By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.

Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite for all $n \in \mathbb{N}$?

Remarks: Let $j:U \to X$ be the open subset given by removing the fibres of $X \to \mathrm{Spec} \, \mathbb{Z}$ lying above those primes dividing $n$.

  1. I can easily show that $H_{et}^1(X,\mathbb{Z}/n\mathbb{Z})$ is finite. For example, it follows from the Leray spectral sequence that $H_{et}^1(X,\mathbb{Z}/n\mathbb{Z}) \to H_{et}^1(U,\mathbb{Z}/n\mathbb{Z})$ is injective, and the latter group is finite by a special case of [1, Proposition II.7.1].
  2. If $n$ invertible on $X$, then the answer is yes and this is again a special case of [1, Proposition II.7.1]. So I'm interested in the case where $n$ is not invertible on $X$. (In particular, this shows that $H_{et}^2(U,\mathbb{Z}/n\mathbb{Z})$ is finite).
  3. If $\dim X = 1$ then the answer if yes by [1,Theorem II.3.1].
  4. Another application of the Leray spectral sequence argument shows that is suffices to show that $H^0(X, R^1j_* \mathbb{Z}/n\mathbb{Z})$ is finite, but I don't see why this should be the case.
  5. The analogous question concerning finiteness of $H_{et}^i(X,\mathbb{Z}/n\mathbb{Z})$ is also interesting, but for my application I only need $i = 2$.


[1] Milne - Arithmetic duality theorems.

  • 7
    $\begingroup$ @JasonStarr: In the papers of Deligne and/or Esnault surely the relevant torsion sheaves have torsion orders invertible on that base (and section 6 of Weil II assumes properness). With "invertible" torsion-orders one can use Deligne's "generic base change" expose from SGA 4.5 to see that the higher direct images on ${\rm{Spec}}(\mathbf{Z})$ are constructible, and it is a general theorem (perhaps due to some combination of Artin, Mazur, Zink?) that the cohomology of any constructible abelian sheaf (no hypothesis on torsion-orders) on the ring of $S$-integers of a number field is finite. $\endgroup$ – nfdc23 Oct 25 '17 at 13:16
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    $\begingroup$ @JasonStarr: The purity theorems (such as in Milne's notes that you mention) require the closed complement to satisfy regularity properties (which $X \bmod p$ may not) and more importantly assume torsion-orders for the sheaf are invertible on the scheme being considered (and avoiding the latter assumption is the main thrust of the question here). $\endgroup$ – nfdc23 Oct 25 '17 at 13:19
  • 3
    $\begingroup$ I have one more comment. For every flat, regular, affine $\mathbb{Z}$-scheme $X$ whose reduction $X_p$ over $\mathbb{Z}/p\mathbb{Z}$ is regular of dimension $\geq 1$, the cohomology $H^1_{\text{et}}(X_p,\mathbb{Z}/p\mathbb{Z})$ is infinite, because of the many different Artin-Schreier covers. If there were a long exact sequence, presumably this would imply that $H^2_{\text{et}}(X,\mathbb{Z}/p\mathbb{Z})$ is also infinite. That might be evidence that there is no long exact sequence, or it might be evidence that the cohomology is infinite. $\endgroup$ – Jason Starr Oct 25 '17 at 15:22
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    $\begingroup$ @JasonStarr: as you know, there's always an excision sequence relating the cohomology of a scheme to that of an open subscheme with the extra term being cohomology with supports along the closed complement (rather than cohomology on that complement), so the Artin-Schreier stuff doesn't directly intervene (but there's likely no purity theorem for this situation with torsion-orders not invertible; with reasonable torsion-orders I think Gabber has proved purity theorems for abstract regular schemes). $\endgroup$ – nfdc23 Oct 25 '17 at 15:46
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    $\begingroup$ A couple of references: the purity result of Gabber referred to is here arxiv.org/abs/1207.3648 Exposé XVI, and indeed assumes the torsion order is invertible. For p-torsion there is some kind of purity theorem in this article of Sato arxiv.org/abs/math/0610426 but I don't know whether it is any use to you. $\endgroup$ – Martin Bright Oct 30 '17 at 14:52

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