**Date & Time:** Tuesday, September 08, 2009, 16:00-17:00.

**Venue:** Ramanujan Hall

**Title:** Matijevic-Roberts Type Theorems for $F$-singularities

**Speaker:** M. Hashimoto, Nagoya University

**Abstract:** Let $C$ be a property of noetherian local rings.
Let $R$ be a noetherian $\Bbb Z^n$-graded ring, $P$ its prime ideal,
and $P^*$ the prime ideal generated by the all homogeneous elements
of $P$. Assume that $R_{P^*}$ satisfies $C$. Then does $R_P$ satisfy $C$?
This question is called Matijevic-Roberts type theorem.

This was asked by Nagata for $C=\text{Cohen--Macaulay}$ and $n=1$, and solved by Hochster--Ratliff and Matijevic--Roberts independently. It has been proved that M--R theorem is true for Cohen--Macaulay, Gorenstein, complete intersection, and regular properties.

$F$-singularities (such as $F$-regularity and $F$-rationality) are important ring theoretic properties in characteristic $p$ defined via Frobenius maps. It has been shown that $F$-singularities are closely related to singularity theory in characteristic zero.

In this talk, under mild condition, we prove M--R theorem for several $F$-singularities. On the way, we define $F$-purity of homomorphisms, and discuss strong $F$-regularity without $F$-finiteness.