𝐹-regularity, test elements, and smooth base change

Hochster, Melvin; Huneke, Craig

doi:10.1090/S0002-9947-1994-1273534-X

$F$-regularity, test elements, and smooth base change
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by Melvin Hochster and Craig Huneke

Trans. Amer. Math. Soc. 346 (1994), 1-62

DOI: https://doi.org/10.1090/S0002-9947-1994-1273534-X

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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 $\S 6$ and is then applied in $\S 7$ 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 $\S 6$ 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 $\S 7$ 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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