Monday, February 6, 2017 11:15am to 12:30pm
About this Event
30 Pryor Street, Atlanta, GA
Speaker name: Robin Baidya, Department of Mathematics & Statistics, Georgia State University
Talk title: Characterizations of Regular Local Rings of Positive Prime Characteristic - Part 2
Talk abstract: et $R$ be a commutative Noetherian local ring of positive prime characteristic $p$; let $F$ denote the Frobenius endomorphism of $R$ sending each element $r$ of $R$ to its $p$th power $r^p$; and, for all positive integers $e$, let $^eR$ denote $R$ as a module over itself via the structural map $F^e$, the $e$th power of the Frobenius endomorphism. At first glance, the $R$-modules $^1R, ^2R, ^3R, ...$ may not seem to have any special significance apart from what $R$-modules in general already reveal about $R$. On the contrary, results due to Kunz and Rodicio certify that flat resolutions of these modules, in fact, negotiate the regularity of $R$. In particular, Kunz proved in 1969 that $R$ is regular if and only if $^eR$ is flat over $R$ for some positive integer $e$ if and only if the particular module $^1R$ is flat over $R$. A later result of Rodicio verifies that $R$ is regular even if we assume only that $^1R$ has finite flat dimension. In this talk, I will present proofs of both of these remarkable characterizations of regular local rings of positive prime characteristic.
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