how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:37:45Z http://mathoverflow.net/feeds/question/15072 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15072/how-good-an-approximation-to-the-equivariant-derived-category-is-given-by-the-gra how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space? Ben Webster 2010-02-12T05:08:12Z 2010-02-13T20:49:26Z <p>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 $X/GL_n$).</p> <p>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 $p_m:X_m\to X/GL_n$. </p> <p>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 $m$.</p> <blockquote> <p>Can anything precise be said about how fast this kernel shrinks? </p> </blockquote> <p>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 <code>$Hom_{X_m}(p_m^*F,p_m^*G)=H^*(Gr(m,n))$</code>, the cohomology of the Grassmannian of $n$-planes in $m$-space. In this case the kernel is pretty well understood.</p> <p>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.</p> http://mathoverflow.net/questions/15072/how-good-an-approximation-to-the-equivariant-derived-category-is-given-by-the-gra/15208#15208 Answer by Geordie Williamson for how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space? Geordie Williamson 2010-02-13T20:49:26Z 2010-02-13T20:49:26Z <p>Something related (but not exactly what you are asking) is covered in some detail in Bernstein-Lunts (around 2.2.3).</p> <p>Call a map $\pi : P \to X$ $n$-acyclic if for any <em>sheaf</em> $F$ the truncated adjunction morphism $F \to \tau_{\le n} \pi_* \pi^* F$ is an isomorphism.</p> <p>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).</p> <p>Then, if $F$ and $G$ have cohomology sheaves concentrated in an interval $I$ with $|I| &lt; n$ 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).</p> <p>Two comments:</p> <ul> <li>(depending on your definition) $Hom(F,G)$ is <em>defined</em> to be $\varprojlim Hom(p_m^* F, p_m^* G)$ and so the statement "is injective" is a bit misleading!</li> <li>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!</li> </ul>