split semisimple braided Verlinde category Ver_{p^{n}}(K) (the semisimplification of T L(ζ)_{K}, see [3], Subsection 3.3 and references therein), with simple objects T_{i}^{ζ} = W_{i}^{ζ} = L_{i}^{ζ}, 0 ≤ i ≤ p^{n} − 2, the Weyl modules over the quantum group (which are also tilting and simple modules). Thus the objects T̃_{i}, p^{n−1} − 1 ≤ i ≤ p^{n} − 2 are splitting objects of T (ζ).

Remark 4.9. Note that for p > 2, since ζ is a root of unity of odd order, the category T L(ζ)_{K} is not modular (unlike the situation of [3], Subsection 3.3); rather, its symmetric (Müger) center is sVec_{K} generated by T_{0}^{ζ} = 1 and T_{p^{n}−2}^{ζ} = ψ. In fact, we have Ver_{p^{n}}(K) = Ver_{p^{n}}^{+}(K)⊠sVec_{K} as braided categories, where Ver_{p^{n}}^{+}(K) is the (modular) category generated by the objects of Ver_{p^{n}}(K) with even highest weight.

Remark 4.10. The construction of this subsection gives another proof of Theorem 4.2(i) which works for any p and does not use [5]. Namely, Theorem 4.2(i) follows from Corollary 2.61, since the category Ver_{p^{n}}(K) is semisimple.

4.5. The Frobenius-Perron dimension and the Grothendieck ring of Ver_{p^{n}}.

Proposition 4.11. (i) We have

FPdim(Ver_{p^{n}}) = \frac{p^{n}}{2 sin^{2}(\frac{π}{p^{n}})}.

(ii) The Cartan matrix of Ver_{p^{n}} is given by C = DD^{T}, where D is its decomposition matrix (with respect to its lift to characteristic zero).

Proof. (i) This follows from Proposition 2.60(i) since

FPdim(Ver_{N}(K)) = ∑_{k=1}^{N−1} \frac{sin^{2}(\frac{πk}{N})}{sin^{2}(\frac{π}{N})} = \frac{N}{2 sin^{2}(\frac{π}{N})}.

(ii) This follows from Corollary 2.61, as the category Ver_{p^{n}}(K) is semisimple. □

Corollary 4.12. If p > 2 then we have Ver_{p^{n}} = Ver_{p^{n}}^{+} ⊠ sVec. Thus the Grothendieck ring of Ver_{p^{n}} is given by Gr(Ver_{p^{n}}) = O_{n,p}ℤ/2 = O_{n,p}[g]/(g^{2} − 1).

Proof. The first statement holds because it holds for the category Ver_{p^{n}}(K) (cf. Remark 4.9). Namely, the nontrivial invertible object of Ver_{p^{n}}(K) has to descend to an invertible object over k (as it has FPdim = 1). The second statement then follows from Theorem 4.5(iv). □

4.6. The Cartan matrix and the decomposition matrix of Ver_{p^{n}}. Now we can compute the decomposition matrix D (and hence the Cartan matrix C) of Ver_{p^{n}}. Recall that the indecomposable projectives of Ver_{p^{n}} are P_{i} = F (T_{i}) for p^{n−1} − 1 ≤ i ≤ p^{n} − 2. So we may label the simple objects by the same integers and denote them by L_{i}; thus P_{i} is the projective cover of L_{i}. Also we have the extended decomposition matrix D̃, a square matrix of size p^{n} − 1 with entries d_{ij}, 0 ≤ i, j ≤ p^{n} − 2, where d_{ij} := [(T̃_{i})_{K} : W_{j}^{ζ}] is the multiplicity of W_{j}^{ζ} in the reduction of T̃_{i} to K. Then D is the submatrix of D̃ consisting of the rows labeled by p^{n−1} − 1 ≤ i ≤ p^{n} − 2.

Note that the characters of W_{j}^{ζ} are [j + 1]_{x} = \frac{x^{j+1}−x^{−j−1}}{x−x^{−1}} . Thus if χ_{i}(x) is the character of T_{i} as an SL_{2}(k)-module then the entries d_{ij} of D̃ (hence D) are determined from the formula

(4.5)  χ_{i}(x) = ∑_{j} d_{ij} \frac{x^{j+1} − x^{−j−1}}{x − x^{−1}} .

Proposition 4.13. (i) We have

∑_{j} d_{ij}x^{j+1} = (χ_{i}(x)(x − x^{−1}))_{+},

where the subscript + denotes the part with positive powers of x.

(ii) The decomposition matrix D has maximal rank p^{n−1}(p − 1);

(iii) The entries of the Cartan matrix C are c_{ij} = (χ_{i}, χ_{j}) (the inner product of SL_{2}- characters).

(iv) The Cartan matrix C of Ver_{p^{n}} is positive definite, hence nondegenerate.

Proof. (i) is immediate from formula (4.5). (ii) follows since d_{ij} = 0 if j > i. (iii) follows from (i) and Proposition 4.11(ii). Finally, (iv) follows from (ii), (iii). □

Corollary 4.14. The Frobenius-Perron dimensions d_{i} = FPdim(L_{i}) of simple objects of Ver_{p^{n}} are determined uniquely by the system of equations

Cd = p,

where d = (d_{i}), and p = (p_{i}) is the vector of Frobenius-Perron dimensions p_{i} of P_{i}, p^{n−1}−1 ≤ i ≤ p^{n} − 2. That is,

d = C^{−1}p.

Proof. This follows from Proposition 4.13(iv). □

The vector p can be computed from the formula p = Dw, where w_{j} = [j + 1]_{q} = \frac{sin(π(j+1)/p^{n})}{sin(π/p^{n})} are the Frobenius-Perron dimensions of the Weyl modules W_{j}^{ζ} in Ver_{p^{n}}(K). Thus we get

d = C^{−1}Dw = (DD^{T})^{−1}Dw

So we obtain

Proposition 4.15. We have d = (DD^{T})^{−1}Dw, where w_{j} = [j + 1]_{q} for 0 ≤ j ≤ p^{n} − 2 and d_{ij} are the coefficients of χ_{i}(x)(x − x^{−1}) for p^{n−1} − 1 ≤ i ≤ p^{n} − 2.

This determines the Grothendieck ring of Ver_{p^{n}} with ℤ_{+}-basis since the Frobenius-Perron dimension defines an isomorphism of this ring with O_{n,p}.

Remark 4.16. We indicate here an alternative approach to the results of Subsections 4.4, 4.6 which avoids using the lifting to characteristic zero. Namely one can compute the entries of the Cartan matrix directly as

c_{ij} = dim Hom_{Ver_{p^{n}}}(P_{i}, P_{j}) = dim Hom_{T_{p}}(T_{i}, T_{j})

using Proposition 3.5 and Theorem 4.2. However, it is well known that dim Hom_{T_{p}}(T_{i}, T_{j}) = (χ_{i}, χ_{j}), see for example [2, Section 3A], so we get the formula C = DD^{T} as in Proposition 4.11 (ii) where D is defined by equation (4.5). This implies that Propositions 4.13 and 4.15 hold and an easy computation gives the formula from Proposition 4.11 (i).

The matrix D̃ (hence D) can now be easily computed via Donkin’s explicit algorithm, see Proposition 3.1(iii) and [29, E.9]. However, there is an even more explicit formula for D̃, given in [33]. Namely, given a number a ∈ [p^{n−1}, p^{n} − 1] written in base p as \overline{a_{1}...a_{n}} (so a_{1} ≠ 0), we say that b is a descendant of a if b = a_{1}p^{n−1} ± a_{2}p^{n−2} ± ... ± a_{n} for some choice of signs (clearly, in this case b ∈ [1, p^{n} − 1]). Thus, every number a has 2^{ℓ} distinct descendants, where ℓ is the number of nonzero digits among a_{2}, ..., a_{n}.

Proposition 4.17. [33] One has d_{ij} = 1 if j + 1 is a descendant of i + 1, otherwise d_{ij} = 0.

This yields a very explicit description of the Cartan matrix of Ver_{p^{n}}.

Corollary 4.18. The entry c_{ij} of the Cartan matrix is the number of common descendants of i + 1 and j + 1.

Proof. This follows from Proposition 4.11(ii) and Proposition 4.17. □

4.7. An explicit formula for the Cartan matrix of Ver_{p^{n}}. Let us now derive an explicit formula for the Cartan matrix C = C_{n,p} of the category Ver_{p^{n}}. We first consider the case p = 2. In this case let C_{n,2}^{+} be the Cartan matrix of Ver_{2^{n}}^{+}. Then it is shown in [5], Proposition 4.2 that

C_{n+1,2}^{+} = (\begin{matrix}2C_{n,2}^{+} & C_{n,2}^{+} \\ C_{n,2}^{+} & 2C_{n−1,2} \end{matrix})

and

C_{n,2} = C_{n,2}^{+} ⊕ C_{n−1,2},

for an appropriate labeling of rows and columns, where C_{1,2} = 1, C_{2,2}^{+} = 2. Hence

(\begin{matrix}C_{n+1,2}^{+} \\ C_{n,2} \end{matrix}) = (\begin{matrix}2 1 1 0 & 0 0 0 2 \\ 1 0 0 0 & 0 0 0 1 \end{matrix}) ⊗ (\begin{matrix}C_{n,2}^{+} \\ C_{n−1,2} \end{matrix}) .

Thus we get

Proposition 4.19. One has

(\begin{matrix}C_{n+1,2}^{+} \\ C_{n,2} \end{matrix}) = (\begin{matrix}2 1 1 0 & 0 0 0 2 \\ 1 0 0 0 & 0 0 0 1 \end{matrix})^{⊗n−1} ⊗ (\begin{matrix}2 \\ 1 \end{matrix}) .

This can also be derived from Corollary 4.18.

Now let us pass to the case p > 2, and compute C_{n,p} in a somewhat different way. For two numbers a, b with exactly m + 1 digits in the base p expansion let X_{ab}(m) be the number of common descendants of a, b. By Corollary 4.18, C_{n+1,p} = X(n) (where we label simple objects of Ver_{p^{n}} by highest weight of the projective cover). Let Y_{ab}(m) be the number of descendants ã of a such that b̃ := ã − 1 is a descendant of b. We will often write X_{ab}(m), Y_{ab}(m) as X_{ab}, Y_{ab} respectively, using that m can be recovered as the number of digits of a and b. It is clear that for m = 0 we have X_{ab} = δ_{ab} and Y_{ab} = δ_{a,b+1}.

Let a = \overline{a_{m}...a_{1}a_{0}}, b = \overline{b_{m}...b_{1}b_{0}} where a_{m}, b_{m} ≠ 0, and let a′ = \overline{a_{m}...a_{1}}, b′ = \overline{b_{m}...b_{1}}.

Proposition 4.20. (i) If a_{0} = b_{0} = 0 then X_{ab} = X_{a′b′}.

(ii) If a_{0} = b_{0} ≠ 0 then X_{ab} = 2X_{a′b′}.

(iii) If a_{0} + b_{0} = p then X_{ab} = Y_{a′b′} + Y_{b′a′}.

(iv) Otherwise X_{ab} = 0.

Proof. (i), (ii), (iv) are easy, let us prove (iii). Having a common descendant means that ã′p ± a_{0} = b̃′p ± b_{0} for some choice of signs. The possible choices are +, −, which gives ã′ − b̃′ = −1 and −, +, which gives ã′ − b̃′ = 1, as desired. □

Proposition 4.21. (i) If a_{0} − b_{0} = ±1 then Y_{ab} = X_{a′b′}.

(ii) If a_{0} − b_{0} = p − 1 then Y_{ab} = Y_{a′b′}.

(iii) If a_{0} + b_{0} = p + 1 then Y_{ab} = Y_{b′a′}.

(iv) Otherwise Y_{ab} = 0.

Proof. Similar to Proposition 4.20. □

Let Z = Y + Y^{T}.

Corollary 4.22. (i) If a_{0} = b_{0} = 0 then X_{ab} = X_{a′b′}.

(ii) If a_{0} = b_{0} ≠ 0 then X_{ab} = 2X_{a′b′}.

(iii) If a_{0} + b_{0} = p then X_{ab} = Z_{a′b′}.

(iv) Otherwise X_{ab} = 0.

Corollary 4.23. (i) If a_{0} − b_{0} = ±1 then Z_{ab} = 2X_{a′b′}.

(ii) If a_{0} + b_{0} = p − 1 then Z_{ab} = Z_{a′b′}.

(iii) If a_{0} + b_{0} = p + 1 then Z_{ab} = Z_{a′b′}.

(iv) Otherwise Z_{ab} = 0.

Define the following p×p matrices A, B, S, D with rows and columns labeled by 0, ..., p−1. Let A be defined by A_{ij} = δ_{i,j−1} + δ_{i,j+1}, B be defined by B_{ij} = δ_{i+j,p−1} + δ_{i+j,p+1}, S be the matrix with S_{ij} = δ_{i+j,p}, and D := diag(1, 2, 2, ..., 2).

Corollary 4.24. We have X(0) = Id_{p−1}, Z(0) = \overline{A}, where \overline{A} is obtained from A by deleting the 0-th row and column, and for m ≥ 1

X(m) = D ⊗ X(m − 1) + S ⊗ Z(m − 1), Z(m) = 2A ⊗ X(m − 1) + B ⊗ Z(m − 1).

Thus we get

(\begin{matrix}X(m) \\ Z(m) \end{matrix}) = (\begin{matrix}D & S \\ 2A & B \end{matrix}) ⊗ (\begin{matrix}X(m − 1) \\ Z(m − 1) \end{matrix}) .

So we get

Proposition 4.25. One has

(\begin{matrix}C_{n,p} \\ Z(n − 1) \end{matrix}) = (\begin{matrix}D & S \\ 2A & B \end{matrix})^{⊗n−1} ⊗ (Id_{p−1} ; \overline{A})

Example 4.26. For p = 3 we have

(\begin{matrix}C_{n,3} \\ Z(n − 1) \end{matrix}) = (\begin{matrix}1 0 0 0 2 0 0 0 2 & 0 0 0 0 0 1 0 1 0 \\ 0 2 0 2 0 2 0 2 0 & 0 0 1 0 1 0 1 0 1 \end{matrix})^{⊗n−1} ⊗ (\frac{1 0 0 1}{0 1 1 0}) .

Thus, for n = 2 we get

C_{2,3} = (\begin{matrix}1 0 0 0 2 0 0 0 2 & 0 0 0 0 0 1 0 1 0 \\ 0 0 0 0 0 1 0 1 0 & 1 0 0 0 2 0 0 0 2 \end{matrix})

which after permuting rows and columns falls into a direct sum of the Cartan matrices for

the four blocks of Ver_{3^{2}}, two copies of 1 and two copies of (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}).

Corollary 4.27. All entries of the Cartan matrix C_{n,p} are either 0 or 2^{m} with 0 ≤ m ≤ n−1.

Proof. It follows from Propositions 4.19, 4.25 that while computing entries of C_{n,p}, we never have to add nonzero numbers and only have to multiply them (for example, for p > 2 this is so because the matrices D and S don’t have nonzero entries in the same position, and A, B don’t either). This implies the statement. □

Example 4.28. The Cartan matrix entry c_{𝟙,𝟙} of C = C_{n,p} equals 2^{n−1}.

Remark 4.29. 1. For p = 2 Corollary 4.27 is proved in [5], Proposition 4.2.

2. Here is another proof of Corollary 4.27. Namely, by Proposition 3.1(iii), it suffices to show that for p − 1 ≤ a, c ≤ 2p − 2 the space

Hom(T_{a} ⊗ T_{b}^{(1)}, T_{c} ⊗ T_{d}^{(1)}) = Hom(T_{a} ⊗ T_{c}, (T_{b} ⊗ T_{d})^{(1)})

has dimension 0 or 2^{m}, 0 ≤ m ≤ n − 1. But among the direct summands of T_{a} ⊗ T_{c} only T_{2p−2} and T_{3p−2} have quotients on which the Frobenius kernel acts trivially (note that T_{3p−2} appears only if a and c have different parities and the corresponding quotient is T_{1}^{(1)}). Thus we have the following recursive formulas:

(4.6) dim Hom(T_{a} ⊗ T_{b}^{(1)}, T_{c} ⊗ T_{d}^{(1)}) = \{ \begin{matrix}2 dim Hom(T_{b}, T_{d}) & if & a = c ≠ p − 1, \\ dim Hom(T_{b}, T_{d}) & if & a = c = p − 1, \\ dim Hom(T_{b}, V ⊗ T_{d}) & if & a + c = 3p − 2. \\ 0 otherwise. \end{matrix}

We can combine formula (4.6) with (3.3) and (3.2). One observes that in all cases the number dim Hom(T_{b}, V ⊗ T_{d}) equals to dim Hom(T_{b}, T_{d±1}) or dim Hom(T_{b±1}, T_{d}) as the summands in (3.3) are typically in different linkage classes. This implies that all nonzero entries of the Cartan matrix are powers of 2, and the power is ≤ n − 1.

4.8. The objects 𝕋_{i} = F (T_{i}). Let us now discuss the properties of the objects 𝕋_{i} := F (T_{i}). Recall that for p^{n−1} − 1 ≤ i ≤ p^{n} − 2 the objects 𝕋_{i} = P_{i} are precisely the indecomposable projectives of the category Ver_{p^{n}}.

Next consider the case i < p. Note that for such i we have 𝕋_{i} = S^{i}𝕋_{1}.

Lemma 4.30. For i < p the objects 𝕋_{i} are simple. The projective cover of 𝕋_{i} is 𝕋_{2p^{n−1}−2−i}.

Proof. We claim that if, for j ∈ [p^{n−1} − 1, p^{n} − 2], we have Hom(T_{j}, T_{i}) ≠ 0 then it is 1-dimensional and j = 2p^{n−1} − 2 − i. Indeed,

Hom(T_{j}, T_{i}) = Hom(1, T_{i} ⊗ T_{j}^{∗}) = Hom(1, T_{i} ⊗ T_{j}).

Thus by Lemma 3.6 Hom(T_{j}, T_{i}) ≠ 0 if and only if T_{i} ⊗ T_{j} contains T_{2p^{n−1}−2} as a direct summand. Now the claim follows from Proposition 3.13.

It follows that for any 𝕋_{i} there exists a unique indecomposable projective P such that Hom(P, 𝕋_{i}) ≠ 0 and this Hom–space is one dimensional. This implies the statement. □

Lemma 4.31. The object 𝕋_{p} has length 3 and composition series [𝕋_{p−2}, U, 𝕋_{p−2}] for some simple U.

Proof. We have 𝕋_{p} = 𝕋_{1} ⊗ 𝕋_{p−1}. Thus FPdim(𝕋_{p}) = \frac{(q^{2}−q^{−2})(q^{p}−q^{−p})}{(q−q^{−1})^{2}}. But we have nonzero morphisms 𝕋_{p} → 𝕋_{p−2} and 𝕋_{p−2} → 𝕋_{p}, so by Lemma 4.30, 𝕋_{p} has composition series as claimed with some object U. It remains to show that U is simple. To this end, note that

FPdim(U) = FPdim(𝕋_{p}) − 2 FPdim(𝕋_{p−2}) =

= \frac{(q^{2} − q^{−2})(q^{p} − q^{−p}) − 2(q − q^{−1})(q^{p−1} − q^{−p+1})}{(q − q^{−1})^{2}} = q^{p} + q^{−p}.

Thus FPdim(U) < 2, so U is simple. □

4.9. Some properties of the Frobenius functor. Consider the Frobenius functor

𝔽 : Ver_{p^{n}} → Ver_{p^{n}}^{(1)} ⊠ Ver_{p}

defined in [24] (note that Ver_{p^{n}}^{(1)} ≅ Ver_{p^{n}} since the categories T_{n,p} and Ver_{p^{n}} are defined over the prime field).

Proposition 4.32. For n ≥ 2 the functor 𝔽 is not exact, i.e., the category Ver_{p^{n}} is not Frobenius exact in the sense of [24]. Equivalently, Ver_{p^{n}} is not locally semisimple in the sense of [12] (see [24], Remark 7.2).

Proof. Since there are no tensor functors Ver_{p^{n}} → Ver_{p} (Theorem 4.5(v)), by [24, Section 8] the category Ver_{p^{n}} is not Frobenius exact. □

In this section we discuss some properties of 𝔽 restricted to F (T_{n,p}) ⊂ Ver_{p^{n}}. We assume that n > 1; see [32, Example 3.8] for the case n = 1.

Corollary 4.33. We have 𝔽(𝕋_{1}) = U ⊠𝟙 ∈ Ver_{p^{n}} ⊠Ver_{p}, where U is the object from Lemma 4.31.

Proof. We have a decomposition 𝕋_{1}^{⊗p} = ⨁_{i<p−2}𝕋_{i} ⊗ π_{i} ⊕ M, where π_{i} are free ℤ/p-modules and

M = (p − 2)𝕋_{p−2} ⊕ 𝕋_{p} = (p − 2)𝕋_{p−2} ⊕ [𝕋_{p−2}, U, 𝕋_{p−2}],

where the last equality comes from Lemma 4.31. This implies the statement. □

Corollary 4.33 implies that for n > 1 the functor 𝔽 restricted to F (T_{n,p}) lands in Ver_{p^{n}}⊠𝟙 ⊂ Ver_{p^{n}} ⊠ Ver_{p}. In what follows we identify Ver_{p^{n}} ⊠ 𝟙 with Ver_{p^{n}}, so 𝔽 : F (T_{n,p}) → Ver_{p^{n}}.

The following result should be compared with (3.1).

Proposition 4.34. Let n > 1. For p − 1 ≤ a ≤ 2p − 2 and b ∈ ℤ_{≥0} we have

(4.7)  𝕋_{a} ⊗ 𝔽(𝕋_{b}) = 𝕋_{a+pb}.

Proof. The morphisms 𝕋_{p−2} → 𝕋_{p} and 𝕋_{p} → 𝕋_{p−2} from the proof of Lemma 4.31 split after tensoring with St = 𝕋_{p−1} by Proposition 3.2, so we get St ⊗ U = 𝕋_{2p−1} by an easy computation (where we denote F (St) just by St for brevity).

Let T_{n,p}^{1} = I_{1}/I_{n} ⊂ T_{n,p} be the thick ideal generated by St. We prove now that St ⊗ 𝔽(𝕋_{i}) ∈ F (T_{n,p}^{1}) by induction in i. The base case i = 0 is clear as T_{0} = 𝟙. Assume that St ⊗ 𝔽(𝕋_{i}) ∈ F (T_{n,p}^{1}) holds for some i. Then 𝕋_{i+1} is a direct summand of 𝕋_{i} ⊗ 𝕋_{1}, so St ⊗ 𝔽(𝕋_{i+1}) is a direct summand of St ⊗ 𝔽(𝕋_{i}) ⊗ 𝔽(𝕋_{1}) = St ⊗ U ⊗ 𝔽(𝕋_{i}) = 𝕋_{2p−1} ⊗ 𝔽(𝕋_{i}). Since 𝕋_{2p−1} is a direct summand of 𝕋_{p} ⊗ St, we see that St ⊗ 𝔽(𝕋_{i+1}) is a direct summand of 𝕋_{p} ⊗ St ⊗ 𝔽(𝕋_{i}) and thus St ⊗ 𝔽(𝕋_{i+1}) ∈ F (T_{n,p}^{1}) since the category T_{n,p}^{1} is closed under tensor products and direct summands.

For a ≥ p − 1 the object 𝕋_{a} is a direct summand of 𝕋_{a−(p−1)} ⊗ St, so 𝕋_{a} ⊗ 𝔽(𝕋_{b}) is a direct summand of 𝕋_{a−(p−1)} ⊗ St ⊗ 𝔽(𝕋_{b}), so 𝕋_{a} ⊗ 𝔽(𝕋_{b}) ∈ F (T_{n,p}^{1}) by the preceding paragraph.

We now prove the proposition by induction in b with the base case b = 0 being clear. Thus assume that (4.7) holds for all b ≤ m. Let S = S(m) be the multi-subset of ℤ_{≥0} such that

T_{1} ⊗ T_{m} = T_{m+1} ⊕ ⨁_{s∈S}T_{s}.

It is clear that s < m for any s ∈ S, so (4.7) holds for b = s ∈ S. We have

St⊗𝕋_{a}⊗𝔽(𝕋_{m+1}⊕⨁_{s∈S}𝕋_{s}) = St⊗𝕋_{a}⊗𝔽(𝕋_{1}⊗𝕋_{m}) = St⊗U ⊗𝕋_{a}⊗𝔽(𝕋_{m}) = 𝕋_{2p−1}⊗𝕋_{a+pm} =

F (T_{2p−1} ⊗ T_{a+pm}) = F (T_{p−1} ⊗ T_{1}^{(1)} ⊗ T_{a} ⊗ T_{m}^{(1)}) = St ⊗ F (T_{a} ⊗ (T_{1} ⊗ T_{m})^{(1)}) =
St ⊗ F (T_{a} ⊗ (T_{m+1} ⊕ ⨁_{s∈S}T_{s})^{(1)}) = St ⊗ F (T_{a+p(m+1)} ⊕ ⨁_{s∈S}T_{a+ps}) =
= St ⊗ (𝕋_{a+p(m+1)} ⊕ ⨁_{s∈S}𝕋_{a+ps}).

Thus using the induction assumption for b = s ∈ S we get

St ⊗ 𝕋_{a} ⊗ 𝔽(𝕋_{m+1}) = St ⊗ 𝕋_{a+p(m+1)},

that is (4.7) holds for b = m + 1 after tensoring with St.

Luckily we can cancel St: in the split Grothendieck ring K(T_{n,p}) we have K(T_{n,p}^{1}) = [St]K(T_{n,p}) and we get the cancellation by applying the following result to the ring K(T_{n,p}) and a = [St]: in a commutative semisimple ring the equality a^{2}b_{1} = a^{2}b_{2} implies ab_{1} = ab_{2}. □

4.10. The inclusion Ver_{p^{n−1}} ↪ Ver_{p^{n}}. In this section we use (4.7) to prove the following theorem.

Theorem 4.35. For n ≥ 2 we have a natural inclusion H : Ver_{p^{n−1}} ↪ Ver_{p^{n}} as a tensor subcategory.

Proof. Recall that we have a tensor functor 𝔽 : F (T_{n,p}) → Ver_{p^{n}}.

Corollary 4.36. 𝔽(F (St_{n−1})) = 0 and 𝔽(F (St_{n−2})) ≠ 0.

Proof. By (4.7) we have

𝔽(𝕋_{p^{n−1}−1}) ⊗ 𝕋_{p−1} = 𝕋_{p−1+p(p^{n−1}−1)} = St_{n} = 0

and

𝔽(𝕋_{p^{n−2}−1}) ⊗ 𝕋_{p−1} = 𝕋_{p−1+p(p^{n−2}−1)} = St_{n−1} ≠ 0. □

Thus the functor 𝔽 ∘ F induces a faithful monoidal functor H̃ : T_{n−1,p} → Ver_{p^{n}}, see Proposition 3.7 or [5], Lemma 3.7. But T_{n−1,p} has an abelian envelope Ver_{p^{n−1}} by Theorem 4.2. Thus the functor H̃ extends to an (exact) tensor functor H : Ver_{p^{n−1}} → Ver_{p^{n}}. This functor is automatically faithful, see [22, Remark 4.3.10]. We claim that it is also full, i.e., H is an embedding, see [22, 6.3].

Corollary 4.37. The functor H̃ is full. In particular the functor H is full on projective objects of Ver_{p^{n−1}}.

Proof. It is sufficient to show that the injective map Hom(𝟙, 𝕋_{b}) → Hom(𝔽(𝟙), 𝔽(𝕋_{b})) is surjective for any b. Thus by Lemma 3.6 we need to show that Hom(𝟙, 𝔽(𝕋_{b})) is one dimensional for b = 2p^{l} − 2, 0 ≤ l ≤ n − 1 and zero otherwise. But since 𝟙 is a quotient of 𝕋_{2p−2}^{∗}, we have

dim Hom(𝟙, 𝔽(𝕋_{b})) ≤ dim Hom(𝕋_{2p−2}^{∗}, 𝔽(𝕋_{b})) =
= dim Hom(𝟙, 𝕋_{2p−2} ⊗ 𝔽(𝕋_{b})) = dim Hom(𝟙, 𝕋_{2p−2+pb}),

and the result follows from Lemma 3.6. □

It follows that the functor H is full which proves the theorem. □

Remark 4.38. Here is an alternative approach to the existence of the functor H. By Proposition 4.32 the category Ver_{p^{n}} for n ≥ 2 is not Frobenius exact, so by [24], Proposition 7.6, 𝔽(Q) = 0 for every projective Q ∈ Ver_{p^{n}}. This means that the monoidal functor 𝔽 ∘ F : T_{n,p} → Ver_{p^{n}} ⊠ Ver_{p} factors through T_{n−k,p} for some k > 0. But T_{n−k,p} has an abelian envelope Ver_{p^{n−k}}. Thus the functor 𝔽 ∘ F factors through a tensor functor H̃ : Ver_{p^{n−k}} → Ver_{p^{n}} ⊠ Ver_{p}. By Corollary 4.33, this functor lands in Ver_{p^{n}} (as 𝕋_{1} generates Ver_{p^{n−k}}). Recall that the Frobenius-Perron dimensions of the objects of Ver_{p^{n−k}} are contained in the ring O_{n−k,p}, see Theorem 4.5 (iv). However by Corollary 4.33 the image of the functor H̃ contains an object U ⊠𝟙 such that its Frobenius-Perron dimension generates the ring O_{n−1,p}, see the proof of Lemma 4.31. We conclude that k = 1.

In what follows we consider Ver_{p^{n−1}} as a subcategory of Ver_{p^{n}} via the functor H.

Corollary 4.39. Assume n ≥ 2. Let L be a simple object of Ver_{p^{n−1}} with projective cover 𝕋_{r} where r ∈ [p^{n−2} − 1, p^{n−1} − 2]. Then projective cover of L considered as an object of Ver_{p^{n}} is 𝕋_{2p−2+pr}.

Proof. By (4.7) we have 𝕋_{2p−2+pr} = 𝔽(𝕋_{r})⊗𝕋_{2p−2}, so H(𝕋_{r}) = 𝔽(𝕋_{r}) is a quotient of 𝕋_{2p−2+pr} as 𝕋_{2p−2} surjects onto 𝟙. □

Proposition 4.40. The functor X ↦ H(X) ⊗ 𝕋_{p−1} is an (additive) equivalence of Ver_{p^{n−1}} with a direct summand of the category Ver_{p^{n}}.

Proof. It is clear that this functor is exact and faithful. We claim that it sends projective objects to projective objects and that it is full on projective objects. The first assertion is immediate from (4.7) where we substitute a = p − 1. The second assertion is an immediate consequence of (4.6). The proposition is proved. □

4.11. The tensor product theorem for Ver_{p^{n}}. Here is a generalization of Lemma 4.30.

Proposition 4.41. Assume n ≥ 2. Let L be a simple object of Ver_{p^{n−1}} with projective cover 𝕋_{r}, r ∈ [p^{n−2} − 1, p^{n−1} − 2]. Then for any i ∈ [0, p − 1] the object 𝕋_{i} ⊗ L of Ver_{p^{n}} is simple with projective cover 𝕋_{2p−2−i+pr}. Any simple object of Ver_{p^{n}} is of this form.

Proof. We claim that there is a unique j ∈ [p^{n−1} − 1, p^{n} − 2] such that Hom(𝕋_{j}, 𝕋_{i} ⊗ L) ≠ 0 and this Hom–space is one-dimensional. But

Hom(𝕋_{j}, 𝕋_{i} ⊗ L) = Hom(𝕋_{j} ⊗ 𝕋_{i}^{∗}, L) = Hom(𝕋_{j} ⊗ 𝕋_{i}, L)

and by Corollary 4.39 this space is nonzero if and only if 𝕋_{j} ⊗𝕋_{i} contains 𝕋_{2p−2+pr} as a direct summand. The first statement now follows from Proposition 3.13. The second statement is a consequence of the count of the simple modules. □

By iterating Theorem 4.35 we get a sequence of embeddings Ver_{p} ⊂ Ver_{p^{2}} ⊂ … ⊂ Ver_{p^{n}}. For any r ≥ 2 and i ∈ [0, p − 1] let us denote by 𝕋_{i}^{[r]} the simple object of the category Ver_{p^{r}} as in Lemma 4.30 possibly considered as an object of Ver_{p^{s}} with s ≥ r. Similarly, for i ∈ [0, p − 2] let 𝕋_{i}^{[1]} denote the simple object of Ver_{p} possibly considered as an object of Ver_{p^{s}} with s ≥ 1. Then Theorem 4.35 implies

Proposition 4.42. For 0 ≤ i ≤ p − 1 and m ≥ 2, we have 𝔽(𝕋_{i}^{[m]}) = 𝕋_{i}^{[m−1]} except if m = 2 and i = p − 1, in which case 𝔽(𝕋_{i}^{[m]}) = 0.

For a sequence of base p digits a_{1}, a_{2}, …, a_{n} ∈ [0, p − 1] let

\overline{a_{1}a_{2} … a_{n}} = a_{1}p^{n−1} + … + a_{n−1}p + a_{n}

be the corresponding integer in base p. For a base p digit a let a^{∗} := p − 1 − a. By iterating Proposition 4.41 we get the following result (cf. Steinberg’s tensor product theorem [29, 3.17] and [5, Theorem 2.1 (ix)]):

Theorem 4.43. Let i_{1} ∈ [0, p − 2], i_{2}, …, i_{n} ∈ [0, p − 1], and i = \overline{i_{1}...i_{n}} ∈ [0, p^{n−1}(p − 1) − 1] the corresponding integer written in base p. Then the object L_{i} := 𝕋_{i_{1}}^{[1]}⊗𝕋_{i_{2}}^{[2]}⊗…⊗𝕋_{i_{n}}^{[n]} ∈ Ver_{p^{n}} is simple, and any simple object of Ver_{p^{n}} can be uniquely written in this form.^{4} The projective cover of L_{i} is P_{i} := P_{s} = 𝕋_{s} with

s = s(i) := p^{n−1} − 1 + \overline{i_{1}i_{2}^{∗} … i_{n}^{∗}};

in other words, L_{i} = L_{s}. Moreover, the simple objects of Ver_{p^{n}}^{+} are the L_{i} for even i.

Thus, Theorem 4.43 gives a new way of labeling the simple objects of Ver_{p^{n}}, and the transition between the two labelings is given by the formula L_{i} = L_{s(i)}. For example, for p = 2 we have i_{1} = 0, so s(i) = 2^{n} − 2 − i, so L_{i} = L_{2^{n}−2−i} for 0 ≤ i ≤ 2^{n−1} − 1. Also, L_{p^{n−1}(p−2)} = L_{p^{n}−2} is the generator of sVec ⊂ Ver_{p^{n}}.

Remark 4.44. For p = 2, the simple objects L_{1}^{[m]} were denoted in [5] by X_{m−1}, 2 ≤ m ≤ n. Thus for a subset S ∈ [1, n − 1], the object X_{S} ∈ Ver_{2^{n}} = C_{2n−2} from [5] is L_{i} where i = ∑_{k∈S} 2^{n−1−k}.

Corollary 4.45. (i) The categorical dimensions of simple objects of Ver_{p^{n}} are given by the formula

dim(L_{i}) = ∏_{k=1}^{n}(i_{k} + 1) ∈ k.

(ii) The Frobenius-Perron dimensions of simple objects of Ver_{p^{n}} are given by the formula

FPdim(L_{i}) = ∏_{k=1}^{n}[i_{k} + 1]_{q^{p^{n−k}}} .

Thus Ver_{p^{n}}^{+} is a categorification of the ring O_{n,p} with basis b_{i} = ∏_{k=1}^{n}[i_{k} + 1]_{q^{p^{n−k}}} for even i ∈ [0, p^{n−1}(p − 1) − 1].

4The object L_{i} should not be mixed up with the simple module over SL_{2}(k) with highest weight i which we also denoted by L_{i}. We hope this will not lead to confusion since the simple SL_{2}(k)-modules will not be used from this point on.

Proof. This follows since dim (T_{j}) = j + 1 and FPdim(𝕋_{j}) = [j + 1]_{q} in Ver_{p^{n}}. □

Corollary 4.46. The Frobenius functor acts on the simple objects of Ver_{p^{n}} as follows. Let 0 ≤ r ≤ p − 2 and 0 ≤ b ≤ p^{n−1} − 1. Then if b ≥ p^{n−1} − p^{n−2}, we have 𝔽(L_{rp^{n−1}+b}) = 0. Otherwise 𝔽(L_{rp^{n−1}+b}) = L_{b}^{[n−1]} ⊠ L_{r} if r is even and 𝔽(L_{rp^{n−1}+b}) = (L_{p−2}^{[1]} ⊗ L_{b}^{[n−1]}) ⊠ L_{p−2−r} if r is odd.

Proof. This follows from Proposition 4.42, Theorem 4.43, [32, Example 3.8] and the fact that 𝔽 is a monoidal functor, see [24]. □

Remark 4.47. The following remark is a generalization to odd characteristic of [5], Remark 3.14. The category Ver_{p^{n}} is filtered by Serre subcategories Ver_{p^{n}}^{r}, r ≥ 0, whose simple objects are those occurring in V^{⊗j}, j ≤ r (where we recall that V = 𝕋_{1}). These categories are not closed under tensor product, but we have partial tensor products Ver_{p^{n}}^{j} × Ver_{p^{n}}^{r−j} → Ver_{p^{n}}^{r} with associativity isomorphisms satisfying the pentagon relation. Moreover, we claim that the Frobenius functor 𝔽 maps Ver_{p^{n}}^{r} to Ver_{p^{n−1}}^{r} when n is large enough compared to r, preserving this partial tensor product. Indeed, for simple objects this follows from Corollary 4.46, and for arbitrary objects it follows since for a filtered object Y any composition factor of 𝔽(Y) is also one of 𝔽(gr(Y)) (see [24], Proposition 3.6).

Example 4.48. Let p ≤ a ≤ 2p − 2. Then by Theorem 4.43 the projective cover of 𝕋_{1}^{[n−1]} ⊗ 𝕋_{a−p}^{[n]} is 𝕋_{2p^{n−1}−2−a}. We have a nonzero morphism 𝕋_{2p^{n−1}−2−a} ⊗ 𝕋_{a} → 𝕋_{2p^{n−1}−2} → 𝟙, hence a nonzero morphism 𝕋_{2p^{n−1}−2−a} → 𝕋_{a}. Hence 𝕋_{1}^{[n−1]} ⊗ 𝕋_{a−p}^{[n]} is a simple constituent of 𝕋_{a}. Similarly, using the nonzero morphism 𝕋_{a} ⊗ 𝕋_{2p−2−a} → 𝕋_{2p−2} → 𝟙, we see that the simple object 𝕋_{2p−2−a} (see Lemma 4.30) is both a subobject and a quotient object of 𝕋_{a}. By comparing the Frobenius-Perron dimensions we deduce that the object 𝕋_{a} has composition series

𝕋_{a} = [𝕋_{2p−2−a}, 𝕋_{1}^{[n−1]} ⊗ 𝕋_{a−p}, 𝕋_{2p−2−a}],

generalizing Lemma 4.31.

Corollary 4.49. The only braided tensor functor E : Ver_{p^{n}} → Ver_{p^{n}} is isomorphic to the identity. In particular the group of braided autoequivalences of the category V er_{p^{n}} is trivial.

Proof. By Corollary 4.7, braided (= symmetric) tensor functors E : Ver_{p^{n}} → Ver_{p^{n}} correspond to objects X ∈ Ver_{p^{n}} with ∧^{2}X ≅ 𝟙, ∧^{3}X = 0 and Q_{n,p}^{+}(X) ≅ Q_{n,p}^{−}(X). Such an object X must have Frobenius-Perron dimension q +q^{−1}, in particular, it is simple, since this number is less than 2. Thus it follows from Corollary 4.45(ii) that X ≅ L_{1}. Thus there is a unique braided tensor functor E up to an isomorphism, which must therefore be isomorphic to the identity. □

4.12. Tensor product of simple objects. Let us now compute the tensor product of simple objects of Ver_{p^{n}} (in its Grothendieck ring). We start with the following result about the tensor product 𝕋_{m} ⊗ 𝕋_{r} for 0 ≤ m, r ≤ p − 1.

Proposition 4.50. The tensor product 𝕋_{m} ⊗𝕋_{r} for 1 ≤ m, r ≤ p−1 is given by the formula

𝕋_{m} ⊗ 𝕋_{r} = ⨁_{|m−r|≤k≤2(p−2)−m−r}𝕋_{k} ⊕ ⨁_{p−1≤k≤m+r}𝕋_{k},

if m + r ≥ p, and

𝕋_{m} ⊗ 𝕋_{r} = ⨁_{|m−r|≤k≤m+r}𝕋_{k}

if m + r < p, where the summation is over integers k such that m + r − k is even.

Proof. This follows immediately from Lemma 3.10. □

Corollary 4.51. The multiplication of simple objects in the Grothendieck ring Gr(Ver_{p^{n}}) is given by the following recursive rule. Let n ≥ 2, a = a′p + m, b = b′p + r, where 0 ≤ m, r ≤ p − 1 and 0 ≤ a′, b′ ≤ p^{n−2}(p − 1) − 1. Then for m + r ≥ p we have

L_{a}L_{b} =

(  ∑_{|m−r|≤k≤2(p−2)−m−r} L_{k} + ∑_{2(p−1)−m−r≤k≤p−1}(2 − δ_{k,p−1})L_{k}) (L_{a′}L_{b′})^{[n−1]}+

+ ∑_{p≤k≤m+r} L_{k−p}(V L_{a′}L_{b′})^{[n−1]}.

and for m + r < p we have

L_{a}L_{b} = ( ∑_{|m−r|≤k≤m+r} L_{k}) (L_{a′}L_{b′})^{[n−1]},

where V := 𝕋_{1} and the summation is over integers k such that m + r − k is even.

Proof. This follows from Proposition 4.50 and Example 4.48. □

This allows one to quickly compute products in Gr(Ver_{p^{n}}), since the products L_{a′}L_{b′} and V L_{a′}L_{b′} are computed in the smaller category Ver_{p^{n−1}}.

Here is a somewhat more elegant way to reformulate Corollary 4.51. Let A be a unital commutative ℤ_{+}-ring ([22], Subsection 3.1). By a unital ℤ_{+}-ring over A we will mean a ring B ⊃ A with a free A-basis b_{i}, i ∈ I (where b_{0} = 1 for some element 0 ∈ I), such that A is central in B and b_{i}b_{j} = ∑_{k} N_{ij}^{k}b_{k}, where N_{ij}^{k} are nonnegative elements of A. In this case B is a unital ℤ_{+}-ring in the sense of [22], Subsection 3.1, with basis a_{i}b_{j}, where a_{i} is the basis of A.

Corollary 4.52. For n > 1 the ring Gr(Ver_{p^{n}}) is a unital ℤ_{+}-ring over Gr(Ver_{p^{n−1}}) with basis L_{m}, 0 ≤ m ≤ p − 1, and multiplication given by

L_{m}L_{r} = ∑_{|m−r|≤k≤2(p−2)−m−r} L_{k} + ∑_{2(p−1)−m−r≤k≤p−1}(2 − δ_{k,p−1})L_{k} + ∑_{p≤k≤m+r} V L_{k−p}

for m + r ≥ p and

L_{m}L_{r} = ∑_{|m−r|≤k≤m+r} L_{k},

for m + r < p, where V = 𝕋_{1}^{[n−1]} ∈ Gr(Ver_{p^{n−1}}) and the summation is over integers k such that m + r − k is even.

Example 4.53. Since Ver_{3} = sVec, the ring Gr(Ver_{3}) is ℤ[V]/(V^{2} −1) with basis 1, V. Thus by Corollary 4.52 the ring Gr(Ver_{9}) has basis 1, L_{1}, L_{2} over Gr(Ver_{3}) with multiplication given by

L_{1}^{2} = 1 + L_{2}, L_{1}L_{2} = 2L_{1} + V, L_{2}^{2} = 2 + L_{2} + V L_{1}.

So the basis of Gr(Ver_{9}^{+}) is 1, M_{1} = V L_{1}, M_{2} = L_{2} with multiplication given by

M_{1}^{2} = 1 + M_{2}, M_{1}M_{2} = 1 + 2M_{1}, M_{2}^{2} = 1 + M_{1} + M_{2}.

This agrees with the data in Subsection 5.1.

4.13. The blocks of Ver_{p^{n}}.

Proposition 4.54. The objects 𝕋_{i} and 𝕋_{j} are in the same block if and only if the following holds:

(a) i and j have the same parity;

(b) base p expansions of i + 1 and j + 1 have the same number of zeros at the end;

(c) the last nonzero base p digits of i + 1 and j + 1 are either equal or sum up to p.

Proof. This follows from Proposition 4.17. □

It now follows from Proposition 4.54 that Ver_{p^{n}} has the following block structure.

Proposition 4.55. The category Ver_{p^{n}} has n(p − 1) blocks. More specifically, it has p − 1 blocks of size 1 formed by projective (simple) objects 𝕋_{i} with i + 1 divisible by p^{n−1}, p − 1 blocks of size p − 1 formed by projective objects 𝕋_{i} with i + 1 divisible by p^{n−2} but not p^{n−1}, p − 1 blocks of size p^{2} − p formed by projective objects 𝕋_{i} with i + 1 divisible by p^{n−3} but not p^{n−2}, etc. In particular it has p − 1 blocks of maximal size p^{n−1} − p^{n−2} formed by projective objects 𝕋_{i} with i + 1 not divisible by p; all these blocks are equivalent to each other by the translation principle, see [29].

Observe that by Proposition 4.40 the functor T ↦ St ⊗ T^{(1)} from T_{≤p^{n−1}−1} to T_{≤p^{n}−1} is fully faithful; its image is generated by the objects T_{i} such that i + 1 is divisible by p, i.e., by the objects lying in the blocks not of maximal size. Thus we get the following statement:

Proposition 4.56. The blocks of the same size in the categories Ver_{p^{n}} (where n is allowed to vary) are all equivalent to each other; in particular they have the same Cartan matrices.

Remark 4.57. Let p = 2 and let Ver_{2^{n}}^{−} ⊂ Ver_{2^{n}} be the additive subcategory generated by 𝕋_{i} with odd i. Then Ver_{2^{n}}^{+} acts on Ver_{2^{n}}^{−} by left multiplication. At the same time the category Ver_{2^{n−1}} acts on Ver_{2^{n}}^{−} by right multiplication: X, Y ↦ X ⊗ H(Y); clearly these two actions commute. It follows from the discussion above that Ver_{2^{n}}^{−} considered as a module category over Ver_{2^{n−1}} is equivalent to the regular module (i.e., to Ver_{2^{n−1}} considered as a module over itself). This implies that there exists a tensor (but not braided) functor Ver_{2^{n}}^{+} → Ver_{2^{n−1}}; thus the category Ver_{2^{n}}^{+} can be described as the representation category of some triangular Hopf algebra in Ver_{2^{n−1}}. This can be considered as an a posteriori explanation of the construction in [5].

4.14. Ver_{p^{n}} is a Serre subcategory in Ver_{p^{n+k}}.

Proposition 4.58. Let p > 2. Then

(i) Any tensor subcategory C ⊂ Ver_{p^{n+1}} containing Ver_{p^{n}} as a full subcategory coincides with Ver_{p^{n}} or with Ver_{p^{n+1}}.

(ii) The category Ver_{p^{n}} is a Serre subcategory in Ver_{p^{n+k}}.

Proof. (i) Let X = 𝕋_{p^{n−1}−1} ∈ Ver_{p^{n}}. This is a simple projective object. On the other hand, in Ver_{p^{n+1}} it is not projective and has projective cover P_{X}.

Lemma 4.59. We have [P_{X} : X] = 2.

Proof. By Corollary 4.39 we have P_{X} = 𝕋_{p^{n}+p−2}. Now [P_{X} : X] = dim Hom(P_{X}, P_{X}) = 2 by Proposition 4.17. □

Now suppose Ver_{p^{n}} ⊂ C ⊂ Ver_{p^{n+1}}. Let Q_{X} be the projective cover of X in C. Then we have a diagram of surjective morphisms P_{X} → Q_{X} → X. It is clear that Q_{X} is self-dual, so by Lemma 4.59 either Q_{X} = P_{X} or Q_{X} = X. In the first case every indecomposable projective in C, being a direct summand in Y ⊗ Q_{X} for some simple Y ∈ C, is projective in Ver_{p^{n+1}}, which implies that C = Ver_{p^{n+1}} (using the multiplication rule for simple objects from Corollary 4.52). On the other hand, if Q_{X} = X then similarly all projectives of Ver_{p^{n}} are projective in C, so we get C = Ver_{p^{n}}.

(ii) It suffices to prove the statement for k = 1, in which case it follows from (i). □

Corollary 4.60. The only tensor subcategories of Ver_{p^{n}} are Ver_{p^{m}} and Ver_{p^{m}}^{+} for m ≤ n.

Proof. For p = 2 this is proved in [5], Corollary 2.6, so let us consider the case p > 2. It suffices to show that the only tensor subcategories of Ver_{p^{n}}^{+} are Ver_{p^{m}}^{+} for m ≤ n. Let C ⊂ Ver_{p^{n}}^{+} be a tensor subcategory. Let m be the largest integer such that C contains all simple objects of Ver_{m}^{+}. Then by Corollary 4.51, C cannot contain any other simple objects. Thus C is contained in the Serre closure of Ver_{p^{m}}^{+} inside Ver_{p^{n}}^{+}. So it follows from Proposition 4.58 that C = Ver_{p^{m}}^{+}. □

4.15. Ext^{1} between simple objects. Let us describe a recursive procedure for computing Ext^{1} between simple objects of Ver_{p^{n}}. We will focus on the case p > 2 since the case p = 2 is addressed in [5], Subsection 4.5. Recall that by Theorem 4.43 the simple objects of Ver_{p^{n}} look like X ⊗ 𝕋_{i} where X ∈ Ver_{p^{n−1}} and 0 ≤ i ≤ p − 1. Also by block considerations for X, Y ∈ Ver_{p^{n−1}} we have

Ext_{Ver_{p^{n}}}^{m}(X ⊗ 𝕋_{i}, Y ⊗ 𝕋_{j}) = 0, m ≥ 0

unless i = j or i = p − 2 − j. Consider first the case i = j.

Lemma 4.61. For each m we have a canonical isomorphism Ext_{Ver_{p^{n}}}^{m}(X ⊗𝕋_{p−1}, Y ⊗𝕋_{p−1}) ≅ Ext_{Ver_{p^{n−1}}}^{m}(X, Y).

Proof. This is an immediate consequence of Proposition 4.40. □

Lemma 4.62. (i) If 0 ≤ i ≤ p − 2 then we have a canonical isomorphism

Ext_{Ver_{p^{n}}}^{m}(X ⊗ 𝕋_{i}, Y ⊗ 𝕋_{i}) ≅ Ext_{Ver_{p^{n}}}^{m}(X, Y).

(ii) If p > 2 and 0 ≤ i ≤ p − 2 then we have a canonical isomorphism

Ext_{Ver_{p^{n}}}^{1}(X ⊗ 𝕋_{i}, Y ⊗ 𝕋_{i}) ≅ Ext_{Ver_{p^{n−1}}}^{1}(X, Y).

Proof. (i) We have

Ext_{Ver_{p^{n}}}^{m}(X ⊗ 𝕋_{i}, Y ⊗ 𝕋_{i}) = Ext_{Ver_{p^{n}}}^{m}(X, Y ⊗ 𝕋_{i} ⊗ 𝕋_{i}).

Now, 𝕋_{i} ⊗𝕋_{i} can be written as a direct sum of indecomposables, and by block considerations the only direct summand which contributes to Ext^{•} is 𝟙. This implies the statement.

(ii) This follows from (i) and Proposition 4.58(ii). □

Now consider the case j = p − 2 − i. In this case we can again write

Ext_{Ver_{p^{n}}}^{m}(X ⊗ 𝕋_{i}, Y ⊗ 𝕋_{j}) = Ext_{Ver_{p^{n}}}^{m}(X, Y ⊗ 𝕋_{i} ⊗ 𝕋_{j}),

and by block considerations the only indecomposable direct summand in 𝕋_{i} ⊗ 𝕋_{j} that contributes to Ext^{•} is 𝕋_{p−2}. Thus, it suffices to compute

Ext_{Ver_{p^{n}}}^{1}(X, Y ⊗ 𝕋_{p−2}).

To this end, note that 𝕋_{1} ⊗ 𝕋_{p−1} = [𝕋_{p−2}, 𝕋_{1}^{[n−1]}, 𝕋_{p−2}] (Lemma 4.31). Thus,

Ext_{Ver_{p^{n}}}^{m}(X, Y ⊗ [𝕋_{p−2}, 𝕋_{1}^{[n−1]}, 𝕋_{p−2}]) =
= Ext_{Ver_{p^{n}}}^{m}(X, Y ⊗ 𝕋_{1} ⊗ 𝕋_{p−1}) = Ext_{Ver_{p^{n}}}^{m}(X ⊗ 𝕋_{p−1}, Y ⊗ 𝕋_{1}) = 0

by block considerations. From the long exact sequence it then follows that

Ext_{Ver_{p^{n}}}^{1}(X, Y ⊗ 𝕋_{p−2}) ≅ Hom_{Ver_{p^{n}}}(X, Y ⊗ [𝕋_{p−2}, 𝕋_{1}^{[n−1]}]) ≅ Hom_{Ver_{p^{n−1}}}(X, Y ⊗ V).

Finally, this Hom can be computed using the following proposition.

Proposition 4.63. Let X, Y ∈ Ver_{p^{n−1}} and 0 ≤ i, j ≤ p−2. Then Hom(X⊗L_{i}, Y ⊗L_{j}⊗V) = 0 unless |i − j| = 1. In the latter case, Hom(X ⊗ L_{i}, Y ⊗ L_{j} ⊗ V) = Hom(X, Y).

Proof. We have

Hom(X ⊗ L_{i}, Y ⊗ L_{j} ⊗ V) = Hom(L_{i} ⊗ L_{j} ⊗ V, Y ⊗ X^{∗}).

Now, the product L_{i} ⊗ L_{j} ⊗ V decomposes as a direct sum of 𝕋_{j} for 0 ≤ j ≤ 2p − 1. But by block considerations for every object Z ∈ Ver_{p^{n−1}} ⊂ Ver_{p^{n}}, we have Hom(𝕋_{j}, Z) = 0 unless j = 0 or j = 2p − 2, and

Hom(𝕋_{2p−2}, Z) = Hom(𝟙, Z).

This implies the statement. □

Thus, we obtain the following proposition.

Proposition 4.64. Let n ≥ 2, p > 2, a = a′p + m, b = b′p + r, where 0 ≤ m, r ≤ p − 1 and 0 ≤ a′, b′ ≤ p^{n−2}(p − 1) − 1. Then the following hold for Ext^{1} between simple objects of Ver_{p^{n}}.

(i) Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = 0 unless r = m or r = p − 2 − m.

(ii) If r = m then Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = Ext_{Ver_{p^{n−1}}}^{1}(L_{a′}, L_{b′}).

(iii) If r = p − 2 − m then Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = 0 unless the last digit of a′ is < p − 1 and b′ = a′ + 1 or the last digit of a′ is > 0 and b′ = a′ − 1. In the latter two cases, Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = k.

In other words, Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) ≠ 0 if and only if a and b differ only in two consecutive digits, of which the first ones differ by 1 and the second ones add up to p − 2, and in this case Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = k.

Remark 4.65. For p = 2 the fact that for any a, b, Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) is either 0 or k is proved in [5], Proposition 4.12.

Example 4.66. 1. If p > 2 then Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{a}) = 0 for all a.

2. Let n = 2, p > 2. Then Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = 0 unless a = kp+c−1 and b = (k+2)p−c−1 or b = kp+c−1 and a = (k +2)p−c−1 or a = kp+c−1 and b = kp−c−1 or b = kp+c−1 and a = kp − c − 1 for some 1 ≤ c ≤ p − 1. In these cases, Ext_{Ver_{p^{n}}}^{1}(L_{a}, L_{b}) = k. Note that this agrees with the data in Subsection 5.1.

4.16. The Grothendieck ring of the stable category. Given a finite tensor category C, one may consider the Grothendieck ring GrStab(C) of the stable category Stab(C), which by definition is the quotient of Gr(C) by the ideal of classes of projective objects. It is clear that as an abelian group, GrStab(C) is the cokernel of the Cartan matrix C of C. Thus if C is invertible then this ring is finite of order |det(C)|.

In this section we compute the ring GrStab(C) for C = Ver_{p^{n}} and C = Ver_{p^{n}}^{+} for n ≥ 2.^{5} Note that for p > 2 we have GrStab(Ver_{p^{n}}) = GrStab(Ver_{p^{n}}^{+})[u]/(u^{2} − 1), so it suffices to compute GrStab(Ver_{p^{n}}^{+}).

Recall that q = e^{πi/p^{n}} and [a]_{q} = \frac{q^{a}−q^{−a}}{q−q^{−1}} .

Proposition 4.67. (i) For p > 2 the ring GrStab(Ver_{p^{n}}^{+}) is isomorphic to O_{n,p}/([p^{n−1}]_{q}). (ii) The ring GrStab(Ver_{2^{n}}) is isomorphic to O_{n+1,2}/([2^{n−1}]_{q}). (iii) The ring GrStab(Ver_{2^{n}}^{+}) is isomorphic to O_{n,2}/((q + q^{−1})[2^{n−1}]_{q}).

Proof. (i) Recall that Gr(Ver_{p^{n}}^{+}) ≅ O_{n,p}. Also, since

𝕋_{1}^{⊗m} ⊗ 𝕋_{p^{n−1}−1} = 𝕋_{p^{n−1}+m−1} ⊕ ⨁_{i=0}^{m−1}b_{m}𝕋_{p^{n−1}−1+i},

the ideal of projectives is generated by T_{p^{n−1}−1}. Therefore, the corresponding ideal in O_{n,p} is generated by the Frobenius-Perron dimension of T_{p^{n−1}−1}, which equals [p^{n−1}]_{q}. This implies the statement.

(ii) The proof is the same as (i), using that Gr(Ver_{2^{n}}) ≅ O_{n+1,2}.

(iii) The proof is the same as (ii), using that Gr(Ver_{2^{n}}^{+}) ≅ O_{n,2} and replacing T_{2^{n−1}−1} by T_{2^{n−1}} = T_{1} ⊗ T_{2^{n−1}−1} (as T_{2^{n−1}−1} ∉ Ver_{2^{n}}^{+}). □

Now we would like to compute these rings more explicitly, which is now a problem in elementary number theory. First of all, note that [p^{n−1}]_{q} = \frac{q^{p^{n−1}}−q^{−p^{n−1}}}{q−q^{−1}} divides p, since already q^{p^{n−1}} − q^{−p^{n−1}} = e^{πi/p} − e^{−πi/p} divides p. Indeed, for p > 2 the norm of e^{πi/p} − e^{−πi/p} equals −p, while e^{πi/2} − e^{−πi/2} = 2i obviously divides 2.

Moreover, for p = 2 the number (q + q^{−1})[2^{n−1}]_{q} divides 2. Indeed, \frac{2}{(q+q^{−1})[2^{n−1}]_{q}} = −i\frac{q^{2}−1}{q^{2}+1}.

But the number x = \frac{q^{2}−1}{q^{2}+1} is a unit, since q^{2} = \frac{1+x}{1−x}, thus (1 + x)^{2^{n−1}} = −(1 − x)^{2^{n−1}}, i.e., P (x) := \frac{1}{2}((1 + x)^{2^{n−1}} + (1 − x)^{2^{n−1}}) = 0, while P is a monic polynomial with integer coefficients and constant term 1.

Thus, the above rings are 𝔽_{p}-algebras. So in Proposition 4.67(i) the ring R in question is the quotient of 𝔽_{p}[q^{2} + q^{−2}] by the relations

(q − q^{−1})^{p^{n−1}−1} = 0, q^{−(p−1)p^{n−1}}(1 + q^{2p^{n−1}} + ... + q^{2(p−1)p^{n−1}}) = 0.

5We thank R. Rouquier for asking this question.

The second relation can be rewritten in characteristic p as (q − q^{−1})^{(p−1)p^{n−1}} = 0, hence it follows from the first relation. Thus

R = 𝔽_{p}[z]/z^{\frac{p^{n−1}−1}{2}}

with z := (q − q^{−1})^{2}.

In Proposition 4.67(ii) we similarly get

R = 𝔽_{p}[w]/(w^{2^{n−1}−1}, w^{2^{n−1}}) = 𝔽_{p}[w]/w^{2^{n−1}−1},

where w = q + q^{−1}.

Finally, in Proposition 4.67(iii) we get

R = 𝔽_{p}[z]/(z^{2^{n−1}}, z^{2^{n−2}}) = 𝔽_{p}[z]/z^{2^{n−2}}

with z := (q − q^{−1})^{2} = q^{2} + q^{−2}.

Thus we obtain

Proposition 4.68. (i) For p > 2 the ring GrStab(Ver_{p^{n}}^{+}) is isomorphic to 𝔽_{p}[z]/z^{\frac{p^{n−1}−1}{2}} and

the ring GrStab(Ver_{p^{n}}) is isomorphic to 𝔽_{p}[z, g]/(z^{\frac{p^{n−1}−1}{2}} , g^{2} − 1).

(ii) The ring GrStab(Ver_{2^{n}}) is isomorphic to 𝔽_{p}[z]/z^{2^{n−1}−1}.

(iii) The ring GrStab(Ver_{2^{n}}^{+}) is isomorphic to 𝔽_{p}[z]/z^{2^{n−2}}.

Proposition 4.69. The determinant of the Cartan matrix of any block of Ver_{p^{n}} of size p^{m−1}(p − 1) equals p^{p^{m−1}}.

Proof. By Proposition 4.56, all blocks of the same size have the same Cartan matrices. So the determinant in question depends only on m; let us denote it by D_{m}. Then Proposition 4.68 and Proposition 4.55 imply that

(∏_{m=1}^{n−1} D_{m})^{p−1} = p^{p^{n−1}−1}.

Thus D_{m}^{p−1} = p^{p^{m−1}(p−1)}. Also since C_{n,p} is positive definite (Proposition 4.13(iv)), D_{m} is a positive integer. Thus D_{m} = p^{p^{m−1}}, as claimed. □

4.17. Incompressibility. Let D be a (not necessarily braided) tensor category.

Theorem 4.70. (i) Assume p > 2. Let E : Ver_{p^{n}}^{+} → D be a (not necessarily braided) tensor functor. Then E is an embedding (i.e., it is fully faithful).

(ii) Let D be braided and E : Ver_{p^{n}} → D be a braided tensor functor. Then E is an embedding.

Remark 4.71. (i) In the language of [5, Definition 4.3] Theorem 4.70 says that the category Ver_{p^{n}}^{+} is incompressible as a tensor category, and Ver_{p^{n}} is incompressible as a symmetric (or braided) category.

(ii) Theorem 4.70(i) is not true if Ver_{p^{n}}^{+} is replaced by Ver_{p^{n}}: since there exists a tensor functor sVec → Vec, we have a non-injective tensor functor Ver_{p^{n}} = Ver_{p^{n}}^{+} ⊠ sVec → Ver_{p^{n}}^{+}, see Corollary 4.12.

(iii) See [5, Theorem 4.4] for the case p = 2. Note that in this case Ver_{2^{n}} is incompressible as a tensor category while Ver_{2^{n}}^{+} is not. However, both are incompressible as symmetric (or braided) categories.

Proof. (ii) easily follows from (i), so it suffices to prove (i). By replacing D with the image of E we can assume that D is finite and E is surjective. Moreover by Corollary 4.12 we can extend E to a tensor functor R : Ver_{p^{n}} → D ⊠ sVec.

Lemma 4.72. (i) Assume i ∈ [0, p − 1]. Then R(𝕋_{i}) is simple and for i > 0 it is not isomorphic to an object from R(Ver_{p^{n−1}}).

(ii) Let a ∈ [p − 1, 2p − 2] and let L be a simple object from R(Ver_{p^{n−1}}) ⊂ D. Assume Hom(R(𝕋_{a}), L) ≠ 0. Then a = 2p − 2, L = 𝟙 and Hom(R(𝕋_{a}), L) is one-dimensional.

Proof. (i) We employ induction in i. The base case i = 0 is clear. Assume R(𝕋_{i+1}) is not simple. Thus it has a nontrivial subobject N; clearly either FPdim(N) ≤ \frac{1}{2} FPdim(𝕋_{i+1}) or FPdim(R(𝕋_{i+1})/N) ≤ \frac{1}{2} FPdim(𝕋_{i+1}); since R(𝕋_{i+1}) is self-dual, in either case we will have a quotient M with FPdim(M) ≤ \frac{1}{2} FPdim(𝕋_{i+1}). By (3.2) we have a nonzero morphism R(𝕋_{1}) ⊗ R(𝕋_{i}) → M, so we have a nonzero morphism R(𝕋_{i}) → R(𝕋_{1}) ⊗ M. This must be an embedding as R(𝕋_{i}) is simple by the induction assumption. We have

FPdim(R(𝕋_{1}) ⊗ M/R(𝕋_{i})) =

= [2]_{q} FPdim(M) − [i + 1]_{q} ≤ \frac{[2]_{q}[i + 2]_{q}}{2} − [i + 1]_{q} = \frac{[i + 3]_{q} − [i + 1]_{q}}{2} < 1,

so it must vanish and FPdim(M) = \frac{[i+1]_{q}}{[2]_{q}}. But we claim that the number α := \frac{[i+1]_{q}}{[2]_{q}} =

\frac{q^{m+2}−q^{−m−2}}{q^{2}−q^{−2}} cannot be the Frobenius-Perron dimension of any object. Indeed, let g be the Galois group element such that g(q^{4}) = q^{2} (it exists since the order of q^{2} is a power of p). Then g(q^{2m+4}) = (−1)^{m}q^{m+2}, so we have

|g(α)| = |\frac{(−1)^{m}q^{m+2} − 1}{q^{2} − 1}| = |α| · |\frac{q^{2} + 1}{(−1)^{m}q^{m+2} + 1}| > |α|,

as desired. Thus R(𝕋_{i+1}) must be simple and (i) is proved.

(ii) We already know that the statement holds for a = p − 1 by (i). For a ∈ [p, 2p − 2] the object 𝕋_{a} has 3 simple components: 𝕋_{2p−2−a} with multiplicity 2 and 𝕋_{1}^{[n−1]} ⊗ 𝕋_{a−p}, see Example 4.48. If L ≠ 𝟙 or a ≠ 2p − 2 then Hom(R(𝕋_{2p−2−a}), L) = 0, so we must have Hom(R(𝕋_{1}^{[n−1]} ⊗ 𝕋_{a−p}), L) ≠ 0 which implies Hom(R(𝕋_{a−p}), R(𝕋_{1}^{[n−1]}) ⊗ L) ≠ 0 which is impossible for a ≠ p by (i) as R(𝕋_{1}^{[n−1]}) ⊗ L ∈ R(Ver_{p^{n−1}}).

It remains to show that Hom(R(𝕋_{p}), R(𝕋_{1}^{[n−1]})) = 0 and dim Hom(R(𝕋_{2p−2}), 𝟙) = 1. For the former we observe that 𝕋_{p} = 𝕋_{1} ⊗ 𝕋_{p−1} by (3.3), so Hom(R(𝕋_{p}), R(𝕋_{1}^{[n−1]})) ≠ 0 would imply that R(𝕋_{p−1}) ⊂ R(𝕋_{1} ⊗ 𝕋_{1}^{[n−1]}). This is impossible for p ≥ 5 since

FPdim(𝕋_{p−1}) > FPdim(𝕋_{1} ⊗ 𝕋_{1}^{[n−1]});

it is also impossible for p = 3 since 0 < FPdim(𝕋_{1} ⊗ 𝕋_{1}^{[n−1]}) − FPdim(𝕋_{p−1}) < 1 in this case. Finally, Hom(R(𝕋_{2p−2}), 𝟙) is a direct summand of Hom(R(𝕋_{p−1}) ⊗ R(𝕋_{p−1})^{∗}, 𝟙) = Hom(R(𝕋_{p−1}), R(𝕋_{p−1})), so it is one-dimensional by (i). □

Lemma 4.72(i) implies that the restriction of R to Ver_{p} ⊂ Ver_{p^{n}} is an embedding. Thus Theorem 4.70 is implied by the following result.

Proposition 4.73. Let R : Ver_{p^{n}} → D be a surjective tensor functor such that its restriction to Ver_{p} ⊂ Ver_{p^{n}} is an embedding. Then R is an equivalence.

Proof. We proceed by induction in n. The base case n = 1 is immediate. By applying the inductive assumption to the image of the restriction of R to Ver_{p^{n−1}} we can assume that R restricted to Ver_{p^{n−1}} is an embedding.

It is known that the functor R sends projective objects to projective objects, see [22, Theorem 6.1.16]. To prove the proposition it remains to show that R is fully faithful on projectives. Note that it is faithful automatically by [22, Remark 4.3.10], so let us show that it is full on projectives. Since Hom(Q_{1}, Q_{2}) = Hom(Q_{1} ⊗ Q_{2}^{∗}, 𝟙), it is sufficient to show that for any projective P ∈ Ver_{p^{n}} the map Hom(P, 𝟙) → Hom(R(P), 𝟙) is surjective. We can assume that P is indecomposable. Thus it is enough to show that for p^{n−1} − 1 ≤ r ≤ p^{n} − 2 we have Hom(R(𝕋_{r}), 𝟙) = 0 except for the case r = 2p^{n−1} − 2 when Hom(R(𝕋_{r}), 𝟙) should be one-dimensional, see Lemma 3.6. Let us write r = a + pb where p − 1 ≤ 2p − 2 and p^{n−2} − 1 ≤ b ≤ p^{n−1} − 2. By (4.7) we have Hom(R(𝕋_{r}), 𝟙) = Hom(R(𝕋_{a}), R(𝔽(𝕋_{b}))^{∗}) where R(𝔽(𝕋_{b})) ∈ R(Ver_{p^{n−1}}) by definition of the functor H, see Theorem 4.35. Thus by Lemma 4.72 Hom(R(𝕋_{a}), R(𝔽(𝕋_{b}))^{∗}) ≠ 0 implies that a = 2p−2 and any nonzero morphism factorizes through 𝟙. Using the inductive assumption we see that b = 2p^{n−2} − 2 and the nonzero Hom space is one-dimensional. Thus the functor R is full on projectives, which proves Proposition 4.73. □

Theorem 4.70 is proved. □

5. Examples

5.1. The category Ver_{p^{2}}. The category Ver_{p}, as already mentioned, is semisimple. The simple objects are L_{0}, …, L_{p−2}, where L_{0} = 𝟙 is the tensor identity.^{6} The tensor products

are determined by the formula L_{1} ⊗ L_{m} = \begin{cases}L_{1} & m = 0 \\ L_{m−1} ⊕ L_{m+1} & 0 < m < p − 2. \\ L_{p−3} & m = p − 2 \end{cases}

The Frobenius-Perron dimension of L_{m} is \frac{sin(π(m+1)/p)}{sin(π/p)}.

The category Ver_{p^{2}} is not semisimple, but has finite representation type. There are p − 1 semisimple blocks, corresponding to the projective simple modules L_{p−1}, L_{2p−1}, …, L_{p^{2}−1}, and p − 1 non-semisimple blocks, each of whose basic algebra is a Brauer tree algebra with inertial index p − 1, so that there is no exceptional vertex. The tree is a straight line with p vertices. Thus we obtain

Proposition 5.1. The non-semisimple blocks of Ver_{p^{2}} are equivalent to the non-semisimple block of the group algebra of the symmetric group of degree p.

These results can be obtained from Proposition 3.1 and the results of Subsection 3.5. Here is the structure of the tilting modules for p = 3 and p = 5, with n = 2. The ones

we need to use are T_{m} with p^{n−1} − 1 ≤ m ≤ p^{n} − 2. These are the ones below the double line. In these diagrams, we denote by [m] the simple module with highest weight m. This information can be read off from the paper of Doty and Henke [18].

6Note that this labeling differs by a shift from the one in some other papers, for example [24], where the simple objects of Ver_{p} are denoted L_{1} = 𝟙, L_{2}, ..., L_{p−1}.

p = 3 : \begin{aligned}& T_{0} = [0] & T_{1} = [1] \\ T_{2} = [2] & T_{3} = [1] [3] [1] & T_{4} = [0] [4] [0] \\ T_{5} = [5] & T_{6} = [4] [0] [6] [4] & T_{7} = [3] [1] [7] [3] \end{aligned}

p = 5 : \begin{aligned}& T_{0} = [0] & T_{1} = [1] & T_{2} = [2] & T_{3} = [3] \\ T_{4} = [4] & T_{5} = [3] [5] [3] & T_{6} = [2] [6] [2] & T_{7} = [1] [7] [1] & T_{8} = [0] [8] [0] \\ T_{9} = [9] & T_{10} = [8] [0] [10] [8] & T_{11} = [7] [1] [11] [7] & T_{12} = [6] [2] [12] [6] & T_{13} = [5] [3] [13] [5] \\ T_{14} = [14] & T_{15} = [13] [5] [15] [13] & T_{16} = [12] [6] [16] [12] & T_{17} = [11] [7] [17] [11] & T_{18} = [10] [8] [18] [10] \\ T_{19} = [19] & T_{20} = [18] [10] [20] [18] & T_{21} = [17] [11] [21] [17] & T_{22} = [16] [12] [22] [16] & T_{23} = [15] [13] [23] [15] \end{aligned}

In general if p is odd, for p−1 ≤ ap+b−1 ≤ p^{2} −2, with 1 ≤ a ≤ p−1 and 0 ≤ b ≤ p−1, the structure is

T_{ap+b−1} = \begin{matrix}[ap − b − 1] \\ [(a − 2)p + b − 1] [ap + b − 1] \\ [ap − b − 1] \end{matrix}

unless b = 0, in which case T_{ap+b−1} = [ap + b − 1]. Also, if a = 1 then (a − 2)p + b − 1 < 0, and we omit this composition factor so that we are left with a uniserial module.

The simple modules for A = End_{SL_{2}(k)} (⨁_{m=p−1}^{p^{2}−2}T_{m}) correspond to the indecomposable tilting modules T_{m} with p − 1 ≤ m ≤ p^{2} − 2. We label each simple module with the isomorphism type of the top composition factor of the corresponding projective object 𝕋_{m} := F (T_{m}). So the simple corresponding to 𝕋_{ap+b−1} is labelled L_{ap−b−1}. This has the effect that the simples are L_{i} with 0 ≤ i ≤ p^{2} − p − 1 (this agrees with the labeling in Subsection 4.11). The structure of their projective covers is as follows. If i = ap − 1 then L_{i} is a projective simple in a block of its own. The block containing L_{p−b−1} with 1 ≤ b ≤ p − 1 also contains the modules L_{(2a+1)p±b−1}, which are the ones whose highest weights are in the same orbit of the affine Weyl group under the dot action. The Brauer tree is as follows:

• ^{L_{p−b−1}} • ^{L_{p+b−1}} • ^{L_{3p−b−1}} •  •^{L_{(p−2)p−b−1}} •^{L_{(p−2)p+b−1}} •

53

Thus the structure of the projective indecomposables is:

\begin{matrix}L_{p−b−1} \\ L_{p+b−1} \\ L_{p−b−1} \end{matrix}  \begin{matrix}L_{p+b−1} \\ L_{p−b−1} L_{3p−b−1} \\ L_{p+b−1} \end{matrix}  \begin{matrix}L_{3p−b−1} \\ L_{p+b−1} L_{3p+b−1} \\ L_{3p−b−1} \end{matrix} ⋯ \begin{matrix}L_{(p−2)p−b−1} \\ L_{(p−4)p+b−1} L_{(p−2)p+b−1} \\ L_{(p−2)p−b−1} \end{matrix}  \begin{matrix}L_{(p−2)p+b−1} \\ L_{(p−2)p−b−1} \\ L_{(p−2)p+b−1} \end{matrix}

Projective resolutions in this category are formed in the usual way by walking around the Brauer tree, see Green [28]. In particular, we have

Proposition 5.2. If p is odd then the algebra Ext_{Ver_{p^{2}}}^{•}(𝟙, 𝟙) is isomorphic to k[x, ξ] where x is an even generator of degree 2p − 2 and ξ is an odd generator of degree 2p − 3 (thus ξ^{2} = 0). In particular, the Hilbert series of this algebra is \frac{1+t^{2p−3}}{1−t^{2p−2}} .

For p = 3, the tensor products in Ver_{p^{2}} are given by the following table, in which P_{n} denotes the projective cover of the simple module L_{n}.

\begin{matrix}L_{0} & L_{1} & L_{2} & L_{3} & L_{4} & L_{5} \\ L_{1} & L_{0} ⊕ L_{2} & P_{3} & L_{4} & L_{3} ⊕ L_{5} & P_{4} \\ L_{2} & P_{3} & L_{2} ⊕ P_{0} & L_{5} & P_{4} & L_{5} ⊕ P_{3} \\ L_{3} & L_{4} & L_{5} & L_{0} & L_{1} & L_{2} \\ L_{4} & L_{3} ⊕ L_{5} & P_{4} & L_{1} & L_{0} ⊕ L_{2} & P_{3} \\ L_{5} & P_{4} & L_{5} ⊕ P_{3} & L_{2} & P_{3} & L_{2} ⊕ P_{0} \end{matrix}

Note that L_{3} satisfies Λ^{2}(L_{3}) = L_{0} and S^{2}(L_{3}) = 0, so it generates sVec ⊂ Ver_{9}, and tensoring with L_{3} acts as a parity change (recall that in general the generator of sVec ⊂ Ver_{p^{n}} is L_{p^{n−1}(p−2)}).

For p = 5, we only list the tensor products of simples in Ver_{p^{2}}^{+}, together with their tensor products with the parity change module L_{15}, which satisfies Λ^{2}(L_{15}) = L_{0}, S^{2}(L_{15}) = 0.

\begin{matrix}L_{0} & L_{2} & L_{4} & L_{6} & L_{8} & L_{10} & L_{12} & L_{14} & L_{16} & L_{18} \\ L_{2} & \begin{matrix}L_{0} ⊕ L_{2} \\ ⊕ L_{4} \end{matrix} & L_{4} ⊕ P_{2} & L_{6} ⊕ L_{8} & L_{6} ⊕ P_{8} & L_{12} & \begin{matrix}L_{10} ⊕ L_{12} \\ ⊕ L_{14} \end{matrix} & L_{14} ⊕ P_{12} & L_{16} ⊕ L_{18} & L_{16} ⊕ P_{18} \\ L_{4} & L_{4} ⊕ P_{2} & \begin{matrix}L_{4} ⊕ P_{0} \\ ⊕ P_{2} \end{matrix} & P_{8} & P_{6} ⊕ P_{8} & L_{14} & L_{14} ⊕ P_{12} & \begin{matrix}L_{14} ⊕ P_{10} \\ ⊕ P_{12} \end{matrix} & P_{18} & P_{16} ⊕ P_{18} \\ L_{6} & L_{6} ⊕ L_{8} & P_{8} & \begin{matrix}L_{0} ⊕ L_{2} ⊕ \\ L_{10} ⊕ L_{12} \end{matrix} & \begin{matrix}L_{2} ⊕ L_{4} ⊕ \\ L_{12} ⊕ L_{14} \end{matrix} & L_{6} ⊕ L_{16} & \begin{matrix}L_{6} ⊕ L_{8} ⊕ \\ L_{16} ⊕ L_{18} \end{matrix} & P_{8} ⊕ P_{18} & L_{10} ⊕ L_{12} & L_{12} ⊕ L_{14} \\ L_{8} & L_{6} ⊕ P_{8} & P_{6} ⊕ P_{8} & \begin{matrix}L_{2} ⊕ L_{4} ⊕ \\ L_{12} ⊕ L_{14} \end{matrix} & \begin{matrix}L_{0} ⊕ L_{4} ⊕ \\ L_{10} ⊕ L_{14} ⊕ \\ P_{2} ⊕ P_{12} \end{matrix} & L_{8} ⊕ L_{18} & \begin{matrix}L_{6} ⊕ L_{10} ⊕ \\ P_{8} ⊕ P_{18} \end{matrix} & \begin{matrix}P_{6} ⊕ P_{8} ⊕ \\ P_{16} ⊕ P_{18} \end{matrix} & L_{12} ⊕ L_{14} & \begin{matrix}L_{10} ⊕ L_{14} \\ ⊕ P_{12} \end{matrix} \\ L_{10} & L_{12} & L_{14} & L_{6} ⊕ L_{16} & L_{8} ⊕ L_{18} & L_{0} ⊕ L_{10} & L_{2} ⊕ L_{12} & L_{4} ⊕ L_{14} & L_{6} & L_{8} \\ L_{12} & \begin{matrix}L_{10} ⊕ L_{12} \\ ⊕ L_{14} \end{matrix} & L_{14} ⊕ P_{12} & \begin{matrix}L_{6} ⊕ L_{8} ⊕ \\ L_{16} ⊕ L_{18} \end{matrix} & \begin{matrix}L_{6} ⊕ L_{10} ⊕ \\ P_{8} ⊕ P_{18} \end{matrix} & L_{2} ⊕ L_{12} & \begin{matrix}L_{0} ⊕ L_{2} ⊕ \\ L_{4} ⊕ L_{10} ⊕ \\ L_{12} ⊕ L_{14} \end{matrix} & \begin{matrix}L_{4} ⊕ L_{14} ⊕ \\ P_{2} ⊕ P_{12} \end{matrix} & L_{6} ⊕ L_{8} & L_{6} ⊕ P_{8} \\ L_{14} & L_{14} ⊕ P_{12} & \begin{matrix}L_{14} ⊕ P_{10} \\ ⊕ P_{12} \end{matrix} & P_{8} ⊕ P_{18} & \begin{matrix}P_{6} ⊕ P_{8} ⊕ \\ P_{16} ⊕ P_{18} \end{matrix} & L_{4} ⊕ L_{14} & \begin{matrix}L_{4} ⊕ L_{14} ⊕ \\ P_{2} ⊕ P_{12} \end{matrix} & \begin{matrix}L_{4} ⊕ L_{14} ⊕ \\ P_{0} ⊕ P_{2} ⊕ \\ P_{10} ⊕ P_{12} \end{matrix} & P_{8} & P_{6} ⊕ P_{8} \\ L_{16} & L_{16} ⊕ L_{18} & P_{18} & L_{10} ⊕ L_{12} & L_{12} ⊕ L_{14} & L_{6} & L_{6} ⊕ L_{8} & P_{8} & L_{0} ⊕ L_{2} & L_{2} ⊕ L_{4} \\ L_{18} & L_{16} ⊕ P_{18} & P_{16} ⊕ P_{18} & L_{12} ⊕ L_{14} & \begin{matrix}L_{10} ⊕ L_{14} \\ ⊕ P_{12} \end{matrix} & L_{8} & L_{6} ⊕ P_{8} & P_{6} ⊕ P_{8} & L_{2} ⊕ L_{4} & \begin{matrix}L_{0} ⊕ L_{4} \\ ⊕ P_{2} \end{matrix} \\ L_{15} & L_{17} & L_{19} & L_{11} & L_{13} & L_{5} & L_{7} & L_{9} & L_{1} & L_{3} \end{matrix}

5.2. The category Ver_{p^{3}}^{+} in the case p = 3. We now describe the category Ver_{p^{3}}^{+} in the case p = 3. The relevant even indexed tilting modules have the following Loewy series.

\begin{aligned}T_{8} = [8] & T_{10} = [6] [4] [10] [4] [6] & T_{12} = [4] [10] [0] [6] [4] [12] [4] [6] [0] [10] [4] & T_{14} = [2] [14] [2] & T_{16} = [0] [4] [12] [0] [10] [16] [0] [12] [4] [0] \\ & T_{18} = [16] [12] [0] [18] [12] [16] & T_{20} = [14] [2] [20] [14] & T_{22} = [12] [0] [18] [10] [16] [4] [12] [22] [12] [0] [18] [10] [16] [12] & T_{24} = [10] [4] [12] [22] [0] [6] [10] [18] [24] [10] [4] [12] [22] [10] \end{aligned}

Again we label the simples for A by the top composition factors of the tilting modules. So here is a table of the correspondence:

\begin{matrix}L_{0} & L_{2} & L_{4} & L_{6} & L_{8} & L_{10} & L_{12} & L_{14} & L_{16} \\ T_{16} & T_{14} & T_{12} & T_{10} & T_{8} & T_{24} & T_{22} & T_{20} & T_{18} \end{matrix}

These fall into three blocks. One is a semisimple block containing the projective simple L_{8}, one is a block of finite representation type containing L_{2} and L_{14}, and the remaining six simples, L_{0}, L_{4}, L_{6}, L_{10}, L_{12}, L_{16} lie in a single block of wild representation type. The Cartan matrix is

L_{0} L_{4} L_{6} L_{10} L_{12} L_{16} L_{2} L_{14} L_{8}
L_{0} 4 2 0  1  2  1
L_{4} 2 4 2  2  1  0
L_{6} 0 2 2  1  0  0
L_{10} 1 2 1  4  2  0
L_{12} 2 1 0  2  4  2
L_{16} 1 0 0  0  2  2
L_{2}  2  1
L_{14}  1  2
L_{8}  1

This can be easily obtained from the results of Section 4.

The structure of the projective A-modules (obtained using a computer calculation) is as follows (where the diagrams for the non-uniserial modules are in the sense of Alperin [1] or Benson and Carlson [4]):

[figure: L_{0}; L_{4}; L_{12}; L_{0}; L_{10}; L_{16}; L_{0}; L_{12}; L_{4}; L_{0}; \begin{matrix}L_{2} \\ L_{14} \\ L_{2} \end{matrix}; L_{4}; L_{6}; L_{0}; L_{10}; L_{4}; L_{12}; L_{4}; L_{10}; L_{0}; L_{6}; L_{4}; \begin{matrix}L_{6} \\ L_{4} \\ L_{10} \\ L_{4} \\ L_{6} \end{matrix}; L_{8}]

[figure: L_{10}; L_{4}; L_{12}; L_{10}; L_{6}; L_{0}; L_{10}; L_{12}; L_{4}; L_{10}; L_{12}; L_{16}; L_{10}; L_{0}; L_{12}; L_{4}; L_{12}; L_{0}; L_{10}; L_{16}; L_{12}; \begin{matrix}L_{14} \\ L_{2} \\ L_{14} \end{matrix}; \begin{matrix}L_{16} \\ L_{12} \\ L_{0} \\ L_{12} \\ L_{16} \end{matrix}]

\begin{matrix}L_{0} & L_{2} & L_{4} & L_{6} & L_{8} & L_{10} & L_{12} & L_{14} & L_{16} \\ L_{2} & L_{2} ⊕ \begin{matrix}L_{0} \\ L_{4} \\ L_{0} \end{matrix} & \begin{matrix}L_{4} \\ L_{0} L_{6} \\ L_{4} \end{matrix} & L_{8} & P_{6} ⊕ L_{8} & \begin{matrix}L_{10} \\ L_{12} \\ L_{10} \end{matrix} & L_{14} & L_{14} ⊕ \begin{matrix}L_{12} \\ L_{10} L_{16} \\ L_{12} \end{matrix} & P_{16} \\ L_{4} & \begin{matrix}L_{4} \\ L_{0} L_{6} \\ L_{4} \end{matrix} & \begin{matrix}L_{0} ⊕ L_{2} \\ ⊕ L_{6} ⊕ L_{8} \end{matrix} & \begin{matrix}L_{4} \\ L_{10} \\ L_{4} \end{matrix} & P_{4} & L_{12} ⊕ L_{14} & L_{10} ⊕ L_{16} & P_{16} ⊕ \begin{matrix}L_{10} \\ L_{12} \\ L_{10} \end{matrix} & P_{14} ⊕ \begin{matrix}L_{12} \\ L_{0} \\ L_{12} \end{matrix} \\ L_{6} & L_{8} & \begin{matrix}L_{4} \\ L_{10} \\ L_{4} \end{matrix} & L_{6} ⊕ \begin{matrix}L_{0} \\ L_{12} \\ L_{0} \end{matrix} & P_{2} ⊕ L_{8} & L_{16} & \begin{matrix}L_{12} \\ L_{0} \\ L_{12} \end{matrix} & P_{14} & L_{16} ⊕ \begin{matrix}L_{10} \\ L_{4} \\ L_{10} \end{matrix} \\ L_{8} & P_{6} ⊕ L_{8} & P_{4} & P_{2} ⊕ L_{8} & \begin{matrix}P_{0} ⊕ P_{2} \\ ⊕ P_{6} ⊕ L_{8} \end{matrix} & P_{16} & P_{14} & P_{12} ⊕ P_{14} & P_{10} ⊕ P_{16} \\ L_{10} & \begin{matrix}L_{10} \\ L_{12} \\ L_{10} \end{matrix} & L_{12} ⊕ L_{14} & L_{16} & P_{16} & L_{0} ⊕ L_{2} & L_{4} & \begin{matrix}L_{4} \\ L_{0} L_{6} \\ L_{4} \end{matrix} & L_{6} ⊕ L_{8} \\ L_{12} & L_{14} & L_{10} ⊕ L_{16} & \begin{matrix}L_{12} \\ L_{0} \\ L_{12} \end{matrix} & P_{14} & L_{4} & L_{0} ⊕ L_{6} & L_{2} ⊕ L_{8} & \begin{matrix}L_{4} \\ L_{10} \\ L_{4} \end{matrix} \\ L_{14} & L_{14} ⊕ \begin{matrix}L_{12} \\ L_{10} L_{16} \\ L_{12} \end{matrix} & P_{16} ⊕ \begin{matrix}L_{10} \\ L_{12} \\ L_{10} \end{matrix} & P_{14} & P_{12} ⊕ P_{14} & \begin{matrix}L_{4} \\ L_{0} L_{6} \\ L_{4} \end{matrix} & L_{2} ⊕ L_{8} & L_{2} ⊕ L_{8} ⊕ P_{6} ⊕ \begin{matrix}L_{0} \\ L_{4} \\ L_{0} \end{matrix} & P_{4} \\ L_{16} & P_{16} & P_{14} ⊕ \begin{matrix}L_{12} \\ L_{0} \\ L_{12} \end{matrix} & L_{16} ⊕ \begin{matrix}L_{10} \\ L_{4} \\ L_{10} \end{matrix} & P_{10} ⊕ P_{16} & L_{6} ⊕ L_{8} & \begin{matrix}L_{4} \\ L_{10} \\ L_{4} \end{matrix} & P_{4} & L_{16} ⊕ P_{10} ⊕ P_{16} ⊕ \begin{matrix}L_{10} \\ L_{4} \\ L_{10} \end{matrix} \end{matrix}

Table 1. Tensor products in Ver_{3^{3}}^{+}

For the wild block, the basic algebra is given by the following quiver and relations:

\xymatrix{ &  & L_{0} \ar[dl]^{a} \ar[dr]^{β} \\ L_{6} \ar[r]^{e} & L_{4} \ar[l]^{ε} \ar[ur]^{α} \ar[dr]^{d} &  & L_{12} \ar[ul]^{b} \ar[dl]^{γ} \ar[r]^{ϕ} & L_{16} \ar[l]^{f} \\  &  & L_{10} \ar[ul]^{δ} \ar[ur]^{c}}

Relations: αe = 0, γf = 0, εe = 0, εa = 0, ϕc = 0, ϕf = 0, dδd = 0, δdδ = 0, bβb = 0, βbβ = 0, αδ = bc, βα = cd, γβ = da, δγ = ab, eε = aα, fϕ = (1 + βb)cγ.

Note that it follows from this presentation that βbfϕ = βbcγ + βbβbcγ = βbcγ, so fϕ = cγ + βbfϕ. So the last relation can be replaced by cγ = (1 − βb)fϕ. Moreover,

βbcγ = βαδγ = cdab = cγβb

so cγ and hence also fϕ commute with βb.

Table 1 describes the tensor products of the simples in Ver_{3^{3}}^{+}.

References

[1] J. L. Alperin, Diagrams for modules, J. Pure & Applied Algebra 16 (1980), 111–119.

[2] H. H. Andersen, C. Stroppel, D. Tubbenhauer, Cellular structures using U_{q}−tilting modules. Pacific J. Math. 292 (2018), no. 1, 21-59.

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Institute of Mathematics, University of Aberdeen, Fraser Noble Building, King’s College, Aberdeen AB24 3UE, Scotland, UK.

Email address: d.j.benson@abdn.ac.uk

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Email address: etingof@math.mit.edu

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia

Email address: vostrik@uoregon.edu