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$ F$-regularity, test elements, and smooth base change


Authors: Melvin Hochster and Craig Huneke
Journal: Trans. Amer. Math. Soc. 346 (1994), 1-62
MSC: Primary 13A35; Secondary 13B99, 13F40
DOI: https://doi.org/10.1090/S0002-9947-1994-1273534-X
MathSciNet review: 1273534
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Abstract: This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of $ F$-rationality and a treatment of $ F$-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $ \S6$ and is then applied in $ \S7$ to prove that both tight closure and $ F$-regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in $ \S6$ that for a reduced ring $ R$ essentially of finite type over an excellent local ring of characteristic $ p$, if $ c$ is not in any minimal prime of $ R$ and $ {R_c}$ is regular, then $ c$ has a power that is a test element. It is shown in $ \S7$ that if $ S$ is a flat $ R$-algebra with regular fibers and $ R$ is $ F$-regular then $ S$ is $ F$-regular. The general problem of showing that tight closure commutes with smooth base change remains open, but is reduced here to showing that tight closure commutes with localization.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1273534-X
Keywords: Tight closure, $ F$-regular ring, characteristic $ p$, test element, smooth base change
Article copyright: © Copyright 1994 American Mathematical Society

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