Support varieties and cohomology over complete intersections
type: Item
instance of: scholarly article
title: Support varieties and cohomology over complete intersections
zbMATH Open document ID: 0999.13008
author: Luchezar L. Avramov
author: Ragnar-Olaf Buchweitz
published in: Inventiones Mathematicae
publication date: 2001-03-12T00:00:00Z
full work available at URL: http://hdl.handle.net/1807/9930
review text: 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 …
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 1546913
zbMATH Keywords: minimal free resolution
zbMATH Keywords: complexity
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: cohomology
zbMATH Keywords: local complete intersection ring
zbMATH Keywords: Gorensteinness
describes a project that uses: Macaulay2
MaRDI profile type: Publication
OpenAlex ID: W1980527624
DOI: 10.1007/S002220000090
description: scientific article; zbMATH DE number 1546913