Asymptotic vanishing conditions which force regularity in local rings of prime characteristic

Details Asymptotic vanishing conditions which force regularity in local rings of prime characteristic Asymptotic vanishing conditions which force regularity in local rings of prime characteristic

2007-10-22

arxiv.org/abs/0710.4090v1

Mathematics

Commutative Algebra

6 pages

Scientific paper

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8} {\length(H_i(F^e(G_\dt)))}/{p^{ed}}$. We show that if $\gothic t_i(R) = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\Tor_i^R(k,R^+) = 0$ for some $i>0$, then $R$ is regular (where $R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i = 1$ or 2 via completely different methods.

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