Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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
add comment

1 Answer 1

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!
share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.