Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two
type: Item
instance of: scholarly article
title: Huneke-Ulrich almost complete intersections of Cohen-Macaulay type two
zbMATH Open document ID: 0826.13015
author: Andrew Kustin
published in: Journal of Algebra
publication date: 1995-11-29T00:00:00Z
review text: Let \((R, {\mathfrak m}, k)\) be a commutative, local, noetherian ring, \(X_{(2n + 1) \times (2n + 1)}\) be an alternating matrix and \(Y_{1 \times (2n + 1)}\) be a matrix, both with entries from \({\mathfrak m}\), and \(I\) be the ideal generated by the entries of \(YX\). Assume \(n \geq 2\), and \(I\) has grade at least \(2n\). Then \(I\) has grade exactly \(g = 2n\) and \(I\) is minimally generated by \(g + 1\) elements (hence \(I\) is a perfect almost complete intersection). If \(R\) is Gorenstein, then \(I\) has Cohen-Macaulay type two and is in the linkage class of a complete intersection. The author shows that the minimal free resolution of \(R/I\) over \(R\) is a \(DG \Gamma\)-algebra, that is, an associative, differential, graded algebra with divided powers. -- The proof is an explicit construction of a suitable multiplication on a minimal free resolution of \(R/I\). As an application, if \(R\) is regular and \(\text{char} k \geq (n + 2)/2\) or \(\text{char} k = 0\), then the Poincaré series \(P^M_{R/I} (z) = \sum_{i \geq 0} \dim_k \text{Tor}_i^{R/I} (M,k)z^i\) is a rational function for every finitely generated module \(M\). It follows that if \(\text{pd} M\) is infinite, then the Betti numbers of \(M\) eventually increase with strong exponential growth.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 783869
zbMATH Keywords: \(DG\Gamma\)-algebra
zbMATH Keywords: rationality of Poincaré series
zbMATH Keywords: Gorenstein
zbMATH Keywords: linkage class
zbMATH Keywords: complete intersection
zbMATH Keywords: minimal free resolution
zbMATH Keywords: divided powers
zbMATH Keywords: Betti numbers
MaRDI profile type: Publication
OpenAlex ID: W2067188264
DOI: 10.1006/JABR.1995.1132
description: scientific article; zbMATH DE number 783869