Speaker: Mark Walker

Title: Chern characters for dg modules and curved modules

Abstract: In their original setting, Chern classes refer to invariants of vector bundles on a space, taking values in the singular cohomology groups of the space. The Chern character is a ring homomorphism defined in term of these classes, going from the Grothendieck ring of vector bundles to the cohomology ring. The notion of a Chern character has expanded over the years from its original context to include any sort of invariant of bundles on a space or variety, or of modules over a ring, that leads to a ring homomorphism from a Grothendieck ring to a cohomology ring.

In this talk, I describe a Chern character map that is defined for maximal Cohen-Macaulay (MCM) modules over a complete intersection ring. The target of this map is the Hochschild homology groups, suitably defined, of the stable category of MCM modules. The construction builds on the case of Chern characters and Hochschild homology for the stable category of modules over a  hypersurface, which has been studied by many authors of late  (including Carqueville-Murfet, Dyckerhoff, Platt, Polishchuck-Vaintrob, Yu) in the setting of matrix factorizations. I will explain how such a Chern character map is of value to prove results in commutative algebra.