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I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety over $\mathbb{C}$) and a finite group $G$ acting on it. I would like to understand simple things like: what does a coherent sheaf on $[X/G]$ look like? what do the (derived) pushforward functors between the various spaces do? and the pullbacks?

Any pointers would be greatly appreciated!

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A coherent sheaf on $[X/G]$ is a sheaf on the action groupoid corresponding to the action. This is essentially a $G$-equivariant sheaf. –  David Roberts Nov 15 '11 at 23:55

3 Answers 3

A very readable reference for constructible derived categories on quotient stacks is Bernstein-Lunts, Equivariant sheaves and functors. In particular, Bernstein and Lunts construct the bounded derived category of sheaves on $[X/G]$ as the limit of truncated bounded derived categories of the "Borel constructions" $X\times_G E$ where $E$ is a sufficiently acyclic free $G$-space. A key observation is that $E$ does not have to be contractible, only acyclic enough, to know a piece of $D^b([X/G])$. So when one considers algebraic varieties and linear algebraic group actions, one can let $E$ be e.g. the Stiefel manifold of $k$-frames in a vector space of a very large dimension. This allows one to describe all the pushforward and pullback functors one could wish for in terms of the respective functors for algebraic varieties, or at worst, spaces.

There is one important exception: if $X$ and $Y$ are equipped with actions of $G$, respectively $H$, $f:G\to H$ is a group homomorphism and $F:X\to Y$ is a map such that $F(gx)=f(g)F(x)$ for all $g\in G,x\in X$, then there is the pushforward fuctor $F_*:D^+([X/G])\to D^+([Y/H])$. This functor comes up in many applications, but to define it Bernstein and Lunts use infinite-dimensional spaces, and I don't know if there is a way around this.

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Bernstein-Lunts deals with derived categories of topological sheaves; in this context the naive derived category of equivariant sheaves is inadequate. For coherent sheaves I would say the naive notion is the correct one. –  Angelo Nov 16 '11 at 5:51
This is indeed very readable, but deals with constructible sheaves (as pointed out by Angelo). One should also be careful of some inaccuracies. –  Anton Fonarev Nov 17 '11 at 11:37

In case anyone is interested, I've been finding this thesis useful

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Bridgeland, King and Reid, "The McKay correspondence as an equivalence of derived categories" has a nice discussion expanding on David Roberts's comment.

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this is actually the paper I started with. unfortunately I'm a bit dense, and I would really like a reference which spells out some of the details they omit. (if I recall correctly Grothendieck's Tohoku paper, chapter 5, is also a bit terse. also it deals with the topological case) –  Yosemite Sam Nov 16 '11 at 21:12

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