Speaker: Neil Epstein (George Mason University)
Title: The Ohm-Rush content function and its applications.
Abstract: For an $R$-algebra $S$, the (Ohm-Rush) \emph{content} $c(f)$ of an element $f\in S$ is the intersection of all ideals $I$ such that $f\in IS$. If there is always a smallest such ideal (i.e. $f \in c(f)S$), we call $S$ an \emph{Ohm-Rush algebra}. Further content-related properties carry their own names and implications. The theory examines algebraic properties of polynomial extensions $R \rightarrow R[x]$ and what can be generalized from them.
I will report on some results regarding the Ohm-Rush content function, along with applications to apparently disparate areas of commutative algebra. For instance, \begin{itemize}
\item a new criterion for regularity in Noetherian reduced local rings of characteristic $p$
\item Given a regular field extension $L/K$, a Noetherian $K$-algebra $R$, and a zero-divisor $g \in S := R \otimes_K L$, some nonzero element of $R$ kills $g$.
\item (w/Shapiro) With $R$, $S$ as above, if $S$ is locally a UFD, so is $R$.
\item (w/Shapiro) $R \rightarrow \hat R$ ($R$ Noetherian local) is Ohm-Rush if and only if every ideal of $\hat{R}$ is extended from $R$.
\item (w/Carchedi) for any ring map $R \rightarrow S$, an algebraic characterization of when the map of topological spaces Spec $S \rightarrow$ Spec $R$ is open
\item When $R$ is an excellent Noetherian ring of characteristic $p$, the Frobenius endomorphism $R \rightarrow R$ given by $r \mapsto r^p$ is Ohm-Rush.
\end{itemize}
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