Timeline for Relationship between topological cohomology and $\ell$-adic cohomology
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2010 at 5:24 | vote | accept | josh_whitney | ||
| Jul 31, 2010 at 6:10 | answer | added | Torsten Ekedahl | timeline score: 6 | |
| Jul 31, 2010 at 6:07 | comment | added | BCnrd | josh_whitney: when you say "no singularities", do you mean over $\C$ (rather than smoothness over the ring of integers)? Can you specify away from which characteristics the zero scheme is smooth? It is tempting to then use Poincare duality to replace cohomology with compactly-supported variant (for which have an Artin comparison isom.), so can then use higher direct image sheaves with proper supports, for which base change is easier than in "Th. finitude" in SGA 4.5. (algori may have this in mind). But there remains the task of where these constr. sheaves are lisse, so feels ineffective. | |
| Jul 31, 2010 at 5:38 | comment | added | josh_whitney | Algori -- $\Delta$-regularity just means that if you restrict $f$ to the faces of $\Delta$ that there are no singularities of this restriction inside the torus. In particular, $f$ is a face of itself (of dimension n), so this implies that $f$ has no singular points inside the torus. Thanks Richard and BCnrd for the help! | |
| Jul 31, 2010 at 5:00 | comment | added | algori | josh -- it would be nice if you gave more details on the notion of $\triangle$-regular. Does this imply e.g. that $X_{\bar{F}_q}$ is smooth? | |
| Jul 31, 2010 at 4:38 | comment | added | BCnrd | Yes, if you avoid finitely many characteristics (away from $\ell$). Let $A$ be integers of number field, $F:X \rightarrow {\rm{Spec}}(A)$ any sep'td map of finite type. For any prime $\ell$ there is a dense open $U \subseteq {\rm{Spec}}(A)$ s.t. the constructible $\ell$-adic sheaves ${\rm{R}}^i(F_ {\ast})(\mathbf{Q}_ {\ell})$ are lisse and their formation commutes with any base change; see Deligne's "Th. finitude", SGA 4.5. Hence, for all closed pts $u \in U$, ${\rm{H}}^i(X_ {\overline{u}},\mathbf{Q}_ {\ell})$ has same dim. as for complex fibers. Now apply Artin comparison isom. to conclude. | |
| Jul 31, 2010 at 4:28 | answer | added | Richard Borcherds | timeline score: 4 | |
| Jul 31, 2010 at 1:34 | history | asked | josh_whitney | CC BY-SA 2.5 |