# Mathematical Sciences Research Institute

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# Seminar

Commutative Algebra and Algebraic Geometry (Eisenbud Seminar) April 16, 2013 (03:45pm PDT - 04:45pm PDT)
Location: UC Berkeley
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Abstract/Media

Commutative Algebra and Algebraic Geometry Seminar
Tuesdays, 3:45-4:45
939 Evans Hall

Organized by David Eisenbud
Abstract: Let $R$ be a standard graded algebra over a field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let $^{\phi}M$ be the abelian group $M$ with an $R$-module structure induced by the Frobenius endomorphism. The $R$-module $^{\phi}M$ has a natural grading given by $\deg x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this talk, I\\'ll discuss our new characterization of Koszul algebras saying that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that \$\reg_R \up{\phi}M