Claudia Miller

Frobenius Powers of Complete Intersections

Generalizing Kunz' characterization of regular local rings via flatness of
the Frobenius endomorphism $\phi$, Peskine and Szpiro showed that, for a
local ring $R$ of characteristic $p>0$, if an $R$-module $M$ has finite
projective dimension, then ${\textrm{Tor}}_i(M,{}^{\phi^n}R)$ vanishes for
all $i>0$ and all $n>0$, where ${}^{\phi^n}R$ denotes the ring R
considered as a module over itself via the $n$th composition of the
Frobenius map $\phi$. The converse was shown by Herzog, and various
results have appeared since on how many $i$ and $n$ are really needed. We
will talk about joint work with L.\ Avramov showing that if $R$ is a
complete intersection ring, then the vanishing of just one
${\textrm{Tor}}_i(M,{}^{\phi^n}R)$ is needed to deduce that $M$ has finite
projective dimension. We will also discuss the other result, comparing the
growth of the Tors to that of the Betti numbers of $M$ when $M$ does not
have finite projective dimension.