Tight closure with respect to a multiplicatively closed subset of an F-pure local ring
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Rodney Y. Sharp [1]
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[1] University of Sheffield
University of Sheffield
Reino Unido
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- Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 3 ((March 2015)), 2015 (Ejemplar dedicado a: Special Issue in honor of Prof. Hans-Bjørn Foxby), págs. 672-685
- Idioma: inglés
- DOI: 10.1016/j.jpaa.2014.05.021
- Enlaces
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Resumen
Let R be a (commutative Noetherian) local ring of prime characteristic that is F -pure. This paper studies a certain finite set II of radical ideals of R that is naturally defined by the injective envelope E of the simple R -module. This set II contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S -test ideal. It is shown that an ideal aa of R belongs to II if and only if it is the S′S′-test ideal of R for some multiplicatively closed subset S′S′ of R. When R is complete, II is also ‘closed under taking test ideals’, in the following sense: for each proper ideal cc in II, it turns out that R/cR/c is again F -pure, and if gg and hh are the unique ideals of R that contain cc and are such that g/cg/c is the (tight closure) test ideal of R/cR/c and h/ch/c is the big test ideal of R/cR/c, then both gg and hh belong to II. The paper ends with several examples.
