# What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map

$$\varphi^*:\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$$

What are sufficient conditions on $\varphi$ which make $\mathrm{Ext}_A^*(k,k)$ a finite module over $\mathrm{Ext}_B^*(k,k)$?

This question is motivated by the cohomology of finite groups in which the inclusion of group algebras $kH\hookrightarrow kG$ for a subgroup $H\subset G$ induces a map $H^*(G,k)\to H^*(H,k)$ which gives $H^*(H,k)$ the structure of a finite module over $H^*(G,k)$. I'm trying to isolate the properties of the $k$-algebra map $kH\hookrightarrow kG$ which give a finite map in cohomology so that I can generalize to algebras that are not group algebras of finite groups.

• "I'm trying to isolate the properties ..." This seems a bit naive. Did you look into the proof of Evans theorem in the group cohomology case ? It uses Sylow subgroups and the fact that a p-group has a central cyclic subgroup. So the proof from group cohomology is restricted to groups. There is a generalization from groups to finite dimensional cocommutative Hopf algebras by Friedlander and Suslin. But their proof uses quite different techniques from algebraic geometry and is rather long. – tj_ Sep 6 '14 at 11:24