Student Arithmetic Geometry Seminar: The Hochschild-Kostant-Rosenberg Theorem in Characteristic p

Seminar | November 1 | 4:10-5 p.m. | 891 Evans Hall

 Joseph Stahl, UC Berkeley

 Department of Mathematics

The celebrated theorem of Hochschild-Kostant-Rosenberg says that for a smooth algebra $R$ over a ring $k$, the Hochschild homology algebra $HH_*(R/k)$ is isomorphic to the algebra of differential forms $\Omega _{R/k}^*$. Moreover, if $k$ is a $Q$-algebra, then there is a quasi-isomorphism at the level of complexes $HH(R/k)\simeq \bigoplus _i\Omega ^i_{R/k}[i]$, where the right hand side is viewed as a complex with zero differential. This isomorphism can be extended to smooth quasi-projective schemes $X$ in characteristic $0$, but in characteristic $p$ less is known. In this talk, we will discuss a result of Antieau and Vezzosi that if $k$ is a field of characteristic $p$, and $X$ is a smooth proper k-scheme of dimension at most $p$, then a version of the HKR theorem holds for $X/k$ (extending work of Yekutieli). We will also review the basics of Hochschild homology and the classical HKR theorem in characteristic zero.