Timeline for Functors between categories of equivariant sheaves are equivariant sheaves on the product?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 19, 2017 at 12:36 | comment | added | Anton Fetisov | @SaalHardali You take a $G$-equivariant sheaf on $X$ and a $G^{op} \times H$-equivariant sheaf on $X\times Y$, take their total spaces (which are $G$ and $G^{op}\times H$ sets respectively), factor their product by the diagonal action of $G$ and consider the projection to $Y$, getting an object in $Sh_H(Y)$. More geometrically you can describe it in Fourier-Mukai style: pull $F:Sh_G(X)$ back to $X\times Y$, multiply it by the kernel and push forward along $X\times Y \to Y$, which amounts to taking fiberwise coinvariants of $G$ action. | |
| May 19, 2017 at 12:23 | comment | added | Saal Hardali | The point about the colimit preservation was illuminating thank you. Perhaps I don't understand the answer. As I said I'm looking for the explicit (set-theoretic) consturction which takes a $G$-equivariant sheaf on $X$ and a $G^{op}\times H$ equivariant sheaf and gives an $H$-equivariant sheaf. It might hide somewhere in your answer but I can't spot it. | |
| May 19, 2017 at 12:09 | comment | added | Anton Fetisov | @SaalHardali Everything that I described is perfectly explicit and generalizes your construction for the point. Note also that your description also works exclusively for colimit-preserving functors (e.g. right adjoint to restriction of scalars isn't included). What exactly is your question? | |
| May 19, 2017 at 10:07 | comment | added | Saal Hardali | This was helpful but still slightly off as i'm looking for an explicit construction like the one I describe for the case where everything is a point (notice that my construction easily generalizes to the case of where $G$ topological/algebraic/formal etc. group and could work in principle in any category). | |
| May 18, 2017 at 14:06 | history | answered | Anton Fetisov | CC BY-SA 3.0 |