On invariants of complete intersections
type: Item
instance of: scholarly article
title: On invariants of complete intersections
zbMATH Open document ID: 1275.13011
author: Sandra Spiroff
published in: Michigan Mathematical Journal
publication date: 2013-06-06T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.mmj/1363958247
review text: This paper introduces and studies properties of an invariant, denoted \(\Theta^R_c\), for standard graded complete intersections with an isolated singularity. Let \(R = k[x_0, \dots, x_{n+c-1}]/(f_1, \dots, f_c)\), where \(k\) is a separably closed field, \(\deg x_j = 1\) for each \(j\), and each \(f_i\) is a homogeneous polynomial such that \(f_1, \dots, f_c\) forms a regular sequence. In addition, let \(M\) and \(N\) be finitely graded \(R\)-modules. The author shows that ``if the tensor product of a graded pair \(M, N\) has finite length, then \(\Theta^R_c(M, N) = 0\) if and only if \(\dim M + \dim N \leq \dim R\). Moreover, \(\dim M + \dim N \leq \dim R + 1\) regardless of the value of the invariant \(\Theta^R_c(M, N)\).'' This generalizes work on Hochster's invariant \(\theta^R\) for a ring \(R\) which is the quotient of a regular local ring by a regular element, and in particular work found in the papers \textit{H. Dao} [Math. Res. Lett. 15, No. 2--3, 207--219 (2008; Zbl 1229.13014)], \textit{H. Dao} [``Decency and rigidity over hypersurfaces'', \url{arXiv:math/0611568}, to appear in Trans. Am. Math. Soc.], \textit{Y. Kobayashi} [Math. Jap. 24, 643--655 (1980; Zbl 0434.13009)], and \textit{W. F. Moore}, \textit{G. Piepmeyer}, \textit{S. Spiroff} and \textit{M. E. Walker} [Adv. Math. 226, No. 2, 1692--1714 (2011; Zbl 1221.13027)]. The first section of the paper defines \(\Theta_c^R(M, N)\) and provides a geometric description of this invariant. This establishes that \(\Theta^R_c\) shares many of the same properties as Hochster's theta function. A useful reference for these results is given by \textit{W. F. Moore}, \textit{G. Piepmeyer}, \textit{S. Spiroff} and \textit{M. E. Walker} [Math. Z. 273, No. 3--4, 907--920 (2013; Zbl 1278.13013)]. The second section of the paper is dedicated to an investigation of the expected dimension of the intersection of the modules \(M\) and \(N\). An interesting result (Theorem 2.4) relates \(\Theta_c^R(M, N)\) with ``a generali…
reviewed by: Susan Marie Cooper
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14M10
zbMATH DE Number: 6171221
zbMATH Keywords: Hochster's theta function
zbMATH Keywords: complete intersections
zbMATH Keywords: Bézout's Theorem
MaRDI profile type: Publication
OpenAlex ID: W2058737618
cites work: Q4694939
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DOI: 10.1307/MMJ/1363958247
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description: scientific article; zbMATH DE number 6171221