Modules with reducible complexity (Q876340)

Let \(A\) be a commutative noetherian local ring and all modules are supposed to be finitely generated. For an \(A\)-module \(M\), the complexity of \(M\), denoted by \(\text{cx}\,M\), is defined as \(\text{cx}\,M = \inf\{ t \in \mathbb N_0 \mid \exists a \in \mathbb R \text{ such that } \beta_n(M) \leq an^{t-1} \text{ for } n\gg0\}\), where \(\beta_n(M)\) is the \(n\)-th Betti number of \(M\). For \(A\)-modules \(X,Y\) and \(\eta \in \text{Ext}_A^n(X,Y)\), choose a map \(f_\eta : \varOmega_A^n(X) \to Y\) representing \(\eta\) and denote by \(K_\eta\) the pushout of this map and the inclusion \(\varOmega_A^n(X) \to P_{n-1}\), where \(P_\bullet\) is the minimal free resolution of \(X\) and \(\varOmega_A^n(X)\) is the \(n\)-th syzygy of \(X\). Let \(\mathcal C_A\) denote the category of all \(A\)-modules having finite complexity. The subcategory \(\mathcal C_A^r \subseteq \mathcal C_A\) of modules having reducible complexity is defined inductively as follows: (i) Every module of complexity zero belongs to \(\mathcal C_A^r\). (ii) A module \(X \in \mathcal C_A\) with \(\text{cx}\,X > 0\) belongs to \(\mathcal C_A^r\) if there exists an element \(\eta \in \text{Ext}_A^n(X,X)\) with \(n > 0\) such that \(\text{cx}\,K_\eta < \text{cx}\,X\), \(\text{depth}\,K_\eta = \text{depth}\,X\) and \(K_\eta \in \mathcal C_A^r\). (It is said that the element \(\eta\) reduces the complexity of \(M\).) Various properties of \(\mathcal C_A^r\) are given in section\,2, and in section\,3 some results on the vanishing of Ext and Tor functors are shown. For example: Theorem 3.1. Let \(M,N\) be nonzero \(A\)-modules and assume \(M \in \mathcal C_A^r\). Then \(\sup\{\, n \,| \, \text{Ext}_A^n(M,N) \neq 0\} = \text{depth}\,A - \text{depth}\,M\) if \(\text{Ext}_A^n(M,N) = 0\) for \(t \leq n \leq t+s\) with some integers \(t > \text{depth}\,A - \text{depth}\,M\) and \(s\) being described by the degrees of elements reducing \(\text{cx}\,M\) to \(0\). Theorem 3.5. The symmetry in the vanishing of Ext holds for modules belonging to \(\mathcal C_A^r\) if \(A\) is Gorenstein. In section\,4, a generalization is discussed.