how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

2010-02-12T05:08:12Z

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.

Answer by Geordie Williamson for how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

2010-02-13T20:49:26Z

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!

how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space? - MathOverflow

how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

Answer by Geordie Williamson for how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?