Name: Mitsuyasu Hashimoto
Title: $F$-regularity of multi-graded rings
Abstract: We study $F$-regularity of $\Bbb Z^n$-graded rings.
In particular, we prove that if $A$ and $B$ are standard graded
algebras over an algebraically closed field, then the Segre
product ring $A\# B$ is
strongly $F$-regular if and only if both $A$ and $B$ are strongly
$F$-regular.
We also prove that if $G$ is a connected reductive group over $\Bbb C$,
$S$ is a normal semigroup scheme of finite type over $\Bbb C$,
and $\varphi: G\rightarrow S$ is a dominating semigroup homomorphism,
then $S$ is of strongly $F$-regular type.
Some part of the study utilizes the notion of global $F$-regularity.