Commutative Algebra Seminar
Nicholas Switala
University of Minnesota
On the de Rham homology and cohomology of complete local rings
Abstract: Let $R$ be a formal power series ring over a field $k$ of characteristic zero, and let $D$ be the ring of $k$-linear differential operators on $R$. By interpreting the Matlis dual of an $R$-module $M$ in terms of certain $k$-linear maps from $M$ to $k$ (following SGA2), we show that given any left $D$-module, there is a natural structure of left $D$-module on its Matlis dual, whose de Rham cohomology is $k$-dual to that of the original module in the holonomic case. We give an application to Hartshorne's theory of de Rham homology and cohomology for complete local rings, showing that the associated Hodge-de Rham spectral sequences (beginning with their $E_2$ terms) are independent of the embedding used in their definition and consist of finite-dimensional $k$-spaces.
Monday April 4, 2016 at 2:00 PM in SEO 1227