Frobenius powers of non-complete intersections

Abstract

For a commutative ring $R$ of characteristic $p$, let $\phi : R \to R$ be the Frobenius homomorphism and let $^{\phi^r}R$ denote the $R$-module structure on $R$ defined via the $r$-th power of the Frobenius. We show that the Tor functor against the Frobenius module, $\Tor^R_*(-, {^{\phi^r}}R)$, is rigid for a certain class of depth zero rings which includes rings that are not complete intersection. We also show that $\Tor^R_*(-, {^{\phi^r}}R)$ is not rigid (non-vacuously) when $\depth (R) >0$ and $r$ is large enough. This answers a question of Avramov and Miller: does rigidity of $\Tor^R_*(-, {^{\phi^r}}R)$ hold for non-complete intersections?