It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, is equivalent to the category $\text{Vect}(X/G)$ consists of complex vector bundles on the quotient space $X/G$.

On the other hand, we have the correspondence between vector bundles and finitely generated projective modules, i.e Serre-Swan theorem. In more details, let $A:=C(X)$ be the ring of continuous complex functions on $X$ and for $E$ a vector bundle on $X$, the global sections $M:=\Gamma(E)$ is a finitely generated projective $A$-module. Since $G$ acts on $X$, $G$ also acts on $A$ and if we denote $A^G$ for the invariant subring then we have $A^G=C(X/G)$. Moreover $G$ acts on the category $A$-Mod by twist. Explicitly, if $M$ is an $A$-module and $g\in G$, then the twisted module $M^g$ is the same as $M$ as sets but with different $A$-module structure, i.e for $a\in A$, $m\in M$ we have $a\cdot_g m:=g(a)\cdot m$. The category of equivariant $A$-modules consists of $M$ such that $M\cong M^g$ as $A$-modules for any $g\in G$ and the morphisms are $G$- equivariant $A$-module maps.

The statement in the first paragraph now becomes: the category of equivariant finitely generated projective modules of $A$ is equivalent to the category of finitely generated projective modules of $A^G$, given that the action of $G$ is free. The functor $A^G$-mod to equivariant $A$-mod is given by $$ N \mapsto A\otimes_{A^G}N. $$

My question is: is there a proof purely in terms of $A$-modules and $A^G$-modules and Barr-Beck theorem? (We have an algebraic version of all above, say in Section 4.4 of Vistoli's "Notes on Grothendieck topologies, fibered categories and descent theory". Notice that free $G$ action on space $X$ is replaced by $G$-torsor $X$ there. It is essentially Barr-Beck theorem.)

In particular

  1. Is the condition that the action of $G$ on $X$ is free is the same as $A^G\rightarrow A$ is faithfully flat or something similar to faithfully flat?

  2. How to apply Barr-Beck theorem in this case?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.