Regular closure

McDermott, Moira

doi:10.1090/S0002-9939-99-04756-5

Regular closure
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by Moira A. McDermott

Proc. Amer. Math. Soc. 127 (1999), 1975-1979

DOI: https://doi.org/10.1090/S0002-9939-99-04756-5

Published electronically: March 17, 1999

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Abstract:

Regular closure is an operation performed on submodules of arbitrary modules over a commutative Noetherian ring. The regular closure contains the tight closure when both are defined, but in general, the regular closure is strictly larger. Regular closure is interesting, in part, because it is defined a priori in all characteristics, including mixed characteristic. We show that one can test regular closure in a Noetherian ring $R$ by considering only local maps to regular local rings. In certain cases, it is necessary only to consider maps to certain affine algebras. We also prove the equivalence of two variants of regular closure for a class of rings that includes $R =K[[x,y,z]]/(x^{3}+y^{3}+z^{3})$.

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