On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity

Abstract

This paper provides a method to get a noetherian equicharacteristic local UFD with an isolated singularity from a given noetherian complete equicharacteristic local ring, preserving certain properties. This is applied to invesitgate the (non)vanishing of Ext modules. It is proved that there exist a Gorenstein local UFD $A$ having an isolated singularity such that ${\mathrm{Ext}}_A^{\gg 0}(M,N)=0$ does not imply ${\mathrm{Ext}}_A^{\gg 0}(N,M)=0$, a Gorenstein local UFD $B$ having an isolated singularity such that ${\mathrm{Tor}}_{>0}^B(M,N)=0$ does not imply $\mathrm{depth}(M{\otimes }_{B}N)=\mathrm{depth}M+\mathrm{depth}N-\mathrm{depth}B$, and a Cohen-Macaulay local UFD $C$ having an isolated singularity such that ${\mathrm{Ext}}_C^{>0}(M,C)=0$ does not imply the total reflexivity of $M$.

v1: 7 pages. v2: 10 pages, results on the depth formula added