2012/08/30 by Hailong Dao, Dao, Hailong, Kazuhiko Kurano +1
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AC #math.KT
paper · pdf · doi:10.48550/arxiv.1208.6083
arxiv created 2012/08/30 · arxiv updated 2012/08/31
Let (A,\m) be a local hypersurface with isolated singularity. We show that Hochster's theta pairing vanishes on elements that are numerically equivalent to zero in the Grothendieck group of A under the mild assumption that \spec A admits a resolution of singularity. We also prove that when dim A =3, the Hochster's theta pairing is positive semidefinite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. Our method involves showing that theta gives a bivariant class for the morphism \spec A/\m → \spec A. It also follows that if A is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group of A is finitely generated torsion-free.