THE AVRUNIN-SCOTT THEOREM FOR QUANTUM COMPLETE INTERSECTIONS

PETTER ANDREAS BERGH & KARIN ERDMANN

Abstract. We prove the Avrunin-Scott theorem for quantum complete intersections, relating the rank variety of a module to its support variety.

1. Introduction

Inspired by the impact of the theories of varieties for modules over group algebras, similar theories have been studied for other classes of algebras. For example, using Hochschild cohomology, Snashall and Solberg developed a theory of support varieties for finite dimensional algebras in [SnS]. As shown in [EHSST], this theory is very powerful when the cohomology of the algebra satisfies sufficient finite generation, and for selfinjective such algebras the theory shares many of the properties of that for group algebras. However, support varieties are difficult to compute. In [Car], Carlson introduced rank varieties for modules over group algebras of elementary abelian groups, varieties defined without using cohomology. Given a module over such an algebra, its rank variety is very explicit and easy to compute. Moreover, Avrunin and Scott proved in [AvS] that the support variety of a module is in fact isomorphic to its rank variety. Needless to say, this result has had important consequences (see, for example, the introduction in [AvS] or [ErH]).

Motivated by this, the second author and Holloway introduced in [ErH] rank varieties for truncated polynomial algebras in which the generators square to zero. Such algebras also have support varieties, and it was shown that these two varieties are related by an analogue of the Avrunin-Scott theorem.

In this paper, we study rank varieties for quantum complete intersections, a class of algebras originating from work by Manin and Avramov, Gasharov and Peeva (cf. [Man] and [AGP]). When the defining parameters are roots of unity, then these algebras have support varieties; it was shown in [BeO] that finite generation of cohomology holds. However, certain quantum complete intersections also have rank varieties, and one would therefore like to know if and how the support and rank varieties are related. This is the purpose of this paper; we prove an analogue of the Avrunin-Scott theorem, relating rank varieties to support varieties.

2. Varieties

Throughout this section, let k be an algebraically closed field. All modules considered are assumed to be left modules and finitely generated. We start by recalling the definitions and some results on support varieties; details can be found in [EHSST] and [SnS].

Let Λ be a finite dimensional k-algebra with Jacobson radical 𝔯. We denote by Λ^{e} the enveloping algebra Λ ⊗_{k} Λ^{op} of Λ. The nth Hochschild cohomology group of Λ, denoted HH^{n}(Λ), is the vector space Ext_{Λ^{e}}^{n}(Λ, Λ) of n-fold bimodule extensions of Λ with itself. The Yoneda product turns HH^{∗}(Λ) = ⊕_{n=0}^{∞} HH^{n}(Λ) into a graded k-algebra, the Hochschild cohomology ring of Λ. This algebra is graded commutative, that is, given two homogeneous elements η, θ ∈ HH^{∗}(Λ), the equality ηθ = (−1)^{|η||θ|}θη holds. In particular, the even part HH^{2∗}(Λ) = ⊕_{n=0}^{∞} HH^{2n}(Λ) of HH^{∗}(Λ) is a commutative ring.

2000 Mathematics Subject Classification. 16E30, 16E40, 16S80, 16U80, 81R50.

Key words and phrases. Quantum complete intersections, rank varieties, support varieties, Avrunin-Scott theorem.

The first author was supported by NFR Storforsk grant no. 167130.

Every homogeneous element η ∈ HH^{∗}(Λ) can be represented by a bimodule extension

η: 0 → Λ → B_{|η|} → ⋯ → B_{1} → Λ → 0.

Given a Λ-module M, the complex

η ⊗_{Λ} M: 0 → M → B_{|η|} ⊗_{Λ} M → ⋯ → B_{|η|} ⊗_{Λ} M → M → 0

is exact, hence the tensor product − ⊗_{Λ} M induces a homomorphism

HH^{∗}(Λ) ⟶^{φ_{M}} Ext_{Λ}^{∗}(M, M)

of graded k-algebras. Thus, given two Λ-modules M and N, the graded vector space Ext_{Λ}^{∗}(M, N) becomes a graded module over HH^{∗}(Λ) in two ways; either via φ_{M} or via φ_{N}. These two module structures are equal up to a graded sign, that is, given homogeneous elements η ∈ HH^{∗}(Λ) and θ ∈ Ext_{Λ}^{∗}(M, N), the equality φ_{N}(η) ∘ θ = (−1)^{|η||θ|}θ ∘ φ_{M}(η) holds, where “∘” denotes the Yoneda product.

Let H be a commutative graded subalgebra of HH^{∗}(Λ). The support variety of an ordered pair (M, N) of Λ-modules (with respect to H), denoted V_{H}(M, N), is defined as

V_{H}(M, N) =^{def} {𝔪 ∈ MaxSpec H | Ann_{H} Ext_{Λ}^{∗}(M, N) ⊆ 𝔪},

where MaxSpec H is the set of maximal ideals of H. There are equalities

V_{H}(M, Λ / 𝔯) = V_{H}(M, M) = V_{H}(Λ / 𝔯, M),

and this set is defined to be the support variety V_{H}(M) of M.

In general, support varieties do not contain any homological information on the modules involved. For example, if HH^{∗}(Λ) is finite dimensional, that is, if HH^{n}(Λ) = 0 for n ≫ 0, then the support variety of any pair of modules is trivial. However, under certain finiteness conditions, the situation is quite different. Suppose H is Noetherian and Ext_{Λ}^{∗}(M, N) is a finitely generated H-module for all Λ-modules M and N (this is equivalent to Ext_{Λ}^{∗}(Λ / 𝔯, Λ / 𝔯) being finitely generated over H). In this case, the dimension of V_{H}(M, N) equals the polynomial rate of growth of Ext_{Λ}^{∗}(M, N). In particular, the dimension of the support variety of a module equals its complexity, hence a module has finite projective dimension if and only if its support variety is trivial. Moreover, the dimension of the support variety of a module is one if and only if its minimal projective resolution is bounded. Under the finite generation hypothesis given, this happens precisely when the module is eventually periodic, that is, when its minimal projective resolution becomes periodic from some step on.

When the algebra Λ, in addition to satisfying the finite generation hypothesis, is also selfinjective, then the projective support variety of an indecomposable module is connected. Namely, let M be a Λ-module whose support variety decomposes as V_{H}(M) = V_{1} ∪ V_{2}, where V_{1} and V_{2} are closed homogeneous varieties such that V_{1} ∩ V_{2} is trivial. Then there are submodules M_{1} and M_{2} of M, with the property that M = M_{1} ⊕ M_{2} and V_{H}(M_{i}) = V_{i}. In particular, this implies that the support variety of an indecomposable periodic module is a single line.

We now turn to the class of algebras for which we will prove the Avrunin-Scott theorem. These are analogues of truncated polynomial rings. Let c ≥ 1 be an integer, and let q = (q_{ij}) be a c × c commutation matrix with entries in k. That is, the diagonal entries q_{ii} are all 1, and q_{ij}q_{ji} = 1 for i ≠ j. Furthermore, let a_{c} = (a_{1}, …, a_{c}) be an ordered sequence of c integers with a_{i} ≥ 2. The quantum complete intersection A_{q}^{a_{c}} determined by these data is the algebra

A_{q}^{a_{c}} =^{def} k⟨x_{1}, …, x_{c}⟩/(x_{i}^{a_{i}}, x_{i}x_{j} − q_{ij}x_{j}x_{i}),

which is selfinjective and finite dimensional of dimension ∏ a_{i}. It was proved in [BeO] that if all the commutators q_{ij} are roots of unity, then HH^{2∗}(A_{q}^{a_{c}}) is Noetherian, and Ext_{A_{q}^{ac}}^{∗} (M, N) is a finitely generated HH^{2∗}(A_{q}^{a_{c}})-module for all A_{q}^{a_{c}}-modules M and N. Thus, in this case, the support varieties with respect to HH^{2∗}(A_{q}^{a_{c}}) detect projective and periodic modules, as we saw above. The Krull dimension of HH^{2∗}(A_{q}^{a_{c}}) is c, that is, the number of generators defining the quantum complete intersection. Therefore, the support varieties are homogeneous affine subsets of k^{c}.

Now fix an integer a ≥ 2, and define a′ by

a′ =^{def} \{ \begin{matrix}a/ gcd(a, char k) & if char k > 0 \\ a & if char k = 0. \end{matrix}

Let q ∈ k be a primitive a′th root of unity, let q be the commutation matrix with q_{ij} = q for i < j, and let a_{c} be the c-tuple (a, …, a). Then we denote the corresponding quantum complete intersection A_{q}^{a_{c}} by A_{q}^{c}, i.e.

A_{q}^{c} = k⟨x_{1}, …, x_{c}⟩/({x_{i}^{a}}, {x_{i}x_{j} − qx_{j}x_{i}}_{i<j}).

Given any c-tuple λ = (λ_{1}, …, λ_{c}) ∈ k^{c}, denote the element λ_{1}x_{1} + ⋯ + λ_{c}x_{c} ∈ A_{q}^{c} by u_{λ}, and let k[u_{λ}] be the subalgebra of A_{q}^{c} generated by this element. Then u_{λ}^{a} = 0 by [BEH, Lemma 2.3], and the rank variety of an A_{q}^{c}-module M, denoted V_{A_{q}^{c}}^{r}(M), is defined as

V_{A_{q}^{c}}^{r}(M) =^{def} {0} ∪ {0 ≠ λ ∈ k^{c} | M is not a projective k[u_{λ}]-module}.

When λ is nonzero, then since u_{λ}^{a} = 0, the subalgebra k[u_{λ}] is isomorphic to the truncated polynomial ring k[x]/(x^{a}). Therefore, the requirement that an A_{q}^{c}-module M is not k[u_{λ}]-projective is equivalent to the requirement that the k-linear map M ⟶^{·u_{λ}} M satisfies

dim Im(·u_{λ}) < ((n − 1)/n) dim M.

This explains the choice of terminology and shows that, like support varieties, rank varieties are homogeneous affine subsets of k^{c}.

Our aim is to relate the rank variety V_{A_{q}^{c}}^{r}(M) of an A_{q}^{c}-module M to its support variety V_{HH^{2∗}(A_{q}^{c})}(M). We do this by exploiting the structure of the Ext-algebra of the simple A_{q}^{c}-module k. These algebras were determined for all quantum complete intersections in [BeO, Theorem 5.3]. For A_{q}^{c}, it is given by

Ext_{A_{q}^{c}}^{∗}(k, k) = k⟨y_{1}, …, y_{c}, z_{1}, …, z_{c}⟩/I,

where I is the ideal generated by the relations

y_{j}y_{i} + qy_{i}y_{j} for all i < j
y_{i}z_{j} − z_{j}y_{i} for all i, j
z_{i}z_{j} − z_{j}z_{i} for all i, j
y_{i}^{2} − z_{i}  for all i if a = 2
y_{i}^{2}  for all i if a ≠ 2

and where the homological degrees of y_{i} and z_{i} are |y_{i}| = 1 and |z_{i}| = 2. We see that the commutative subalgebra of Ext_{A_{q}^{c}}^{∗}(k, k) generated by z_{1}, …, z_{c} is the polynomial ring k[z_{1}, …, z_{c}]. Now our algebra A_{q}^{c} is ℤ^{c}-graded, with the generator x_{i} in degree e_{i}, where e_{i} denotes the ith unit vector (0, …, 0, 1, 0, …, 0). This grading induces an internal grading on Ext_{A_{q}^{c}}^{n}(k, k) for all n, and from the paragraph preceding [Opp, Corollary 3.5] it follows that the internal degree of z_{i} in Ext_{A_{q}^{c}}^{2}(k, k) is −ae_{i}. By [Opp, Corollary 3.5], for each 1 ≤ i ≤ c, every element in Ext_{A_{q}^{c}}^{2}(k, k) of internal

degree −ae_{i} is contained in the image of the ring homomorphism HH^{∗}(A_{q}^{c}) ⟶^{φ_{k}} Ext_{A_{q}^{c}}^{∗}(k, k). Therefore there exists a polynomial subalgebra H = k[η_{1}, …, η_{c}] of HH^{2∗}(A_{q}^{c}), with the property that |η_{i}| = 2 and φ_{k}(η_{i}) = z_{i} for all i.

Let H be such a polynomial subalgebra of HH^{2∗}(A_{q}^{c}). Then for all A_{q}^{c}-modules M, the support varieties V_{H}(M) and V_{HH^{2∗}(A_{q}^{c})}(M) are isomorphic. To see this, denote by R the polynomial subalgebra of Ext_{A}^{∗}(k, k) generated by the z_{i}. For each A-module M, the set

{𝔪 ∈ MaxSpec R | Ann_{R} Ext_{A}^{∗}(M, k) ⊆ 𝔪}

is an affine subset of MaxSpec R, namely the projective relative support variety of M with respect to R (introduced in [BeS]). Since φ_{k}(H) = φ_{k}(HH^{2∗}(A)) = R, it follows from [BeS, Proposition 3.4] that the above affine subset of MaxSpec R is isomorphic to both V_{H}(M) and V_{HH^{2∗}(A)}(M). This shows that V_{H}(M) and V_{HH^{2∗}(A)}(M) are isomorphic.

3. The Avrunin-Scott theorem

Throughout this section, we let k, a, c and q be as in the previous section, and we denote the algebra A_{q}^{c} by A. We fix a polynomial subalgebra H = k[η_{1}, …, η_{c}] of HH^{2∗}(A), with η_{i} ∈ HH^{2}(A) and φ_{k}(η_{i}) = z_{i} ∈ Ext_{A}^{2}(k, k). Moreover, we identify the maximal ideals of H with the points of k^{c}. Finally, given a nonzero point λ = (λ_{1}, …, λ_{c}) ∈ k^{c}, we denote the corresponding line in k^{c} by ℓ_{λ}, and the element λ_{1}x_{1} + ⋯ + λ_{c}x_{c} ∈ A by u_{λ}.

As we saw in the previous section, the support varieties V_{H}(M) and V_{HH^{2∗}(A)}(M) are isomorphic. The analogue of the Avrunin-Scott theorem we prove relates V_{A}^{r}(M) to V_{H}(M) for all A-modules M, and therefore also V_{A}^{r}(M) to V_{HH^{2∗}(A)}(M). We do this is in a number of steps, the first of which is the following result. It provides a stable map description of rank varieties.

Proposition 3.1. Given a nonzero point λ ∈ k^{c}, the implications

ℓ_{λ} ⊆ V_{A}^{r}(M) ⇔ \underline{Hom}_{A}(Au_{λ}, M) ≠ 0 ⇔ \underline{Hom}_{A}(Au_{λ}^{a−1}, M) ≠ 0

hold for every A-module M.

Proof. Applying Hom_{A}(−, M) to the right exact sequence

A ⟶^{·u_{λ}^{a−1}} A ⟶^{·u_{λ}} Au_{λ} → 0

gives the left exact sequence

0 → Hom_{A}(Au_{λ}, M) → M ⟶^{·u_{λ}^{a−1}} M

of vector spaces. From this sequence we obtain the isomorphism Hom_{A}(Au_{λ}, M) ≃

Ker(M ⟶^{·u_{λ}^{a−1}} M), and interchanging the roles of the two elements u_{λ} and u_{λ}^{a−1}, we obtain the isomorphism Hom_{A}(Au_{λ}^{a−1}, M) ≃ Ker(M ⟶^{·u_{λ}} M). Next, consider the two exact sequences

0 → Au_{λ}^{a−1} ↪ A ⟶^{·u_{λ}} Au_{λ} → 0

0 → Au_{λ} ↪ A ⟶^{·u_{λ}^{a−1}} Au_{λ}^{a−1} → 0,

which show that the modules Au_{λ}^{a−1} and Au_{λ} are syzygies of each other. Applying Hom_{A}(−, M) to the first sequence, we obtain the long exact sequence

0 → Hom_{A}(Au_{λ}, M) → M → Hom_{A}(Au_{λ}^{a−1}, M) → Ext_{A}^{1}(Au_{λ}, M) → 0

of vector spaces. The isomorphisms we obtained above, together with the isomorphisms Ext_{A}^{1}(Au_{λ}, M) ≃ \underline{Hom}_{A}(Ω_{A}^{1}(Au_{λ}), M) ≃ \underline{Hom}_{A}(Au_{λ}^{a−1}, M), give the equality

dim Ker(·u_{λ}^{a−1}) + dim Ker(·u_{λ}) = dim M + dim \underline{Hom}_{A}(Au_{λ}^{a−1}, M).

Similarly, by applying Hom_{A}(−, M) to the other short exact sequence, we obtain the equality

dim Ker(·u_{λ}) + dim Ker(·u_{λ}^{a−1}) = dim M + dim \underline{Hom}_{A}(Au_{λ}, M).

By [ErH, Remarks 3.2(ii)], the requirement that M is not a projective k[u_{λ}]-module is equivalent to the requirement

dim Ker(·u_{λ}) + dim Ker(·u_{λ}^{a−1}) > dim M,

and so the result follows. □

Our next aim is to determine the support variety V_{H}(Au_{λ}) for all nonzero points λ ∈ k^{c}. We start with the following general result. Recall that a k-algebra Λ is Frobenius if, as a left module over itself, it is isomorphic to D(Λ), where D denotes the vector space dual Hom_{k}(−, k).

Lemma 3.2. Let Λ and Γ be two finite dimensional k-algebras, with Λ selfinjective and Γ Frobenius. Furthermore, let B be a Γ-Λ-bimodule which is projective both as a Γ-module and as a Λ-module. Then for every Λ-module X and every Γ-module Y, there is a natural isomorphism

Ext_{Λ}^{n}(X, D(B) ⊗_{Γ} Y) ≃ Ext_{Γ}^{n}(B ⊗_{Λ} X, Y)

for each n ≥ 0.

Proof. Adjointness gives a natural isomorphism

Hom_{Λ}(X, Hom_{Γ}(B, Y)) ≃ Hom_{Γ}(B ⊗_{Λ} X, Y)

for every Λ-module X and every Γ-module Y. The functor Hom_{Γ}(B, −) is isomorphic to Hom_{Γ}(B, Γ ⊗_{Γ} −), which in turn is isomorphic to Hom_{Γ}(B, Γ) ⊗_{Γ} − since B is a projective Γ-module. Now Γ, being Frobenius, is isomorphic as a left Γ-module to D(Γ), and so adjointness gives

Hom_{Γ}(B, Γ) ≃ Hom_{Γ}(B, D(Γ)) ≃ D(Γ ⊗_{Γ} B) ≃ D(B).

Therefore, the functor Hom_{Γ}(B, −) is isomorphic to D(B) ⊗_{Γ} −, proving the case n = 0. For the case n > 0, note that since B is a projective Λ-module, the Λ-module D(B) ⊗_{Γ} P is projective for every projective Γ-module P. However, since both Λ and Γ are selfinjective, this means that D(B)⊗_{Γ}I is an injective Λ-module for every injective Γ-module I. Moreover, given a Γ-module Y with an injective resolution I, the complex D(B) ⊗_{Γ} I is an injective resolution of the Λ-module D(B) ⊗_{Γ} Y, since D(B) is a projective right Γ-module. Therefore the isomorphisms

Ext_{Γ}^{n}(B ⊗_{Λ} X, Y) ≃ H^{n} (Hom_{Γ}(B ⊗_{Λ} X, I))
≃ H^{n} (Hom_{Λ}(X, D(B) ⊗_{Γ} I))
≃ Ext_{Λ}^{n}(X, D(B) ⊗_{Γ} Y)

hold for every Λ-module X. □

In order to determine the support variety of Au_{λ}, we exploit some nice properties of certain bimodules arising from elements of the Hochschild cohomology ring. Let

ζ be a homogeneous element of HH^{∗}(A), represented by a map Ω_{A^{e}}^{|ζ|}(A) ⟶^{f_{ζ}} A, say. Then ζ corresponds to the bottom short exact sequence in the exact commutative pushout diagram

[figure: 0; Ω_{A^{e}}^{|ζ|}(A); f_{ζ}; P_{|ζ|−1}; Ω_{A^{e}}^{|ζ|−1}(A); 0; 0; A; K_{ζ}; Ω_{A^{e}}^{|ζ|−1}(A); 0]

where P_{|ζ|−1} is the projective cover of Ω_{A^{e}}^{|ζ|−1}(A). If ζ is an element of H and M is an A-module, then by [EHSST, Proposition 4.3] the support variety of the A-module K_{ζ} ⊗_{A} M is given by

V_{H}(K_{ζ} ⊗_{A} M) = V_{H}(ζ) ∩ V_{H}(M),

where V_{H}(ζ) = {α ∈ k^{c} | ζ(α) = 0}. In the following result, we use the pushout bimodule K_{ζ} to give a criterion for when two lines in k^{c} are perpendicular. Given a nonzero point μ ∈ k^{c}, we denote the set {α ∈ k^{c} | ∑ α_{i}μ_{i} = 0} by ℓ_{μ}^{⊥}; this is the hyperplane perpendicular to the line ℓ_{μ}.

Lemma 3.3. Let M be a periodic A-module of period 1, and suppose V_{H}(M) is a single line ℓ_{α}, where α ∈ k^{c} is a nonzero point. Then, given any nonzero point μ ∈ k^{c}, the implication

\underline{Hom}_{A}(M, K_{ζ} ⊗_{A} k) ≠ 0 ⇒ ℓ_{α} ⊆ ℓ_{μ}^{⊥}

holds, where ζ = ∑ μ_{i}η_{i} ∈ H^{2}.

Proof. Since the bimodule K_{ζ} is projective both as a left and as a right A-module, we see from the previous lemma that \underline{Hom}_{A}(M, K_{ζ} ⊗_{A} k) is naturally isomorphic to \underline{Hom}_{A}(D(K_{ζ})⊗_{A}M, k). Thus \underline{Hom}_{A}(M, K_{ζ} ⊗_{A}k) is nonzero if and only if the same holds for \underline{Hom}_{A}(D(K_{ζ}) ⊗_{A} M, k), and this is equivalent to D(K_{ζ}) ⊗_{A} M not being a projective A-module. The latter happens if and only if V_{H}(D(K_{ζ}) ⊗_{A} M) ≠ 0.

Using the previous lemma once more, we see that the H-modules Ext_{A}^{∗}(D(K_{ζ})⊗_{A} M, k) and Ext_{A}^{∗}(M, K_{ζ} ⊗_{A} k) are isomorphic. Therefore

V_{H}(D(K_{ζ}) ⊗_{A} M) = V_{H}(M, K_{ζ} ⊗_{A} k)
⊆ V_{H}(M) ∩ V_{H}(K_{ζ} ⊗_{A} k)
= V_{H}(M) ∩ V_{H}(ζ)
= ℓ_{α} ∩ ℓ_{μ}^{⊥},

and so if \underline{Hom}_{A}(M, K_{ζ} ⊗_{A} k) is nonzero then ℓ_{α} ⊆ ℓ_{μ}^{⊥}. □

Next, define a map k^{c} ⟶^{F} k^{c} of affine spaces by

(α_{1}, …, α_{c}) ↦ (α_{1}^{a}, …, α_{c}^{a}).

Our aim now is to show that V_{H}(Au_{λ}) = ℓ_{F (λ)} for every nonzero point λ ∈ k^{c}. In order to prove this, we need the following lemma.

Lemma 3.4. Let μ and λ be nonzero points in k^{c} with ℓ_{μ} ⊆ ℓ_{F (λ)}^{⊥}, and denote the element ∑ μ_{i}η_{i} in H by ζ. Then there exists a monomorphism

Au_{λ} → K_{ζ} ⊗_{A} k,

and consequently \underline{Hom}_{A}(Au_{λ}, K_{ζ} ⊗_{A} k) is nonzero.

Proof. Note that we have implicitly assumed that c is at least 2. Denote the radical of A by 𝔯. The second syzygy Ω_{A}^{2}(k) is the kernel of the projective cover A^{c} → 𝔯, which maps the ith generator ε_{i} of the free A-module A^{c} to x_{i}. The generators of Ω_{A}^{2}(k) are the two sets

{x_{i}^{a−1}ε_{i}}_{i=1}^{c}, {qx_{j}ε_{i} − x_{i}ε_{j}}_{i<j},

and the element z_{i} ∈ Ext_{A}^{2}(k, k) is represented by the homomorphism Ω_{A}^{2}(k) ⟶^{f_{z_{i}}} k mapping x_{i}^{a−1}ε_{i} to 1 and all the other generators to zero. In the diagram

\xymatrix{0 \ar[r] & Ω_{A}^{2}(k) \ar[d]^{f_{z_{i}}} \ar[r] & A^{c} \ar@{.>}[d]^{f_{i}} \ar[r] & Ω_{A}^{1}(k) \ar@{.>}[d]^{Ω_{A}^{−1}(f_{z_{i}})} \ar[r] & 0 \\ 0 \ar[r] & k \ar[r]^{·(∏ x_{i}^{a−1})} & A \ar[r]^{π} & Ω_{A}^{−1}(k) \ar[r] & 0}

the map f_{i}, given by

e_{j} ↦ \{ \begin{matrix}q^{i−1} ∏_{n≠i} x_{n}^{a−1} & when j = i \\ 0 & when j ≠ i, \end{matrix}

provides the first step in a lifting of f_{z_{i}}. Thus the map Ω_{A}^{−1}(f_{z_{i}}) factorizes as Ω_{A}^{−1}(f_{z_{i}}) = πg_{i}, where A ⟶^{π} Ω_{A}^{−1}(k) is the natural quotient map and Ω_{A}^{1}(k) ⟶^{g_{i}} A is the map given by

x_{j} ↦ \{ \begin{matrix}q^{i−1} ∏_{n≠i} x_{n}^{a−1} & when j = i \\ 0 & when j ≠ i. \end{matrix}

Now we use the fact that K_{ζ} ⊗_{A} k is the pullback of

\xymatrix{ & Ω_{A}^{1}(k) \ar[d]^{Ω_{A}^{−1}(∑ μ_{i}f_{z_{i}})} \\ A \ar[r]^{π} & Ω_{A}^{−1}(k)}

to obtain a monomorphism Au_{λ} → K_{ζ}⊗_{A}k. We do this by constructing a monomor-

phism Au_{λ} ⟶^{f} A ⊕ Ω_{A}^{1}(k) whose composition with A ⊕ Ω_{A}^{1}(k) \underrightarrow{(π,Ω_{A}^{−1}(∑ μ_{i}f_{z_{i}}))} Ω_{A}^{−1}(k) is zero. Namely, define f by

u_{λ} ↦ (∑_{i=1}^{c} μ_{i}g_{i}(u_{λ}), −u_{λ}).

To show that this map is well defined, we need to check that f(u_{λ}^{a}) = 0, that is, that the element u_{λ}^{a−1}f(u_{λ}) is zero. Computing directly, we see that

u_{λ}^{a−1} ∑_{i=1}^{c} μ_{i}g_{i}(u_{λ}) = u_{λ}^{a−1} ∑_{i=1}^{c} (μ_{i}λ_{i}q^{i−1} ∏_{n≠i} x_{n}^{a−1})

= ∑_{i=1}^{c} λ_{i}^{a−1}x_{i}^{a−1} (μ_{i}λ_{i}q^{i−1} ∏_{n≠i} x_{n}^{a−1})

= (∑_{i=1}^{c} μ_{i}λ_{i}^{a}) ∏_{j=1}^{c} x_{j}^{a−1}
= 0

since ℓ_{μ} ⊆ ℓ_{F (λ)}^{⊥}. Thus the map f is well defined, and it is obviously a monomorphism. □

We are now ready to prove that V_{H}(Au_{λ}) = ℓ_{F (λ)} for every nonzero point λ ∈ k^{c}.

Proposition 3.5. If λ ∈ k^{c} is nonzero, then V_{H}(Au_{λ}) = ℓ_{F (λ)}.

Proof. Denote by T the A-module Au_{λ} ⊕ Au_{λ}^{a−1}. This is a periodic module of period 1, and its support variety equals that of Au_{λ} since Au_{λ} and Au_{λ}^{a−1} are syzygies of each other. Moreover, the module Au_{λ} is indecomposable, hence its support variety is a single line. Let therefore α ∈ k^{c} be a nonzero point such that V_{H}(Au_{λ}) = ℓ_{α}. Furthermore, let μ ∈ k^{c} be a nonzero point such that ℓ_{μ} ⊆ ℓ_{F (λ)}^{⊥}, and denote the element μ_{1}η_{1} + ⋯ + μ_{c}η_{c} ∈ H by ζ. By the previous lemma, the space \underline{Hom}_{A}(T, K_{ζ} ⊗_{A}k) is nonzero, and so from Lemma 3.3 we obtain the inclusion ℓ_{α} ⊆ ℓ_{μ}^{⊥}. Therefore ℓ_{α} = ℓ_{F (λ)}. □

Using Proposition 3.1 and Proposition 3.5, we now prove our main result, namely the quantum complete intersection analogue of the Avrunin-Scott theorem. It shows that the map k^{c} ⟶^{F} k^{c} maps the rank variety of a module onto its support variety.

Theorem 3.6. F (V_{A}^{r}(M)) = V_{H}(M) for every A-module M.

Proof. The result is clearly true if M is projective. If M is not projective, let λ ∈ k^{c} be a nonzero point whose corresponding line ℓ_{λ} is contained in V_{A}^{r}(M). Then \underline{Hom}_{A}(Au_{λ}, M) and \underline{Hom}_{A}(Au_{λ}^{a−1}, M) are both nonzero by Proposition 3.1, and so Ext_{A}^{i}(Au_{λ}, M) ≠ 0 for i > 0. This implies that V_{H}(Au_{λ}, M) is nontrivial, and since this variety is contained in V_{H}(Au_{λ}) ∩ V_{H}(M), we see from Proposition 3.5 that the line ℓ_{F (λ)} must be contained in V_{H}(M). Consequently, the inclusion F (V_{A}^{r}(M)) ⊆ V_{H}(M) holds.

To prove the reverse inclusion, we argue by induction on the dimension of the support variety of M. We may assume that M is indecomposable, since the variety (rank or support) of a direct sum is the union of the varieties of the summands. Suppose first that the dimension of V_{H}(M) is one, i.e. that M is periodic. Then V_{H}(M) is a line, and we proved above that this variety contains F (V_{A}^{r}(M)). If F (V_{A}^{r}(M)) = 0, then V_{A}^{r}(M) = 0, which is not the case since M is not projective. Hence F (V_{A}^{r}(M)) is nontrivial, and so F (V_{A}^{r}(M)) = V_{H}(M) in this case. Next, suppose that dim V_{H}(M) > 1, and let μ ∈ k^{c} be a nonzero point such that dim (ℓ_{μ}^{⊥} ∩ V_{H}(M)) < dim V_{H}(M). Denote the element μ_{1}η_{1} + ⋯ + μ_{c}η_{c} ∈ H by

ζ, and let Ω_{A^{e}}^{2}(A) ⟶^{f_{ζ}} A be a bimodule map representing this element. Since the bottom exact sequence in the pushout diagram

[figure: 0; Ω_{A^{e}}^{2}(A); f_{ζ}; P_{1}; Ω_{A^{e}}^{1}(A); 0; 0; A; K_{ζ}; Ω_{A^{e}}^{1}(A); 0]

splits as right A-modules, it stays exact after tensoring with M. This gives a short exact sequence

0 → M → K_{ζ} ⊗_{A} M → Ω_{A}^{1}(M) ⊕ P → 0,

in which P is projective, and so

V_{A}^{r}(K_{ζ} ⊗_{A} M) ⊆ V_{A}^{r}(M) ∪ V_{A}^{r}(Ω_{A}^{1}(M) ⊕ P) = V_{A}^{r}(M).

Now since V_{H}(K_{ζ} ⊗_{A} M) = V_{H}(ζ) ∩ V_{H}(M) = ℓ_{μ}^{⊥} ∩ V_{H}(M), induction gives V_{H}(K_{ζ} ⊗_{A} M) ⊆ F (V_{A}^{r}(K_{ζ} ⊗_{A} M)). Therefore ℓ_{μ}^{⊥} ∩ V_{H}(M) ⊆ F (V_{A}^{r}(M)), and since this inclusion holds for every nonzero point μ ∈ k^{c} such that dim (ℓ_{μ}^{⊥} ∩ V_{H}(M)) is strictly less than dim V_{H}(M), we see that V_{H}(M) ⊆ F (V_{A}^{r}(M)). □

Consequently, the dimension of the rank variety of an A-module is the complexity of the module. In particular, an indecomposable module is periodic if and only if its rank variety is of dimension one.

Corollary 3.7. For every A-module M, the dimension of V_{A}^{r}(M) is the complexity of M.

Corollary 3.8. An indecomposable A-module M is periodic if and only if the dimension of V_{A}^{r}(M) is one.

References

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Petter Andreas Bergh, Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no

Karin Erdmann, Mathematical Institute, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom

   E-mail address: erdmann@maths.ox.ac.uk