Finiteness of $\bold{\bigcup_e \Ass F^e(M)}$ and its connections to tight closure

Abstract

The paper shows that if the set of associated primes of Frobenius powers of ideals or a closely related set of primes is finite then if tight closure does not commute with localisation one can find a counter-example where $R$ is complete local and we are localizing at a prime ideal $P \subset R$ with $\dim (R/P)=1$. If one assumes further that for any local ring $(R,m)$ of prime characteristic $p$ and every finitely generated $R$-module $\bar M$ the set $ \bigcup_e \Ass G^e (\bar M) $ has finitely many maximal elements and, in addition, for every $R$-module $\bar M$ there exists a positive integer $B>0$ such that $m^{qB}$ kills $\H_m^0(F^e(\bar M))$ (or $\H_m^0(G^e(\bar M))$) then it is shown tight closure commutes with localization. The author then produces an example of an ideal in an hypersurface whose union of sets associated primes of all its Frobenius powers form an infinite set.