Timeline for Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
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Jul 10 at 10:32 | comment | added | Z. M | @R.vanDobbendeBruyn There is also a more recent paper by Porta–Teyssier which generalizes the topological exodromy to non-paracompact not-necessarily-locally-of-singular-shape spaces in the same fashion (Thm 1.4 & Cor 1.5). Maybe something like this works for perfectoid spaces as well. | |
Jul 9 at 22:52 | comment | added | R. van Dobben de Bruyn | Ah, thanks for the pointer; I was vaguely aware of that paper but didn't actually read it. Yes, I think what you said sounds right to me. | |
Jul 9 at 13:06 | comment | added | Z. M | @R.vanDobbendeBruyn In addition, the homotopy invariance that you mentioned has an exodromic generalization in Thm 0.2 of Haine–Porta–Teyssier, which says that the category of locally hyperconstant sheaves in strongly homotopy invariant in the sense that, the projection $X\times Y\to X$ induces a categorical equivalence for every weakly contractible and locally weakly contractible spaces $Y$ (which contains the long line as a special case). | |
Jul 8 at 14:16 | comment | added | Z. M | @R.vanDobbendeBruyn Let me check: non-paracompact manifolds are locally of singular shape, thus even local systems are the same as local systems on the singular complex. In particular, this seems to imply the Artin vanishing for local systems. Do I understand correctly? | |
Jul 7 at 23:18 | comment | added | R. van Dobben de Bruyn | @Z.M although Dmitri Pavlov gave a better argument, let me add some details. Homotopy invariance is proved using the proper base change theorem plus the fact that $[0,1]$ is acyclic. Replacing $[0,1]$ by the closed long ray $[0,\omega_1]$, you can run the same argument for $h \colon X \times [0,\omega_1]\to X$ such that $h(-,0)=*$ and $h(-,\omega_1)=\operatorname{id}$, assuming $[0,\omega_1]$ is acyclic. Bredon argues that any compact totally ordered space with the order topology is acyclic. A long homotopy $[0,\omega_1)\times [0,\omega_1]\to [0,\omega_1)$ could be $(a,b)\mapsto\min(a,b)$. | |
Jul 7 at 22:51 | comment | added | R. van Dobben de Bruyn | @DmitriPavlov Oh right, that's also true. The locally contractible case (as opposed to cohomologically locally connected) was already known prior to Sella; see for instance Voisin's Hodge theory and complex algebraic geometry I (Theorem 4.47). | |
Jul 7 at 20:48 | comment | added | Tom Goodwillie | Can you get anything from the fact that $M$ is locally compact? | |
Jul 7 at 20:27 | history | edited | Z. M | CC BY-SA 4.0 |
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Jul 7 at 20:17 | comment | added | Dmitri Pavlov | @Z.M: By a theorem of Sella, the sheaf cohomology of a locally contractible space (such as the long line) with coefficients in an abelian group coincides with its singular cohomology. See arxiv.org/abs/2102.06927v3 | |
Jul 7 at 20:08 | comment | added | Z. M | @R.vanDobbendeBruyn Could you please briefly explain what this "long homotopy invariance" is? The book is under paywall and inaccessible to me. | |
Jul 7 at 20:02 | comment | added | R. van Dobben de Bruyn | Related: in case of the long line, it is still true that $H^i(X,\underline{A}) = 0$ for any $A \in \mathbf{Ab}$ and any $i>0$. This uses some sort of 'long homotopy invariance' argument; see e.g. Bredon's Sheaf theory, chapter II, exercise 3 (solution in Appendix B). But this does not say much about the sheaf $C^0(-,\mathbf R)$. (For manifolds, note that both the proof that $C^0(-,\mathbf R)$ is soft and the proof that soft sheaves are acyclic use the paracompactness hypothesis.) | |
Jul 7 at 19:13 | history | edited | Z. M | CC BY-SA 4.0 |
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Jul 7 at 14:51 | history | asked | Z. M | CC BY-SA 4.0 |