Hochster's Theta Pairing and Algebraic Equivalence

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.