The question asks whether the étale cohomology group ( H_{\text{ét}}^2(X, \mathbb{Z}/n\mathbb{Z}) ) is finite for an arithmetic scheme ( X ) (defined as a finite type, flat, regular, integral scheme over ( \operatorname{Spec} \mathbb{Z} )) and any ( n \in \mathbb{N} ).

Key Observations:

1. Arithmetic Schemes: ( X ) is quasi-compact (since it is finite type over ( \operatorname{...

Yes.

Sketch of proof and references:

• Reduce to a prime power. By Chinese remainders, Z/nZ ≅ ⊕p^r∥n Z/p^rZ, so it suffices to treat n = p^r.

• Remove the bad fiber. Let j: U = X[1/p] → X and i: Z = X ×Spec Z Spec Fp → X (a finite disjoint union of fibers over the primes p|n). There is the localization distinguished triangle i_* Ri^! Z/p^r → Z/p^r → Rj_* Z/p^r → . Taking cohomology gives an...

This is an interesting question about the finiteness of étale cohomology groups for arithmetic schemes. Let me analyze this carefully.

The answer to your question is yes - is finite for all when is an arithmetic scheme.

Here's a proof outline:

Key ingredients:

1. Reduction to the case where is a prime power: Sin...

No.
As soon as the arithmetic scheme has dimension ≥ 2 and the coefficient (n) is divisible by a residue characteristic that actually occurs in one of the fibres, the group (H ^2_{\mathrm{ét}}(X,\mathbf Z /n)) is already infinite.

Below we give an explicit counter–example (the affine line over (\mathrm{Spec},\mathbf Z)) and then explain why the same argument works for every arithmetic sc...

Here is what one sees after unraveling the usual “Leray‐localization’’ (or purity) argument: one reduces the finiteness of
H^2_et(X, ℤ/n)
to the finiteness of the zero–th cohomology of a certain sheaf of “wild–vanishing–cycles’’ supported on the fibres of X over the primes dividing n. Concretely, if you set j∶U=X\bigcup_{p|n}X_p and Z=∪_{p|n}X_p, then from the localization triangle
...

First, recall the remarks from the question:

1. ( H_{\text{ét}}^1(X, \mathbb{Z}/n\mathbb{Z}) ) is finite...

Yes.

More precisely, if
 X is any scheme that is separated and of finite type over Spec ℤ
 (nothing else is needed – regularity or flatness is not required),

and if
 F is a constructible étale sheaf of finite abelian groups on X
 (e.g. the constant sheaf ℤ/nℤ),

then every étale cohomology group Hⁱ(X , F) is finite.

Consequently, for an arithmetic scheme X and every positive integer...

Excellent question. Your analysis in the remarks is very good and gets you most of the way to the answer.

The short answer is yes, the group is finite, and in fact this holds for all degrees .

The General Finiteness Theorem

This is a special case of a fundamental theorem in étale cohomology.

Theorem (Grothendieck, Deligne). Let be a schem...

Finiteness of Étale Cohomology for Arithmetic Schemes

This is a deep question in arithmetic geometry concerning the finiteness of étale cohomology groups for arithmetic schemes.

Analysis of the Problem

Let me clarify the setup: We have an arithmetic scheme (finite type, flat, regular, integral over ), and we want to determine if $H^2_{et}(X,\mathbb{Z}/n\m...