Name Kei-ichi Watanabe
title F-rationality and F-regularity of Rees Algebras
Abstracts
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F-rationality and F-regularity of Rees algebras
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~ \\ ~ \\
{\bf Kei-ichi Watanabe} \\
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This is a joint work with Nobuo Hara and Ken-ichi Yoshida.
Let $(A,\frm)$ be a Noetherian local ring of characteristic
$p>0$ and $I$ be an integrally closed $\frm$-primary ideal.
We discuss F-rational and F-regular property of the Rees
algebra $R_A(I)$ of $I$ over $A$. Our method to ivestigate
the two properties are quite different; we use quite geometric
method for F-regularity, while our method for F-rationality
is relatively algebraic.
Our main results are as follows.
(1) In dimension two, if $A$ is F-rational and $I$ is integrally
closed, then the Rees algebra $R(I)$ is F-rational. If $A$ is
a rational singularity (which may not be F-rational), $R(I)$ is
F-rational for some $I$.
(2) If both $A$ and $R(I)$ are F-rational, then so is the
extended Rees algebra $R'(I)$ (F-rationality of $R(I)$ does
not imply F-rationality of $R'(I)$). On the other hand,
$R(I)$ is F-regular (resp. a rational singularity -- in
characteristic 0) if and only if so is $R'(I)$.
(3) Let $X=\Proj(R(I))$. Then $R(I)$ is F-regular if and only if
$X$ is globally F-regular.
(4) Let $A$ be normal local ring of characteristic 0 and
dimension 2. Then $R(I)$ has F-regular type if and only if
for every proper birational morphism $\tilde{X}\to X'$, where
$\tilde{X}$ is the minimal resolution of $\Proj(R(I))$ and $X'$
is normal, $X'$ has only log-terminal singularities (singularities
of F-regular type).
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