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

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.