Timeline for How duality follows from a six functor formalism
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 21, 2022 at 17:24 | comment | added | Marc Hoyois | @KindBubble They apply when $f$ is smooth. When $f$ is proper you have instead $H^n=H^n_c$ and $H^{BM}_n=H_n$. | |
| Jan 20, 2022 at 22:50 | comment | added | user30211 | @MarcHoyois I know it's been a while since you made this answer, but do you think you could spell out "When $f : X \rightarrow S$ is nice"? Are the following two equations meant to apply in the case where $f$ is smooth of relative dimension $d$, or when $f$ is proper? Also, what does "vertical identifications" entail? | |
| Sep 27, 2021 at 12:38 | vote | accept | Gabriel | ||
| Sep 26, 2021 at 10:33 | comment | added | Gabriel | Well, if $S$ is a point, then the category of sheaves of $\mathscr{O}_S$-modules is equivalent to the category of $\Gamma(S,\mathscr{O}_S)$-modules, isn't it? Then $\underline{\hom}_{\mathsf{D}(S)}(\mathscr{O}_S,p_*M[i])$ should be the cohomology of $M$ but seen as an $\Gamma(S,\mathscr{O}_S)$-module, instead of just an abelian group. | |
| Sep 25, 2021 at 12:16 | history | edited | Marc Hoyois | CC BY-SA 4.0 |
typo
|
| Sep 25, 2021 at 12:15 | comment | added | Marc Hoyois | @Gabriel This is what I mean by a duality in $D(S)$. For example $f_*f^!(1)$ is always the [not necessarily strong] dual to $f_!f^*(1)$ in $D(S)$ (this follows directly from the projection formula). I do not know any case where $D(S)$ is the derived category of $\Gamma(S,\mathcal O_S)$-modules. There is the case where $D(S)$ is the quasi-coherent derived category of $S$, but as I mentioned in the answer, this does not actually have all the six functors. | |
| Sep 25, 2021 at 12:02 | history | edited | Marc Hoyois | CC BY-SA 4.0 |
improve Poincaré duality statement
|
| Sep 25, 2021 at 11:46 | comment | added | Gabriel | @MarcHoyois Not even if we consider $\hom_{\mathsf{D}(S)}(\underline{\hom}_{\mathsf{D}(S)}(\mathscr{O}_S,p_*M[i]),\mathscr{O}_S)$ to be the dual of $H^i(X,M)$? This seems very natural for me, since in most (all?) cases $\mathsf{D}(S)$ is the derived category of $\Gamma(S,\mathscr{O}_S)$-modules. | |
| Sep 25, 2021 at 10:55 | comment | added | Marc Hoyois | @Gabriel Yes, homology only makes sense with constant coefficients. The six functor formalism can give you dualities in $D(S)$ but it cannot say anything about the dual of an actual (co)homology group... | |
| Sep 25, 2021 at 10:32 | comment | added | Z. M | A sidenote: $f^!$ is a twisted shift of $f^*$ when $f$ is nice, especially when you talk about the coherent/Serre duality. | |
| Sep 25, 2021 at 10:22 | comment | added | Gabriel | That being said, I'm not sure I understand your definitions of cohomology and compactly supported cohomology (which differ from mine's). In your definition, $M$ has to be an object of $\mathsf{D}(S)$, while in mine's it's an object of $\mathsf{D}(X)$. Does that mean that homology only works with constant coefficients? If so, can't we obtain an isomorphism between cohomology and the dual of compactly supported cohomology as in Poincaré duality in de Rham cohomology? | |
| Sep 25, 2021 at 10:19 | comment | added | Gabriel | Dear @MarcHoyois, while I indeed supposed a relation between $f^*$ and the tensor product (namely, the fact that $f^*$ is strong symmetric monoidal), I forgot to say that I accept the projection formula as well. | |
| Sep 25, 2021 at 7:31 | history | answered | Marc Hoyois | CC BY-SA 4.0 |