• Corpus ID: 118635600

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

  title={Finiteness of \$\bold\{\bigcup\_e \Ass F^e(M)\}\$ and its connections to tight closure},
  author={Mordechai Katzman},
  journal={arXiv: Commutative Algebra},
  • M. Katzman
  • Published 25 September 2002
  • Mathematics
  • arXiv: Commutative Algebra
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… 


Localization of tight closure and modules of finite phantom projective dimension.
The notion of tight closure for an ideal or submodule, both for Noetherian rings of positive prime characteristic p and for finitely generated algebras over a field of characteristic 0, was