-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 | References | Similar Articles | Additional Information
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 -rationality and a treatment of
-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in
and is then applied in
to prove that both tight closure and
-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
that for a reduced ring
essentially of finite type over an excellent local ring of characteristic
, if
is not in any minimal prime of
and
is regular, then
has a power that is a test element. It is shown in
that if
is a flat
-algebra with regular fibers and
is
-regular then
is
-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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1994-1273534-X
Keywords:
Tight closure,
-regular ring,
characteristic
,
test element,
smooth base change
Article copyright:
© Copyright 1994
American Mathematical Society