Hochster's Theta Pairing and Algebraic Equivalence

TL;DR

This paper introduces a modified Hochster's theta pairing, demonstrating its invariance in flat module families over hypersurfaces with isolated singularities, and explores its implications across algebraic equivalence and characteristic settings.

Contribution

It defines a new variant of Hochster's theta pairing and proves its invariance in flat families, linking it to algebraic equivalence and characteristic transfer.

Findings

Theta pairing is constant in flat families over hypersurfaces.

Theta pairing factors through the Grothendieck group modulo algebraic equivalence.

Application to the rigidity of Tor over hypersurfaces.

Abstract

We define a variant of Hochster's theta pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the theta pairing factors through the Grothendieck group modulo algebraic equivalence. Moreover, our result allows us, in certain situations, to translate the properties of the theta pairing in characteristic zero to the characteristic p setting. We also give an application of our result to the rigidity of Tor over hypersurfaces.