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So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves on the stack $X/GL_n$).

For each integer $m$, let $Y_m$ be the space of injective maps of $k^n\to k^m$ and let $X_m=(Y_m\times X)/GL_n$ (with the diagonal action, as usual). Note that we have a map $p_m:X_m\to X/GL_n$.

Now, it's a fact that $Hom_{X/GL_n}(F,G)$ injects into the inverse limit $\varprojlim Hom_{X_m}(p_m^*F,p_m^*G)$, but it usually isn't injective for any given $m$.

Can anything precise be said about how fast this kernel shrinks?

The most boring case is when $F$ and $G$ are both the constant sheaf on a point. Then $Hom_{X/GL_n}(F,G)=H^*(BGL_n)$, the cohomology of the classifying space and $Hom_{X_m}(p_m^*F,p_m^*G)=H^*(Gr(m,n))$, the cohomology of the Grassmannian of $n$-planes in $m$-space. In this case the kernel is pretty well understood.

Ideally, the kernel in general would simply come from this case: i.e. these cohomology rings act on the right and left no matter what $X$ is, and the kernel might be generated by multiplying maps by classes in the kernel of the map from $H^*(BGL_n) \to H^*(Gr(m,n))$, the map from the cohomology of the classifying space to the cohomology of the Grassmannian. This seems like a reasonable statement, but I'm not sure where to look for it.

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This is probably a silly question, but ... What do you mean by $H^*(BGL_n)\rightarrow H*(Gr(m,n))$? Do these depends on $X$ somehow? – Don Stanley Feb 12 '10 at 12:54
I've edited to try to address your question. The answer is: no, but they act on the whole situation no matter what X is. – Ben Webster Feb 12 '10 at 15:24

Something related (but not exactly what you are asking) is covered in some detail in Bernstein-Lunts (around 2.2.3).

Call a map $\pi : P \to X$ $n$-acyclic if for any sheaf $F$ the truncated adjunction morphism $F \to \tau_{\le n} \pi_* \pi^* F$ is an isomorphism.

Given an $n$-acyclic $G$-equivariant map $P \to X$ where $P$ has free $G$-action one has a map $p: P/G \to X/G$ (keeping your notation).

Then, if $F$ and $G$ have cohomology sheaves concentrated in an interval $I$ with $|I| < n$ then the natural map from $Hom (F, G) \to Hom(p^* F, p^* G)$ is an isomorphism. (Here, in contrast to your usage in the question, $Hom$ means only degree zero homomorphisms).

Two comments:

  • (depending on your definition) $Hom(F,G)$ is defined to be $\varprojlim Hom(p_m^* F, p_m^* G)$ and so the statement "is injective" is a bit misleading!
  • in geometric representation theory the objects on $pt/G$ that are being considered are often direct sums of equivariant constant sheaves (eg if one takes the equivariant intersection cohomology of a projective variety) in which case your description of the kernel works just fine!
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I actually realized yesterday that I could use the description of sheaves on $pt/GL_n$, and meant to write up an answer, but didn't get around to it. I think in the situation I'm interested in, I can just use purity to get formality, but even if I couldn't, my description would be morally correct, just in some dg sense. – Ben Webster Feb 13 '10 at 21:22

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