Periodic resolutions over exterior algebras ✩

In this paper we study modules with periodic free resolutions (that is, periodic modules) over an exterior algebra. We show that any module with bounded Betti numbers (that is, a module whose syzygy modules have a bounded number of generators) must have periodic free resolution of period 2, and that for graded modules the period is 1. We also show that any module with a linear Tate resolution is periodic. We give a criterion of exactness for periodic complexes and a parameterization of the set of periodic modules.  2002 Elsevier Science (USA). All rights reserved. Modules over exterior algebras arise in topology, in the study of sheaves on projective spaces, and in the study of free resolutions over polynomial rings (see, for example, [3,5–8]). In this paper we analyze the simplest modules over exterior algebras, and find a parameterization for the set of these objects. Let K be a field, let V be a K vector space of dimension n + 1, and let E =∧V be the exterior algebra. As E is injective as a module over itself it has no modules of finite projective dimension except free modules. Since E is local, any E-module M has a unique minimal free resolution, and thus the syzygy modules of M are well-defined up to isomorphism. We say that M is periodic (of period t) if M is isomorphic to its t th syzygy. The simplest non-free E-modules are the periodic modules. ✩ The author is grateful to the NSF for partial support during the preparation of this work. E-mail address: [email protected] (D. Eisenbud). 0021-8693/02/$ – see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0021-8693(02) 00 51 12 D. Eisenbud / Journal of Algebra 258 (2002) 348–361 349 To get an idea of what might be true, we recall the case of an exterior algebra on two variables over an algebraically closed field, where modules can be classified completely using the Kronecker–Weierstrass classification of matrix pencils ([12]; see also [9, Chapter XII] or [13, Section 3.2]): Theorem 0.1. If K is algebraically closed and dimV = 2 then every indecomposable, module over E =∧V is either (a) free; or (b) a syzygy of the residue class field or its dual; or (c) periodic of period 1. Any indecomposable periodic module M is self-dual and has minimal presentation matrix of the form   a b 0 . . . 0 0 a b . . . 0 .. . . . . . . . . . .. 0 . . . 0 a b 0 . . . 0 0 a   where a, b ∈ V are independent. The module M is determined up to isomorphism, by its number of generators d (the size of the matrix) and the unique 1-dimensional subspace 〈a〉 such that M is not free as module over the subalgebra E′ :=K + 〈a〉 (the element b may be changed at will to any element independent of a). We can interpret this result by saying that, in the two variable case, periodic modules correspond to sheaves of finite length on P (V ∗)∼= P 1: any such sheaf is a direct sum of the modules of the form OP 1,a/mdP 1,a ; and we let this indecomposable module correspond to the cokernel of the matrix of linear forms of size d as pictured in the proposition. In general we show that any E-module M with syzygies of bounded length is periodic of period at most 2. If M is graded, the period is 1. The Bernstein–Gel’fand–Gel’fand correspondence implies that graded periodic modules correspond to complexes of coherent sheaves of finite length on the projective space P (V ∗) (Theorem 3.1). In particular, an indecomposable graded periodic module corresponds to a single point of projective space and a complex of finite length sheaves supported on an infinitesimal neighborhood of that point, and the category of periodic modules splits into a direct product of subcategories corresponding to the points of projective space. Moreover, every graded periodic module is filtered in such a way that the successive quotients have the formE/(ai) for various linear forms ai ∈E. In two variables every E-module can be graded; but in the general case there are interesting differences between the graded and ungraded modules. For ungraded periodic modules, the minimal period may be 2. Graded or not, any 350 D. Eisenbud / Journal of Algebra 258 (2002) 348–361 periodic module is naturally associated to a periodic module with linear resolution (of period 1), which determines a collection of points in P (W) as above. Although we have no Bernstein–Gel’fand–Gel’fand correspondence in the ungraded case, we show by direct arguments that the category of periodic modules still splits as a direct product of categories of periodic modules corresponding, in the sense above, to single points. We give parameterizations for both the graded and ungraded categories (Theorem 3.3). This similarity of the decomposition in the graded and ungraded cases leads to an interesting speculation. Is there some additional structure on a complex of sheaves of finite length on projective space that would correspond to the deformation from a graded periodic module to an ungraded one? The results in Section 2 of this paper are analogous to results about modules over a commutative “complete intersection” S/(f1, . . . , fc), with S a regular local ring, proved in [4]. (See [2] for generalization.) However, the approach there leans heavily on the regular local ring S so the exterior case requires a different idea. 1. Notation and background Throughout this paper we write K for a fixed field, and V,W for dual vector spaces of finite dimension v over K . We give the elements of W degree 1, and elements of V degree −1. We write E = ∧V and S = Sym(W) for the exterior and symmetric algebras; these algebras are graded by their internal degrees whereby Symi (W) has degree i and ∧j V has degree −j . We suppose all E-modules considered to be finitely generated. We often use the fact that the exterior algebra is Gorenstein, which follows from the fact that HomK(E,K)∼=E as E-modules. As a consequence, the E-dual of any exact sequence is exact. There are well-known algebra isomorphisms E ∼= Ext∗S(K,K) and S ∼= Ext∗E(K,K). This “Koszul duality” is a key ingredient in the Bernstein–Gel’fand– Gel’fand correspondence. We next review the version of this correspondence that is explained in [5] and used below. A complex of free modules over E is called minimal if all the maps can be represented by matrices with entries in the maximal ideal. It is linear if all the entries are linear forms. Given any complex F of free modules, we define its linear part lin(F ) to be the result of taking a minimal complex F ′ homotopic to F and erasing all nonlinear terms from the entries of matrices representing the differentials of F . It is easy to see that lin(F ) is a complex, and the operation F → lin(F ) is functorial in a suitable sense [5, Theorem 3.4]. We have Theorem 1.1 [5, Theorem 3.1]. If F is a free resolution, then lin(F ) is eventually exact. D. Eisenbud / Journal of Algebra 258 (2002) 348–361 351 Let {ei} and {xi} be dual bases of V and W , respectively. We write R for the functor from the category of graded S-modules N = ⊕Ni to the category of linear free complexes over E whose value at N is the complex R(N): · · · −→ HomK(E,Ni) φ −→ Hom(E,Ni+1)−→ · · · where φ(α) : e →∑(xiα(eie)). We have Theorem 1.2 [5, Proposition 2.1]. (a) The functor R is an equivalence between the category of graded S-modules and the category of linear free complexes over E. (b) The linear part of the injective resolution of a finitely generated E-module may be written R(N) with N a finitely generated S-module. (c) If N is any finitely generated module, then R(N) is exact starting from HomK(E,Nr+1) where r is the Castelnuovo–Mumford regularity of N . (d) The functor R can be extended to complexes in such a way that lin(R(F ))= R(H∗(F )). For further background, see [5]. 2. Modules with bounded Betti numbers Let M be a finitely generated E-module. It is easy to show by induction on the length of M that Tor• (M,K) is a finitely generated module over Ext•E(K,K) (such arguments go back at least to the work of Quillen on group cohomology). In particular, the function i → dimK(TorEi (M,K)) is the Hilbert function of a finitely generated S-module. It follows that these Betti numbers of M eventually grow polynomially. Thus the Betti numbers are either eventually constant or eventually strictly increasing, and the syzygies of M are of bounded length if and only if and only if the Betti numbers are eventually constant. In this section we show that in the eventually constant case the resolution is periodic of period at most 2. We will make use of a twisting functor τ taking a (possibly ungraded) E-module M to the E-module τM which is the same as M as a vector space, but on which elements of odd degree in E act by the negative of their action on M . More formally, τ is the functor given by pullback along the algebra homomorphism E → E induced by the map −1 :V → V . It follows from the definition that τ 2 is the identity. The effect of τ on the matrix representing a map of free modules is to change the signs of all the odd degree terms of the entries of the matrix. As τ is exact, it follows that a presentation matrix of a module τM may be obtained from a presentation matrix for M by changing the signs of all odd degree terms of entries in the matrix. The following was pointed out to me by Frank Schreyer: 352 D. Eisenbud / Journal of Algebra 258 (2002) 348–361 Proposition 2.1. If M is a graded E-module then τM ∼=M . Proof. If we change the signs of all the odd degree columns and all the odd degree rows in a presentation matrix for τM , we get the presentation matrix of M; but this change of signs does not change the isomorphism class of the cokernel. ✷ On the other hand, if M is the ungraded module E/(a+ bc), where a, b, c are independent linear forms, then τM ∼=E/(a− bc). This module is not isomorphic to M: it has a different annihilator. Theorem 2.2. Let M be a finitely generated E-module with no free summands. If the syzygies of M have bounded length then the first syzygy of M is τM . In particular, the minimal free resolution of M is periodic, of period at most 2, and if M is graded, then the period is 1. Proof. By a result of Grothendieck [11, Proposition 2.5.8] two modules that become isomorphic after a ground field extension are isomorphic already, so we may assume that the ground field K is infinite. Let F : · · · φ2 −→ F−1 φ1 −→ F 0 −→M −→ 0 be the minimal free resolution of M . For w ∈W , let w⊥ ⊂ V be the annihilator of w. We have an extension ηw(K): 0 →K →E /( w⊥ )→K → 0. Tensoring over K with M and using the diagonal E → E ⊗K E to define the module structure on E/(w⊥)⊗K M we get an extension ηw(M): 0 → τM → ( E /( w⊥ ))⊗K M →M → 0, from which we may derive a map (defined up to homotopy) from F to the shift τF [1] of the resolution of τM: F : · · ·