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I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I wonder to what extent it could be compared with the non-affineness in algebraic geometry? For example, as an analogue of "coherent" coefficients in algebraic geometry,

(cf. Serre vanishing) Does the higher cohomology of the sheaf $\underline{\mathbb R}:=C(-,\mathbb R)$ of real-valued continuous functions vanish on non-paracompact Hausdorff manifolds?

On the other hand, as an analogue of étale coefficients in algebraic geometry,

(cf. Artin vanishing) What is the cohomological dimension of non-paracompact Hausdorff manifolds? Or rather, if we only look at local systems of abelian groups, is the cohomology bounded by the dimension of the non-paracompact Hausdorff manifold?

I would be happy to see references for these. If they are too hard for the moment, I would also be glad to hear anything about the very example of the long line.

Update: As R. van Dobben de Bruyn and Dmitri Pavlov mentioned, for non-paracompact manifolds, the sheaf cohomology with coefficients in constant sheaves is equivalent to the singular cohomology, due to a theorem by Sella. This is also covered in Dustin Clausen's Lecture notes on algebraic de Rham cohomology, Lecture 3, Theorem 12. It might extend to local systems as well. And for singular cohomology, if I understand correctly, we do have the vanishing result.

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    $\begingroup$ 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.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 7 at 20:02
  • $\begingroup$ @R.vanDobbendeBruyn Could you please briefly explain what this "long homotopy invariance" is? The book is under paywall and inaccessible to me. $\endgroup$
    – Z. M
    Commented Jul 7 at 20:08
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    $\begingroup$ @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 $\endgroup$
    – Dmitri Pavlov
    Commented Jul 7 at 20:17
  • $\begingroup$ Can you get anything from the fact that $M$ is locally compact? $\endgroup$
    – Tom Goodwillie
    Commented Jul 7 at 20:48
  • $\begingroup$ @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). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 7 at 22:51

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