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This is a question related to the two dimensional McKay correspondence. Let $R = mathbbC[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with a $G$ action, such that the multiplication map $R otimes M to M$ is $G$-equivariant, i.e., $g *(rm) = (gcdot r) (g *m)$.

It is well known that the category of such modules, $Rtext-Mod_G$, is equivalent to the category of modules over the (noncommutative, unital) skew group algebra $R # G$, which is the algebra on the underlying vector space $R otimes_mathbbC mathbbC[G]$, and multiplication defined by $(rotimes g)(r' otimes g') = r(gcdot r')otimes gg'$.

I would like to know what the projective objects are in the latter category. I know that Quillen-Suslin implies that finitely generated projective $R$-modules are free, and so I suspect that all the finitely generated projective equivariant $R$-modules are also free. However I would like to know a proof of this in the setting of the skew group ring. I am aware that the group ring $mathbbC[G]$ is semisimple, so one would hope that $R # G$ also has a similar decomposition, but I can only see this as vector spaces, rather then algebras. Moreover I'm still not sure how I would use this to get projective objects in $R# G$-Mod.

A first observation is that the centre $Z=Z(R#G)$ is isomorphic to the fixed point ring $R^G$. Next, we can define a ring homomorphism
$$ R# G to mathrmEnd_Z(R), quad rotimes g mapsto (tmapsto rg(t)). $$
Auslander showed that this is actually an isomorphism.

We therefore have an equivalence between the category $mathrmproj(R# G)$ of finitely generated projective $R# G$-modules, and the category $mathrmadd_Z(R)$ of $Z$-module direct summands of $R^n$ for $ngeq1$.

We also have an equivalence between $mathrmproj(R# G)$ and the semisimple category $mathrmrep_mathbb C,G=mathrmmod,mathbb C[G]$.

This sends a $G$-representation $V$ to $Rotimes_mathbb CV$, with its natural $R#G$-module structure. This will be projective since there exists some $W$ such that $Voplus W$ is a free $mathbb C[G]$-module. Alternatively it is extension of scalars along the inclusion $mathbb C[G]subset R#G$.

Conversely it sends a projective $R#G$-module $P$ to the quotient $P/(xP+yP)$. Alternatively this is extension of scalars along the quotient map $R#Gtomathbb C[G]$ which sends $x,y$ to zero.

This latter equivalence shows that the isomorphism classes of indecomposable projective $R#G$-modules are in bijection with the isomorphism classes of irreducible $G$-representations.

Update: The paper by Auslander is 'On the purity of the branch locus' Amer J Math 84 (1962), and note that he takes $R=[[x,y]]$ with $G$ acting linearly.

A good reference is chapter 5 of 'Cohen-Macaulay Representations' by Leuschke and Wiegand.

Note that if we pass to the quotient fields, then this is just Galois theory. Let $L=mathbb C(x,y)$ and $K=L^G$. Then $L/K$ is a Galois extension with Galois group $G$, and so $L#Gcong mathrmEnd_K(L)$.