Timeline for Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Current License: CC BY-SA 4.0
10 events
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| Jul 25, 2022 at 15:18 | vote | accept | Jakob Werner | ||
| Jul 25, 2022 at 15:18 | comment | added | Jakob Werner | Thanks, I didn't know of this use of the notation $H^0$ for presheaves. | |
| Jul 21, 2022 at 15:25 | comment | added | Dmitri Pavlov | @JakobWerner: If H^0 refers to the ordinary sheaf cohomology, then it sheafifies its argument first. A priori this may produce a different answer. However, looking at Section 5 (Example 5.2) of the arXiv version (which is where the local-to-global spectral sequence is mentioned), he says explicitly that the $k$th hypercohomology group vanishes for $k<p$. In particular, the $p$th hypercohomology presheaf is a sheaf of abelian groups (and not just a presheaf), so its 0th sheaf cohomology coincides with its global sections. | |
| Jul 21, 2022 at 14:36 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Jul 21, 2022 at 8:08 | comment | added | Jakob Werner | Before Lemma 2.4 it says: “By Lemma 2.2(3) (vanishing of some cohomology groups) and the local to global spectral sequence, we obtain a canonical isomorphism $H^0(X, \underline{H}^p(X, i_{z!} i_Z^! \mathscr{K}_X^p)) \cong H^p(X, i_{Z!} i_Z^! \mathscr{K}_X^p)$”. | |
| Jul 21, 2022 at 8:06 | comment | added | Jakob Werner | Thanks for your answer! I'm still a bit confused. If I understand it correctly, hypercohomology generalizes sheaf cohomology from sheaves to complexes of sheaves, but for complexes concentrated in degree zero, it coincides with ordinary sheaf cohomology. So if I view $\underline{H}^p(X, F)$ as the presheaf sending $U$ to $p$-th hypercohomology of $F$ over $U$, I should have $H^0(X, \underline{H}^p(X, F)) = H^p(X, F)$ for purely formal reasons, no? But in the paper cited by me, this seems to be a non-trivial fact, holding only in certain situations: | |
| Jul 20, 2022 at 18:06 | comment | added | Dmitri Pavlov | @DenisNardin: Added more details. Homology classes form a presheaf, of course. | |
| Jul 20, 2022 at 18:02 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Jul 20, 2022 at 17:03 | comment | added | Denis Nardin | I'm confused: if you just take the homology of the resolution (as a chain complex of sheaves) don't you get just zero? | |
| Jul 20, 2022 at 16:53 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |