Support varieties and cohomology over complete intersections
This paper develops geometric methods for the study of finite modules over a local complete intersection ring \(R\). For any pair of finite \(R\)-modules \(M\) and \(N\), the authors introduce a support variety \(V_R^*(M,N) \subset \widetilde k^c\), where \(\widetilde k\) denotes an algebraic closure of the residue field \(k\) and \(c\) is the codimension of \(R\). This notion enjoys the following interesting properties: (1) \(V_R^*(M,M) = V_R^*(M,k) = V_R^*(k,M) =: V_R^*(M)\), (2) \(V_R^*(M,N) = [0]\) if and only if \(\text{Ext}_R^n(M,N) = 0\) for \(n \gg 0\), (3) \(V_R^*(M,N) = V_R^*(M) \cap V_R^*(N) = V_R^*(N,M).\) The case \(M = N\) has been already dealt with by the first author [\textit{L. L. Avramov}, Invent. Math. 96, No. 1, 71-101 (1989; Zbl 0677.13004)]. In the paper under review, the authors show that the defining equations of \(V_R^*(M)\) can be computed from a finite free resolution of \(M\) over a regular ring. The meaning of the support variety lies in the fact that its dimension measures the size of \(\text{Ext}_R^*(M,N)\). In fact, if we define the complexity of the pair \((M,N)\) as the number \(\text{cx}_R(M,N) := \inf\{b \in {\mathbb N}\mid v_R(\text{Ext}_R^n(M,N)) \leq an^{b-1}\) for some real number \(a\) and for all \(n \gg 0\},\) then \(\text{cx}_R(M,N) = \dim V_R^*(M,N)\). Hence the above properties can be translated into numerical data as: \((1')\) \(\text{cx}_R(M,M) = \text{cx}_R(M,k) = \text{cx}_R(k,M) =: \text{cx}_RM\), \((2')\) \(\text{cx}_R(M) = 0\) if and only if \(\text{Ext}_R^n(M,N) = 0\) for \(n \gg 0\), \((3')\) \(\text{cx}_R(M)+\text{cx}_R(N)-\text{codim} R \leq \text{cx}_R(M,N) = \text{cx}_R(N,M) \leq \min\{\text{cx}_RM,\text{cx}_RN\}\). As a consequence, \(\text{Ext}_R^n(M,N) = 0\) for \(n \gg 0\) if and only if \(\text{Ext}_R^n(N,M) = 0\) for \(n \gg 0\). This is a remarkable property of finite modules over a local complete intersection ring. Since this property implies the Gorensteinness, it leads to the question whether the class of rings satisfying this property is equal to either of these two classes of rings. To compare the asymptotic vanishing of Ext's and Tor's the authors show that \(\text{Tor}_n^R(M,N) = 0\) for \(n \gg 0\) if and only if \(V_R^*(M) \cap V_R^*(N) = {0}\). The `only if' part of this result has been proved by \textit{D. A. Jorgensen} [J. Algebra 195, No. 2, 526-537 (1997; Zbl 0898.13008)]. It also extends to arbitrary codimension an earlier result of \textit{C. Huneke} and \textit{R. Wiegand} [Math. Scand. 81, No. 2, 161-183 (1997; Zbl 0908.13010)]. Many results of this paper are established in the more general context of modules of finite CI-dimension introduced by \textit{L. L. Avramov, V. N. Gasharov}, and \textit{I. V. Peeva} [Publ. Math. Inst. Hautes. Étud. Sci. 86, 67-114 (1997; Zbl 0918.13008)]. However, there are known obstacles to extend these results for arbitrary finite modules over local rings.
scientific article; zbMATH DE number 1546913