Timeline for Deequivariantisation of indecomposable sheaves
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 22, 2021 at 16:08 | comment | added | Damian Rössler | @David Ben-Zvi. I see! Thank you for the explanation. | |
| Nov 22, 2021 at 14:13 | answer | added | rvk | timeline score: 3 | |
| Nov 22, 2021 at 13:32 | comment | added | David Ben-Zvi | @DamianRössler Unlike in the coherent setting G-equivariant constructible complexes (or local systems) on a point are not algebraic reps of G (which doesn't necessarily make sense since k is unrelated to field of definition) - we need to impose local constancy on the G action on the representation. The abelian category is reps of the component group of G. The derived version is reps of the k-homotopy type of G, or k-valued chains on G (see my answer below) | |
| Nov 22, 2021 at 7:41 | comment | added | Damian Rössler | @David Ben-Zvi. I don’t follow… why eg would $G_m$ (a connected group) have « no representations in the usual sense »? | |
| Nov 21, 2021 at 21:16 | answer | added | rvk | timeline score: 4 | |
| Nov 21, 2021 at 20:16 | history | became hot network question | |||
| Nov 21, 2021 at 19:56 | answer | added | David Ben-Zvi | timeline score: 6 | |
| Nov 21, 2021 at 19:38 | comment | added | David Ben-Zvi | @rvk I agree nothing is happening on the level of hearts (ie faithful as you say) but the question asks about general indecomposable objects, where there's plenty being forgotten even for $X=pt$ and $G=G_m$. | |
| Nov 21, 2021 at 19:22 | comment | added | rvk | @DamianRössler The OP has connected group in the question. Local systems are reps of $\pi_1(BG) = \pi_0(G) = 1$. | |
| Nov 21, 2021 at 19:18 | comment | added | rvk | G is connected. The forgetful functor on the heart of the perverse or ordinary t-structure is faithful (note: this is a statement about $Ext^0$ not higher Ext). | |
| Nov 21, 2021 at 14:12 | comment | added | David Ben-Zvi | I agree - G is connected, so there aren’t any representations of G in the classical sense - the setting is constructible derived categories so only see reps of the homotopy type of G, but there are lots of these. Eg take any class in H^*(BG,k), it represents a nontrivial extension of k by a shift of k as G representation. | |
| Nov 21, 2021 at 12:36 | comment | added | Damian Rössler | What is a decomposable object? One that can be written as a non-trivial direct sum? In that case, any non zero irreducible representation of $G$ over $k$ (pulled-back to $X$ would do the trick (?). | |
| Nov 21, 2021 at 12:16 | history | asked | Peter McNamara | CC BY-SA 4.0 |