THE PRIME SPECTRA OF RELATIVE STABLE MODULE CATEGORIES

SHAWN BALAND, ALEXANDRU CHIRVASITU, AND GREG STEVENSON

To Dave Benson on the occasion of his 60th birthday

Abstract. For a finite group G and an arbitrary commutative ring R, Benson, Iyengar and Krause have defined a Frobenius exact structure on the category of finitely generated RG-modules by letting the exact sequences be those that split on restriction to the trivial subgroup. The corresponding stable category has a tensor triangulated structure. In this paper we examine the case where the coefficient ring R is ℤ/p^{n}, showing that the prime ideal spectrum (in the sense of Balmer) of the relative stable category of RG is a disjoint union of n copies of that for 𝔽_{p}G.

Contents

1. Introduction 1 2. Notation and preliminary calculations 4 3. Localisation sequences and splitting of spectra 5 4. The tensor ideal thick_{D_{n}}^{⊗}(X_{n}) 6 5. A monoidal equivalence 9 6. An example: cyclic groups of prime order 11 References 15

1. Introduction

Let G be a finite group and k a field whose characteristic divides the order of G. One of the main goals in modular representation theory is to try to understand the stable module category stmod kG of kG. The objects in stmod kG are the same as those in the category mod kG of finitely generated kG-modules. If M and N are finitely generated kG-modules, then the morphisms from M to N in stmod kG are elements of the quotient

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N),

where PHom_{kG}(M, N) denotes the set of kG-module homomorphisms M → N that factor through some projective kG-module. In this case mod kG is a Frobenius category, and so stmod kG is triangulated. Recall that the suspension of an object M in stmod kG is defined to be the cosyzygy Ω^{−1}M, i.e., the cokernel of the inclusion of M into its injective hull. Distinguished triangles

The first and third authors are grateful to Universität Bielefeld and were partially supported by CRC 701 during a portion of the period in which this research was conducted. The third author was partially supported by a fellowship from the Alexander von Humboldt Foundation.

\xymatrix{M′ \ar[r] & M \ar[r] & M′′ \ar[r] & Ω^{−1}M′}

in stmod kG are, by definition, induced by short exact sequences

\xymatrix{0 \ar[r] & M′ \ar[r] & M \ar[r] & M′′ \ar[r] & 0}

in mod kG. Further details may be found in [6], for example. Moreover, for every object N in stmod kG, the functor − ⊗_{k} N takes exact triangles to exact triangles. The symmetric monoidal structure induced by tensoring over k therefore gives stmod kG the structure of a tensor triangulated category.

Now suppose that we replace k with an arbitrary commutative ring R. Can one hope to study the category mod RG by mimicking the above setup? The first obstruction to doing so is the fact that mod RG may no longer be Frobenius, in which case stmod RG would fail to be triangulated in the usual way. Even if mod RG were Frobenius, there would still be no guarantee that tensoring over R would be exact, that is, stmod RG might not be tensor triangulated.

Recently, Benson, Iyengar and Krause [5] have examined an alternative exact structure on mod RG. Let ι: R ↪ RG denote the inclusion of the ground ring R and define a short exact sequence

\xymatrix{0 \ar[r] & M′ \ar[r] & M \ar[r] & M′′ \ar[r] & 0}

of RG-modules to be admissible if the restriction

\xymatrix{0 \ar[r] & ι_{∗}M′ \ar[r] & ι_{∗}M \ar[r] & ι_{∗}M′′ \ar[r] & 0}

is a split short exact sequence of R-modules. In this paper we will denote the category of finitely generated RG-modules endowed with this relatively split exact structure by rel RG. It was shown in [5] that rel RG is a Frobenius category, so its stable category strel RG is triangulated. It turns out that the injective/projective objects in rel RG are precisely the direct summands of those RG-modules lying in the image of the induction functor

\xymatrix{ι^{∗}: mod R \ar[r] & mod RG.}

As in [5], we shall call such modules weakly projective. The suspension of an object M in strel RG is the cokernel ΣM in the R-split short exact sequence

\xymatrix{0 \ar[r] & M \ar[r] & ι^{∗}ι_{∗}M \ar[r] & ΣM \ar[r] & 0,}

where the left hand map M ↪ ι^{∗}ι_{∗}M = RG ⊗_{R} M is given by m ↦ ∑_{g∈G} g ⊗ g^{−1}m. A nice feature of the relative stable category that follows directly from the definition is that in the special case where R = k is a field, strel kG and stmod kG coincide.

Another motivation for appealing to this relative exact structure is that tensoring over R preserves R-split short exact sequences, so strel RG is tensor triangulated with unit the trivial module R. The relative stable category therefore provides a setting in which one may exploit a monoidal structure to study representations of G over arbitrary commutative rings.

With the relative stable category in mind, we now recall that Balmer [1] has developed a general framework for studying the coarse structure of essentially small tensor triangulated categories in terms of supports, namely the yoga of tensor triangular geometry. Let (K, ⊗, 1) be an essentially small tensor triangulated category. A thick subcategory I of K is a tensor ideal of K if K ⊗ I ⊆ I. Continuing the analogy with commutative algebra, Balmer defines a proper tensor ideal P of K to be prime if whenever x and y are objects in K satisfying x ⊗ y ∈ P, then x ∈ P or y ∈ P. The collection of prime ideals of K is called the (prime ideal) spectrum of K, denoted Spc K.

The prime ideal spectrum has a Zariski topology whose closed basis elements are of the form

supp(x) = {P ∈ Spc K | x ∉ P} for all x in K.

The closed subset supp(x) is called the support of x. The topological space Spc K, along with the assignment x ↦ supp(x), is something Balmer calls a universal support data for K. Roughly speaking, this means that Spc K is the best abstract setting in which to ask support theoretic questions about the category K. For example, one may express the universality of Spc K by interpreting it as the space dual to the lattice of (radical) thick tensor ideals; therefore any question about the structure of the lattice of tensor ideals is really a question about Spc K.

To see this in action, we recall a famous result due to Benson, Carlson and Rickard [4], which states that the thick tensor ideals of stmod kG are in one to one correspondence with the specialisation closed subsets of Proj H^{∗}(G, k), where H^{∗}(G, k) denotes group cohomology with coefficients in k, i.e., Ext_{kG}^{∗}(k, k). Statements that relate the structure of stmod kG to the geometry of Proj H^{∗}(G, k) constitute the realm of support theory. For example, the above correspondence assigns to each thick tensor ideal I in stmod kG the subset ⋃_{M∈I} V_{G}(M) of Proj H^{∗}(G, k), where V_{G}(M) is the support variety of M. (See [3, Chapter 5] for details.) Viewed in Balmer’s framework, this result may be reinterpreted as saying that (Proj H^{∗}(G, k), V_{G}(−)) is the classifying support data for stmod kG. (See [1, Section 5].) In particular, Spc(stmod kG) is homeomorphic to Proj H^{∗}(G, k).

We remark that little is known about relative stable categories in general. The goal of this paper is to determine the prime ideal spectrum of strel RG in perhaps the most basic non-trivial case, namely that in which R is the ring ℤ/p^{n}, where p is a prime number and n ≥ 0. Our main result is the following.

Theorem 1.1. Letting R_{n} denote the ring ℤ/p^{n}, there is a decomposition of spectra

Spc(strel R_{n}G) ≅ ∐_{i=1}^{n} Spc(strel 𝔽_{p}G).

In other words, the prime ideal spectrum of strel R_{n}G is homeomorphic to n disjoint copies of the prime ideal spectrum of strel 𝔽_{p}G = stmod 𝔽_{p}G.

As mentioned above, the spectrum of stmod 𝔽_{p}G is known, so the theorem gives a complete description of the spectrum of strel R_{n}G.

This computation is also valuable from the point of view of more abstract tensor triangular geometry. There are many examples in which the spectrum has been computed, but they all tend to share a common feature—the tensor triangulated category in question is rigid. It is known that relative stable categories tend not to be rigid, hence they provide a new family of examples that can be fed back into the abstract theory. For instance, if the spectrum of a rigid tensor triangulated category is a disjoint union of subspaces, then the category itself decomposes into a direct sum of subcategories indexed by those subspaces. However, in our example the relative stable category is indecomposable. The information encoded in the triviality of the topology of the spectrum must therefore manifest in more subtle ways that would be interesting to explore.

2. Notation and preliminary calculations

Let p be a prime number, n ≥ 1 and R_{n} = ℤ/p^{n}. Throughout this paper G will denote a finite group. For convenience, we denote the group algebra R_{n}G by A_{n}. Keeping the notation from the introduction, we denote the inclusion of the base ring by ι: R_{n} ↪ A_{n}. If i ≤ n, then the canonical surjection A_{n} → A_{i} makes A_{i} into an A_{n}-module. We write Ω_{i} for the syzygy, with respect to the usual abelian structure, taken in mod A_{i}. We remark that, in general, this is not the suspension in the stable category strel A_{i}.

We shall not recall many details concerning the relative stable categories beyond those given in the introduction (in fact we shall not need much technology). Further details on these categories may be found in [5].

We begin by determining the injective/projective objects in rel A_{n}, that is, the weakly projective modules in mod A_{n}.

Proposition 2.1. The indecomposable weakly projective A_{n}-modules are precisely those of the form A_{i} for 1 ≤ i ≤ n.

Proof. As mentioned in the introduction, the weakly projective A_{n}-modules are precisely those in the additive closure of

{ι^{∗}S | S ∈ mod R_{n}}.

(Recall that ι^{∗}S is the induced module A_{n} ⊗_{R_{n}} S.) The result now follows by noting that the indecomposable R_{n} modules are of the form R_{i} = ℤ/p^{i} for 1 ≤ i ≤ n. □

Now let k denote the field 𝔽_{p}, that is, the ring ℤ/p. Our main object of focus will be the cosyzygy Ω_{n}^{−1}k, which is the cokernel in the short exact sequence of A_{n}-modules

\xymatrix{0 \ar[r] & k \ar[r] & A_{n} \ar[r] & Ω_{n}^{−1}k \ar[r] & 0,}

the left hand map being given by 1 ↦ p^{n−1} ∑_{g∈G} g. We first study the behaviour of Ω_{n}^{−1}k under base change.

Lemma 2.2. Ω_{n}^{−1}k ⊗_{R_{n}} R_{n−1} ≅ A_{n−1}.

Proof. By the third isomorphism theorem we have

\frac{Ω_{n}^{−1}k}{p^{n−1}Ω_{n}^{−1}k} ≅ \frac{A_{n}/k}{p^{n−1}A_{n}/k} ≅ \frac{A_{n}}{p^{n−1}A_{n}} ≅ A_{n−1}.  □

The following computational ingredient will ensure that a very special triangle exists in strel A_{n}.

Lemma 2.3. There exists an R_{n}-split short exact sequence of A_{n}-modules

(1)  \xymatrix{0 \ar[r] & Ω_{n}R_{n} \ar[r] & Ω_{n}k \ar[r] & R_{n−1} \ar[r] & 0.}

Proof. The submodule Ω_{n}k of A_{n} is the collection of elements ∑_{g∈G} r_{g}g satisfying

∑_{g∈G} r_{g} ∈ pR_{n}.

Consider the surjective A_{n}-module homomorphism ϕ: Ω_{n}k → R_{n−1} given by

∑_{g∈G} r_{g}g ⟼ \frac{1}{p} ∑_{g∈G} r_{g} (mod p^{n−1}).

The kernel of ϕ is the collection of elements ∑_{g∈G} r_{g}g satisfying ∑_{g∈G} r_{g} = 0, which we identify with Ω_{n}R_{n}. This establishes the existence of the short exact sequence (1). The fact that it is R_{n}-split follows from the fact that Ω_{n}R_{n} is injective as an R_{n}-module. □

Prior to giving our next result, we remark that the duality functor Hom_{R_{n}}(−, R_{n}) preserves R_{n}-split short exact sequences in mod A_{n}.

Lemma 2.4. There exists an R_{n}-split short exact sequence of A_{n}-modules

(2)  \xymatrix{0 \ar[r] & R_{n−1} \ar[r] & Ω_{n}^{−1}k \ar[r] & Ω_{n}^{−1}R_{n} \ar[r] & 0.}

Proof. Apply Hom_{R_{n}}(−, R_{n}) to the R_{n}-split short exact sequence (1). □

3. Localisation sequences and splitting of spectra

We now turn our attention to the stable category strel A_{n}, which we shall denote by D_{n} for the sake of brevity. We will denote the tensor unit R_{n} ∈ D_{n} by 1_{n}, and the suspension in D_{n} will be denoted by Σ. Observe that it is not necessary to specify for which category Σ is suspension: if i ≤ n and X ∈ D_{i}, then the suspension of X in D_{i} is equal to the suspension of X in D_{n}. This follows from the fact that a short exact sequence in mod A_{i} is R_{i}-split if and only if it is R_{n}-split.

An essential tool used in proving Theorem 1.1 will be that of a semi-orthogonal decomposition of a tensor triangulated category. Recall that a localisation sequence

[figure: R; ψ_{∗}; ψ^{!}; S; ϕ^{∗}; ϕ_{∗}; T .]

of triangulated categories is a couple of adjoint pairs

ψ_{∗} ⊣ ψ^{!} and ϕ^{∗} ⊣ ϕ_{∗}

such that ψ_{∗} and ϕ_{∗} are fully faithful exact functors satisfying

ϕ_{∗}T = (ψ_{∗}R)^{⊥} and ψ_{∗}R = ^{⊥}(ϕ_{∗}T ).

This is sometimes called a semi-orthogonal decomposition of S.

For our purposes we also require that S is an essentially small tensor triangulated category and that R and T are tensor ideals of S under ψ_{∗} and ϕ_{∗}, respectively. In this very special situation one also obtains a decomposition of the prime ideal spectrum of S. If C is any subcategory of S, we define the support of C to be the specialisation closed subset

supp_{S} C = ⋃_{x∈C} supp_{S} x

of Spc S.

Theorem 3.1 ([5], Theorem A.5). The subsets Spc R = supp_{S} R and Spc T = supp_{S} T of Spc S are open and closed, and there is a decomposition

Spc S = Spc R ∐ Spc T .

We now return to the categories D_{i} = strel A_{i}, i ≥ 1. As explained in [5, Remark 6.10], each ring epimorphism ϕ_{i}: A_{i} → A_{i−1} induces a localisation sequence

[figure: K_{i}; ^{ψ_{i}^{∗}}; _{ψ_{i}_{!}}; D_{i}; ^{ϕ_{i}^{∗}}; _{ϕ_{i}_{∗}}; D_{i−1}]

where ϕ_{i}^{∗} = − ⊗_{R_{i}} R_{i−1} and ϕ_{i}_{∗} is restriction of scalars. Note that the functors ϕ_{i}^{∗} and ψ_{i}^{!} are strong monoidal, hence ψ_{i}_{∗}K_{i} and ϕ_{i}_{∗}D_{i−1} are thick tensor ideals in D_{i}. As usual, we identify K_{i} and D_{i−1} with their images under these fully faithful embeddings. A slight generalisation of the conclusion of [5, Remark 6.10] is the following.

Theorem 3.2. Setting K_{1} = D_{1}, there is a decomposition of spectra

Spc D_{n} = ∐_{i=1}^{n} Spc K_{i}.

Proof. This follows by induction on n and the appropriate use of Theorem 3.1. □

In order to prove Theorem 1.1, it therefore suffices to show that each Spc K_{i} is homeomorphic to Spc D_{1}. To establish this fact, we first show that each K_{i} is equal to the thick tensor ideal

thick_{D_{i}}^{⊗}(Ω_{i}^{−1}k)

of D_{i} generated by Ω_{i}^{−1}k. We then exhibit a monoidal equivalence between thick_{D_{i}}^{⊗}(Ω_{i}^{−1}k) and D_{1}.

4. The tensor ideal thick_{D_{n}}^{⊗}(X_{n})

Given its central role in the rest of the paper, we now denote the object Ω_{n}^{−1}k in D_{n} by X_{n} for the sake of brevity.

Lemma 4.1. The thick tensor ideal thick_{D_{n}}^{⊗}(X_{n}) is equal to the kernel of the base change functor − ⊗ 1_{n−1} : D_{n} → D_{n−1}. That is, in the notation of Section 3 we have

K_{n} = thick_{D_{n}}^{⊗}(X_{n}).

Proof. By Lemma 2.2 we have X_{n} ⊗ 1_{n−1} = 0 in D_{n} so that thick_{D_{n}}^{⊗}(X_{n}) ⊆ K_{n}. Conversely, let X ∈ K_{n}. The R_{n}-split short exact sequence (2) induces a triangle

[figure: 1_{n−1}; X_{n}; Σ1_{n}]

in D_{n}. Since X ∈ K_{n}, tensoring this triangle with X yields a triangle of the form

[figure: 0; X ⊗ X_{n}; ΣX].

It follows that ΣX ≅ X ⊗ X_{n} so that X ∈ thick_{D_{n}}^{⊗}(X_{n}). □

Lemma 4.2. If X lies in the kernel of − ⊗ 1_{n−1}, then X is isomorphic in D_{n} to an object in mod A_{n} whose abelian group decomposition contains only summands of the form R_{n} and R_{n−1}.

Proof. Viewing X as an object in mod A_{n}, consider the abelian group decomposition ι_{∗}X = ⨁_{i=1}^{n} R_{i}^{⊕r_{i}}. Let S = R_{n}^{⊕r_{n}} ⊕ R_{n−1}^{⊕r_{n−1}} and T = ⨁_{i=1}^{n−2} R_{i}^{⊕r_{i}} so that ι_{∗}X = S ⊕ T. Let π: X → X ⊗_{R_{n}} R_{n−1} denote the base change homomorphism and let K = ker π. Notice that K ⊆ S since p^{n−1} annihilates T. We then have

ι_{∗}(X ⊗_{R_{n}} R_{n−1}) = (S/K) ⊕ T.

By the assumption that X ⊗ 1_{n−1} = 0 in D_{n}, we also know that X ⊗_{R_{n}} R_{n−1} is weakly projective. We may therefore write

X ⊗_{R_{n}} R_{n−1} ≅ A_{n−1}^{⊕s_{n−1}} ⊕ ⨁_{i=1}^{n−2} A_{i}^{⊕s_{i}}

as A_{n}-modules. Comparing abelian group structures, we must have

ι_{∗}(A_{n−1}^{⊕s_{n−1}}) ≅ S/K and ι_{∗}( ⨁_{i=1}^{n−2} A_{i}^{⊕s_{i}}) ≅ T.

These abelian group isomorphisms allow us to place A_{n}-module structures { \atop S/K} and T̃ on S/K and T, respectively, through which X ⊗_{R_{n}} R_{n−1} = { \atop S/K} ⊕ T̃.

Now consider the short exact sequence of A_{n}-modules

(3)  \xymatrix{0 \ar[r] & X′ \ar[r] & X \ar[r] & T̃ \ar[r] & 0,}

where the right hand map is the composition

\xymatrix{X \ar[r]^{π} & X ⊗_{R_{n}} R_{n−1} \ar[r]^{^{pr^{2}}} & T̃.}

We have X′ = {m ∈ X | π(m) ∈ ker pr_{2}} = S as abelian groups. Applying ι_{∗} to the sequence (3) therefore yields the split short exact sequence of abelian groups

\xymatrix{0 \ar[r] & S \ar[r] & ι_{∗}X \ar[r] & T \ar[r] & 0.}

This means that (3) gives rise to a triangle

[figure: X′; X; T̃]

in D_{n}. Because T̃ ≅ ⨁_{i=1}^{n−2} A_{i}^{⊕s_{i}} is weakly projective in mod A_{n}, we have T̃ = 0 in D_{n}. It follows that X′ ≅ X in D_{n}. □

Recall that D_{1} is just the usual stable module category stmod kG of kG.

Lemma 4.3. If X lies in K_{n} = thick_{D_{n}}^{⊗}(X_{n}), then Ω_{n}X lies in D_{1} = thick_{D_{n}}^{⊗}(k).

Proof. We wish to show that multiplication by p annihilates Ω_{n}X. By Lemma 4.2 we can assume that the abelian group decomposition of ι_{∗}X is of the form R_{n}^{⊕r} ⊕ R_{n−1}^{⊕s} for some non-negative integers r and s. We then have

(4)  ι_{∗}(X ⊗_{R_{n}} R_{n−1}) ≅ R_{n−1}^{⊕(r+s)}.

On the other hand, X ⊗_{R_{n}} R_{n−1} is known to be weakly projective. The decomposition (4) reveals that we must have X ⊗_{R_{n}} R_{n−1} ≅ A_{n−1}^{⊕t}, forcing r + s = |G|t.

Now let π: X → X ⊗_{R_{n}} R_{n−1} and

\xymatrix{ϕ: A_{n}^{⊕t} \ar[r] & A_{n−1}^{⊕t} ≅ X ⊗_{R_{n}} R_{n−1}}

denote the A_{n}-module homomorphisms induced by extension of scalars from R_{n} to R_{n−1}. Since π is surjective and A_{n}^{⊕t} is projective, ϕ lifts to an A_{n}-module homomorphism

\xymatrix{ψ: A_{n}^{⊕t} \ar[r] & X}

satisfying π ∘ ψ = ϕ. Thus ψ ⊗_{R_{n}} R_{n−1} recovers the isomorphism A_{n−1}^{⊕t} ≅ X ⊗_{R_{n}} R_{n−1}. Because ψ ⊗_{R_{n}} R_{n−1} is surjective, it follows by Nakayama’s lemma that ψ is surjective. The syzygy Ω_{n}X therefore fits into a short exact sequence

\xymatrix{0 \ar[r] & Ω_{n}X \ar[r] & A_{n}^{⊕t} \ar[r]^{ψ} & X \ar[r] & 0.}

But Ω_{n}X is also contained in the kernel of the composition π ∘ ψ, which is isomorphic to the canonical surjection A_{n}^{⊕t} → A_{n−1}^{⊕t}. The kernel of the latter is annihilated by p, hence the same is true of its submodule Ω_{n}X. □

Lemma 4.4. If X lies in thick_{D_{n}}^{⊗}(k), then Ω_{n}^{−1}X lies in thick_{D_{n}}^{⊗}(X_{n}).

Proof. Suppose X ∈ D_{1} as in the statement and consider the short exact sequence

\xymatrix{0 \ar[r] & X \ar[r]^{ϕ} & A_{n}^{⊕t} \ar[r]^{ψ} & Ω_{n}^{−1}X \ar[r] & 0}

defining the cosyzygy of X over A_{n}. Since X lies in D_{1}, its underlying abelian group is an 𝔽_{p}-vector space, so the image of ϕ is contained in p^{n−1}A_{n}^{⊕t}. Thus ϕ ⊗_{R_{n}} R_{n−1} = 0, which implies that ψ ⊗_{R_{n}} R_{n−1} is an isomorphism. This shows that Ω_{n}^{−1}X ⊗_{R_{n}} R_{n−1} ≅ A_{n−1}^{⊕t} is weakly projective so that Ω_{n}^{−1}X is in K_{n} as claimed. □

The main theorem of this section follows by the observation that the syzygy Ω_{n} induces a (non-monoidal) exact autoequivalence of D_{n}. Indeed, it is straightforward to check that Ω_{n} takes R_{n}-split short exact sequences in mod A_{n} to R_{n}-split short exact sequences. The same is then of course true of its quasi-inverse Ω_{n}^{−1}.

Theorem 4.5. There is an equivalence of triangulated categories

thick_{D_{n}}^{⊗}(X_{n}) ≅ thick_{D_{n}}^{⊗}(k) = stmod 𝔽_{p}G.

Proof. As mentioned prior to the theorem, Ω_{n} and Ω_{n}^{−1} are quasi-inverse autoequivalences of D_{n}. The restriction of Ω_{n} to thick_{D_{n}}^{⊗}(X_{n}) is therefore an equivalence onto its essential image. By Lemmas 4.3 and 4.4, the latter is precisely thick_{D_{n}}^{⊗}(k). □

Although we have shown that K_{n} = thick_{D_{n}}^{⊗}(X_{n}) is triangle equivalent to D_{1}, we have not produced an equivalence of tensor triangulated categories. In fact, these categories are also equivalent as tensor triangulated categories, as we show in Theorem 5.6.

5. A monoidal equivalence

In Theorem 4.5 we learned that thick_{D_{n}}^{⊗}(X_{n}) and D_{1} are equivalent as triangulated categories, and the latter is known to be monoidal. The localisation sequences in Section 3 show that thick_{D_{n}}^{⊗}(X_{n}) = K_{n} is equal to D_{n}/D_{n−1} (as tensor ideals in D_{n}), so thick_{D_{n}}^{⊗}(X_{n}) is also monoidal.

In this section we produce a monoidal exact equivalence between thick_{D_{n}}^{⊗}(X_{n}) and D_{1}, namely the restriction of the functor

\xymatrix{P: D_{n} \ar[r] & D_{1}}

induced by multiplication by p^{n−1}.

Specifically, if M is an A_{n}-module, then as an abelian group, P (M) is defined to be the A_{n}-submodule p^{n−1}M of M. Identifying A_{1} with A_{n}/(p), we obtain an action of A_{1} on p^{n−1}M via

\overline{a}(p^{n−1}m) = p^{n−1}am for all a ∈ A_{n}.

This action is well defined since p annihilates p^{n−1}M. If ϕ: M → N is a homomorphism of A_{n}-modules, then so is ϕ|_{p^{n−1}M}: p^{n−1}M → N. Moreover, for m ∈ M we have

ϕ(p^{n−1}m) = p^{n−1}ϕ(m) ∈ p^{n−1}N,

hence ϕ|_{p^{n−1}M} induces a map p^{n−1}M → p^{n−1}N. We therefore set P (ϕ) = ϕ|_{p^{n−1}M}.

Observe that multiplication by p^{n−1} preserves R_{n}-split short exact sequences in mod A_{n}, so P is exact. We claim that P is also monoidal. To see this, let M and M′ be A_{n}-modules and consider the map

\xymatrix{ϕ: P (M) ⊗_{R_{1}} P (M′) \ar[r] & P (M ⊗_{R_{n}} M′)}

given by p^{n−1}m ⊗ p^{n−1}m′ ↦ p^{n−1}(m ⊗ m′). One readily verifies that ϕ is a well defined isomorphism of abelian groups and that the action of x commutes with ϕ.

Our strategy in proving that the restriction of P to thick_{D_{n}}^{⊗}(X_{n}) is an equivalence will be to show that the functor

\xymatrix{F = Ω_{n}^{−1}Ω_{1} : D_{1} \ar[r] & thick_{D_{n}}^{⊗}(X_{n}).}

is a quasi-inverse.

Lemma 5.1. The composition P F: D_{1} → D_{1} is naturally isomorphic to the identity functor.

Proof. Let M be an object in D_{1} and consider the short exact sequence of A_{1}-modules

\xymatrix{0 \ar[r] & Ω_{1}M \ar[r] & A_{1}^{⊕r} \ar[r] & M \ar[r] & 0}

defining the syzygy Ω_{1}M in mod A_{1}. A cosyzygy of Ω_{1}M in mod A_{n} is then obtained via the short exact sequence

\xymatrix{0 \ar[r] & Ω_{1}M \ar[r] & A_{n}^{⊕r} \ar[r] & Ω_{n}^{−1}Ω_{1}M \ar[r] & 0,}

and we have a diagram

[figure: 0; Ω_{1}M; A_{1}^{⊕r}; M; 0; 0; Ω_{1}M; A_{n}^{⊕r}; F M; 0,]

where the right two vertical arrows are those induced by multiplication by p^{n−1}. The right hand arrow therefore identifies M with the submodule P F M. □

Corollary 5.2. The functor P: thick_{D_{n}}^{⊗}(X_{n}) → D_{1} is essentially surjective and full.

Proof. This is immediate from Lemma 5.1, combined with Lemma 4.4 which implies that the essential image of F is contained in thick_{D_{n}}^{⊗}(X_{n}). □

Corollary 5.3. The functor P: thick_{D_{n}}^{⊗}(X_{n}) → D_{1} is monoidal.

Proof. We know from the discussion preceding Lemma 5.1 that P is monoidal from D_{n} to D_{1}, thus its restriction to thick_{D_{n}}^{⊗}(X_{n}) respects tensor products. Any such functor that is also essentially surjective will automaticaly be monoidal. □

Lemma 5.4. The kernel of P: thick_{D_{n}}^{⊗}(X_{n}) → D_{1} is trivial.

Proof. Let X be an object in thick_{D_{n}}^{⊗}(X_{n}) with P X weakly projective. By Lemma 4.2, we may assume that ι_{∗}X ≅ R_{n}^{⊕r} ⊕ R_{n−1}^{⊕s} for some non-negative integers r and s. Since P X is weakly projective, we have P X ≅ A_{1}^{⊕t} where r = t|G|. It follows that the injective hull of P X in mod A_{n} is A_{n}^{⊕t}. Because A_{n} is injective as an A_{n}-module, the natural embedding P X ↪ X extends to a morphism X → A_{n}^{⊕t}. The latter is a surjection, as its restriction to the R_{n}-free component of X is a homomorphism of free R_{n}-modules that is an isomorphism on socles. The projectivity of A_{n} allows us to split off a direct summand A_{n}^{⊕t} from X and assume that ι_{∗}X ≅ R_{n−1}^{⊕s}. We then have X ⊗_{R_{n}} R_{n−1} isomorphic to X. But X was assumed to be in thick_{D_{n}}^{⊗}(X_{n}), the kernel of − ⊗_{R_{n}} R_{n−1}, thus X ≅ 0 in D_{n}. □

The final ingredient in proving that P is an equivalence is the following bit of folklore.

Lemma 5.5 ([2], Proposition 3.18). Let G : K → L be an essentially surjective full exact functor between triangulated categories having trivial kernel. Then G is an equivalence.

Theorem 5.6. The functor P: thick_{D_{n}}^{⊗}(X_{n}) → D_{1} is a monoidal equivalence of triangulated categories.

Proof. By Corollary 5.2 we know that P is full and essentially surjective. Lemma 5.4 also tells us that P has trivial kernel. It follows by Lemma 5.5 that P is an equivalence. It is monoidal by Corollary 5.3. □

We are now in a position to prove the main result.

Theorem (1.1). For every positive integer n there is a decomposition of spectra

Spc D_{n} ≅ ∐_{i=1}^{n} Spc D_{1}.

Proof. Setting K_{1} = D_{1}, Theorem 3.2 tells us that there is a decomposition of spectra

Spc D_{n} = ∐_{i=1}^{n} Spc K_{i}.

By Lemma 4.1 we have K_{i} = thick_{D_{i}}^{⊗}(X_{i}). Theorem 5.6 shows that the functor

\xymatrix{P: thick_{D_{i}}^{⊗}(X_{i}) \ar[r] & D_{1}}

induced by multiplication by p^{i−1} is an equivalence of tensor triangulated categories. Putting this all together, we have Spc K_{i} ≅ Spc D_{1} for all 1 ≤ i ≤ n. □

The following corollary summarises the consequences of our results for D_{n}.

Corollary 5.7. The relative stable module category D_{n} = strel R_{n}G admits a semi-orthog-onal decomposition into n tensor ideals

strel R_{n}G = (stmod 𝔽_{p}G, …, stmod 𝔽_{p}G),

where the ith copy embeds as K_{i} = thick_{D_{n}}^{⊗}(Ω_{i}^{−1}k).

Proof. This follows from the discussion in Section 3, along with Theorem 5.6 and Lemma 4.1. □

6. An example: cyclic groups of prime order

In this section we give a more detailed description of Spc(strel R_{n}G) in the case where G = C_{p}, the cyclic group of prime order p. In particular, we give concrete generators for all of the prime tensor ideals.

The first handful of results actually hold for any finite group G. Recall that Ω_{i} denotes the syzygy taken in mod A_{i} = mod R_{i}G under the usual abelian category structure.

Proposition 6.1. If G is any finite group, then the A_{n}-modules Ω_{i}^{−1}k for i ≤ n generate D_{n} = strel R_{n}G as a thick tensor ideal.

Proof. This is immediate from Corollary 5.7. □

Lemma 6.2. For all 1 ≤ i < j ≤ n we have Ω_{i}^{−1}k ⊗_{R_{n}} Ω_{j}^{−1}k ≅ A_{i−1} ⊕ A_{i}^{⊕(p−1)}.

Proof. Since p^{j} annihilates both R_{i} and R_{j}, we have

Ω_{i}^{−1}k ⊗_{R_{n}} Ω_{j}^{−1}k ≅ Ω_{i}^{−1}k ⊗_{R_{j}} Ω_{j}^{−1}k.

We therefore compute the right hand term in mod A_{j}. Applying − ⊗_{R_{j}} Ω_{i}^{−1}k to the short exact sequence

\xymatrix{0 \ar[r] & k \ar[r]^{p^{j−1}f} & A_{j} \ar[r] & Ω_{j}^{−1}k \ar[r] & 0}

yields the exact sequence

\xymatrix{k ⊗_{R_{j}} Ω_{i}^{−1}k \ar[r]^{p^{j−1}f⊗1} & A_{j} ⊗_{R_{j}} Ω_{i}^{−1}k \ar[r] & Ω_{j}^{−1}k ⊗_{R_{j}} Ω_{i}^{−1}k \ar[r] & 0}

so that Ω_{j}^{−1}k ⊗_{R_{j}} Ω_{i}^{−1}k is isomorphic to A_{j} ⊗_{R_{j}} Ω_{i}^{−1}k modulo the image of p^{j−1}f ⊗ 1. By Frobenius reciprocity and the short exact sequence of Lemma 2.4, one sees that

(5)  A_{j} ⊗_{R_{j}} Ω_{i}^{−1}k ≅ A_{i−1} ⊕ A_{i}^{⊕(p−1)}.

Since i < j, p^{j−1} annihilates the right hand side so that the image of p^{j−1}f ⊗ 1 is zero and

Ω_{j}^{−1}k ⊗_{R_{j}} Ω_{i}^{−1}k ≅ A_{i−1} ⊕ A_{i}^{⊕(p−1)}.  □

We again denote the objects Ω_{i}^{−1}k in D_{n} by X_{i} for the sake of brevity.

Proposition 6.3. Any prime tensor ideal in Spc D_{n} contains at least n − 1 objects in the set {X_{1}, …, X_{n}}.

Proof. Let P ∈ Spc D_{n} and suppose that there are distinct objects X_{i} and X_{j}, both of which do not lie in P. By Lemma 6.2 we have

X_{i} ⊗ X_{j} = 0 ∈ P.

Since P is prime, this forces X_{i} ∈ P or X_{j} ∈ P, a contradiction. □

Motivated by this lemma, we now focus our attention on certain thick tensor ideals of D_{n}. For 1 ≤ i ≤ n, we let

P_{i,n} = thick_{D_{n}}^{⊗}({X_{1}, …, X_{n}} \ {X_{i}}).

Our goal will be to show that these are precisely the prime ideals in D_{n}. (We know from Theorem 1.1 that the spectrum is a disjoint union of n points.)

Lemma 6.4. For all X ∈ P_{i,n} we have X ⊗ 1_{n−1} ∈ P_{i,n−1}, i.e.,

ϕ_{i}^{∗}P_{i,n} ⊆ P_{i,n−1}.

Proof. If i ≤ n − 1 then X_{i} ⊗ 1_{n−1} = X_{i}, whereas X_{n} ⊗ 1_{n−1} = 0 by Lemma 2.2. Hence ϕ_{i}^{∗} sends the generators of P_{i,n} into P_{i,n−1}. Because P_{i,n−1} is thick and ϕ_{i}^{∗} is exact, the lemma follows immediately. (The dubious reader may consult [7, Lemma 3.8].) □

Lemma 6.5. Each tensor ideal P_{i,n} is proper in D_{n}.

Proof. We fix an i and proceed by induction on n, the base case being n = i. For this we need to show that P_{i,i} = thick_{D_{i}}^{⊗}(X_{1}, …, X_{i−1}) is proper. We saw in Section 3 that the restriction functor ϕ_{i−1}_{∗} embeds D_{i−1} as a proper tensor ideal in D_{i}. For 1 ≤ j ≤ i − 1 we have X_{j} ∈ ϕ_{i−1}_{∗}D_{i−1}, so P_{i,i} is contained in ϕ_{i−1}_{∗}D_{i−1} and P_{i,i} is proper in D_{i}. (In fact, P_{i,i} = ϕ_{i−1}_{∗}D_{i−1} by Proposition 6.1.)

Now let n > i and assume that P_{i,n−1} is proper in D_{n−1}. For the sake of contradiction, suppose that P_{i,n} is not proper in D_{n} so that it contains the tensor unit 1_{n}. Then 1_{n−1} = 1_{n} ⊗ 1_{n−1} lies in P_{i,n−1} by Lemma 6.4, a contradiction. □

We are only now forced to specialise to the case where G is the cyclic group C_{p}.

Lemma 6.6. If G = C_{p}, then each P_{i,n} is a maximal tensor ideal in D_{n}.

Proof. We fix i and proceed by induction on n. For the base case n = i we need to show that thick_{D_{i}}^{⊗}(X_{1}, …, X_{i−1}) = D_{i−1} is maximal in D_{i}. Recall that the thick tensor ideals in D_{i} containing D_{i−1} are in bijection with those in the quotient D_{i}/D_{i−1}. By the discussion in Section 3, that quotient is tensor equivalent to K_{i}, which in turn is equivalent to

D_{1} = strel 𝔽_{p}C_{p} = stmod 𝔽_{p}C_{p}

by Theorem 5.6. It is known that the rightmost category has precisely two tensor ideals, namely the zero ideal and the entire category. It follows that the only tensor ideal in D_{i} properly containing D_{i−1} is D_{i} itself, so D_{i−1} is maximal as claimed.

Now let n > i, assume that P_{i,n−1} is maximal in D_{n−1} and choose X ∉ P_{i,n}. Tensoring the short exact sequence of Lemma 2.4 with X produces a triangle

[figure: X ⊗ 1_{n−1}; X ⊗ X_{n}; ΣX]

in D_{n}. The middle term lies in P_{i,n}, but the right hand term does not. This implies that X ⊗ 1_{n−1} cannot lie in P_{i,n}. In particular, it cannot lie in P_{i,n−1} since the latter is contained in the former. By the inductive hypothesis on maximality, this means that

1_{n−1} ∈ thick_{D_{n−1}}^{⊗}({X_{1}, …, X_{n−1}, X ⊗ 1_{n−1}} \ {X_{i}}).

Now consider the triangle

[figure: 1_{n−1}; X_{n}; Σ1_{n}]

in D_{n} induced by the short exact sequence of Lemma 2.4. By the above remarks, the left two terms lie in thick_{D_{n}}^{⊗}({X_{1}, …, X_{n}, X} \ {X_{i}}), whence so does the right hand term. In other words

1_{n} ∈ thick_{D_{n}}^{⊗}({X_{1}, …, X_{n}, X} \ {X_{i}}),

proving that P_{i.n} is maximal. □

It now follows from [1, Proposition 2.3] that each P_{i,n} is a prime tensor ideal, i.e., gives a point in Spc D_{n}.

Lemma 6.7. Each P_{i,n} is a minimal prime.

Proof. Suppose there exists a prime ideal P properly contained in P_{i,n}. Then there is an object in the set {X_{1}, …, X_{n}} \ {X_{i}} not contained in P. Since P is prime, Proposition 6.3 forces us to have X_{i} ∈ P. But this implies that X_{i} ∈ P_{i,n}, so {X_{1}, …, X_{n}} ⊆ P_{i,n}. Proposition 6.1 now tells us that P_{i,n} = D_{n}, contradicting Lemma 6.5. □

We are now ready to give a direct computation of the spectrum, verifying Theorem 1.1 in this case.

Theorem 6.8. The prime ideal spectrum of D_{n} = strel (ℤ/p^{n})C_{p} is a disjoint union of n points.

Proof. If P is an element of Spc D_{n}, then by Proposition 6.3 there is an integer 1 ≤ i ≤ n such that {X_{1}, …, X_{n}} \ {X_{i}} ⊆ P. We then have P_{i,n} ⊆ P. Since P_{i,n} is maximal, this implies that P = P_{i,n}. It follows that Spc D_{n} = {P_{1,n}, …, P_{n,n}} as a set. We now recall from [1, Proposition 2.9] that the closed points in the prime ideal spectrum are precisely the minimal primes. Lemma 6.7 therefore informs us that each point P_{i,n} is closed in the topology of Spc D_{n}. □

We now make a few observations based on and related to the theorem; all of the statements are, more or less, trivial consequences of what we have already done, but are worth making explicit.

Corollary 6.9. Each X_{i} = Ω_{i}^{−1}k is supported at a single point

supp X_{i} = {P_{i,n}}.

Proof. Given the computation of the spectrum and the definition of support, we have

supp X_{i} = {P ∈ Spc D_{n} | X_{i} ∉ P} = {P_{i,n}}.  □

Corollary 6.10. The base change functor

\xymatrix{− ⊗ 1_{m}: D_{n} \ar[r] & D_{m}}

induces an embedding Spc D_{m} ↪ Spc D_{n} with image {P_{i,n} | 1 ≤ i ≤ m}.

Proof. Letting ϕ^{∗} denote the base change functor, it is a general fact (see [1, Proposition 3.11]) that Spc(ϕ^{∗}) is a homeomorphism onto its image. The latter set consists precisely of those prime ideals that contain the kernel of ϕ^{∗}. It follows by repeated applications of Lemma 4.1 that the kernel of ϕ^{∗} is generated by {X_{i} | m + 1 ≤ i ≤ n}. The result follows immediately. □

In the above corollary, one might also consider the corresponding base change functors having source strel ℤC_{p}. The result remains true, except for the description of the image. By invoking [5, Theorem A.5], one may write

Spc(strel ℤC_{p}) ≅ Z_{n} ∐ Spc D_{n}

for all n ≥ 1, where Z_{n} is the spectrum of the kernel. Thus for any n ≥ 1 we can find a disjoint union of n points as an open and closed subspace of the spectrum of strel ℤC_{p}. It would be interesting to fully understand the space Spc(strel ℤC_{p}) and, in particular, how the spectra of the D_{n} sit inside of it. We note that since Spc(strel ℤC_{p}) is quasi-compact, there must be some non-trivial topology on the union of these points in Spc(strel ℤC_{p}).

References

1. P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine & Angew. Math. 588 (2005), 149–168.

2. , Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math. 129 (2007), 1227–1250.

3. D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, 1991, reprinted in paperback, 1998.

4. D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

5. D. J. Benson, S. B. Iyengar, and H. Krause, Module categories for group algebras over commutative rings, with an appendix by Greg Stevenson, K-Theory 11 (2013), no. 2, 297–329.

6. D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Note Series, vol. 119, Cambridge University Press, 1988.

7. G. Stevenson, Support theory via actions of tensor triangulated categories, J. Reine & Angew. Math. 681 (2013), 219–254.

Shawn Baland, Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195

  E-mail address: sbaland@math.washington.edu

Alexandru Chirvasitu, Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195

  E-mail address: chirva@math.washington.edu

Greg Stevenson, Universität Bielefeld, Fakultät für Mathematik, BIREP Gruppe, Postfach 10 01 31, 33501 Bielefeld, Germany.

  E-mail address: gstevens@math.uni-bielefeld.de