The $cl-core$ of an ideal

Details The $cl-core$ of an ideal The $cl-core$ of an ideal

2008-10-16

arxiv.org/abs/0810.3033v3

Math. Proc. Camb. Phil. Soc 149 (2010) 247-262

Mathematics

Commutative Algebra

Final version. Math. Proc. Camb. Phil. Soc 149 (2010) 247-262

Scientific paper

We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core$(I)=$ core$(I)$, if $I$ is an ideal in a one-dimensional domain with infinite residue field or if $I$ is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen--Macaulay normal local domain with infinite perfect residue field, if the analytic spread, $\ell$, is equal to the *-spread and $I$ is $G_{\ell}$ and weakly-$(\ell-1)$-residually $S_2$. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed.

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