On invariants of complete intersections
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 generalized Bézout's theorem relating the degrees of the modules, their associated homology modules, and the ambient ring''. The third section of the paper studies an invariant \(\eta^R_c(M, N)\). This invariant was introduced by Dao and is known to differ from \(\theta^R\). The author also ties \(\eta^R_c\) to a generalized Bézout's theorem. The paper includes useful examples and references to the literature.
scientific article; zbMATH DE number 6171221
scientific article