Localization and test exponents for tight closure

Details Localization and test exponents for tight closure Localization and test exponents for tight closure

2002-11-11

arxiv.org/abs/math/0211173v1

Michigan Math. J. 48 (2000), 305--329

Mathematics

Commutative Algebra

Scientific paper

In this paper we study various equivalent conditions for tight closure to commute with localization. If N is a submodule of a finitely generated module M over a Noetherian commutative ring of characteristic p, then a test exponent for c,N,M is defined to be a power q' of p such that u is in the tight closure of N in M whenever cu^q is in the qth Frobenius power of N for some q \ge q'. We prove that that a test exponent for a locally stable test element c and for N,M as above exists if and only if the tight closure of N in M commutes with localization. Other equivalent conditions are given for tight closure to commute with localization.

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