The vanishing of a higher codimension analog of Hochster's theta invariant

TL;DR

This paper proves that Dao's invariant $ heta_c^R$ vanishes for all module pairs over graded complete intersection rings of codimension greater than one with isolated singularities, implying $c$-$ ext{Tor}$-rigidity.

Contribution

It establishes the vanishing of $ heta_c^R$ for all module pairs over graded complete intersections of codimension > 1 with isolated singularities, a higher codimension analogue of Hochster's invariant.

Findings

$ heta_c^R$ vanishes for all pairs of modules in the specified rings.

All module pairs are $c$-$ ext{Tor}$-rigid in these rings.

The result extends understanding of module behavior over singular complete intersections.

Abstract

We study H. Dao's invariant $\eta_c^R$ of pairs of modules defined over a complete intersection ring $R$ of codimension $c$ having an isolated singularity. Our main result is that $\eta_c^R$ vanishes for all pairs of modules when $R$ is a {\em graded} complete intersection ring of codimension $c > 1$ having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are $c$-$\Tor$-rigid.