Vol. 2, No. 2, 2017

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On the vanishing of Hochster's $\theta$ invariant

Mark E. Walker

Vol. 2 (2017), No. 2, 131–174
Abstract

Hochster’s theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations.

Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster’s theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation.

We give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of even dimension in characteristic p > 0 under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao.

Keywords
matrix factorization, hypersurface, theta invariant
Mathematical Subject Classification 2010
Primary: 13D15, 19M05
Milestones
Received: 30 December 2014
Revised: 2 February 2016
Accepted: 17 February 2016
Published: 14 December 2016
Authors
Mark E. Walker
Department of Mathematics
University of Nebraska-Lincoln
203 Avery Hall
Lincoln, NE 68588
United States