A sheaf

Max Lieblich asked if one could find an abelian sheaf G on the category of schemes in the ├ętale topology such that

  1. G(X) = G(X_{red}),
  2. G(X) = 0 when X has only one point,
  3. G is not zero, and
  4. G is limit preserving.

I’ll tell you why he asked in a minute, but first let me tell you an example: Let A be an abelian group. Let F be the presheaf on the category of schemes which associates to a scheme X the group of constructible functions a : |X| —> A modulo locally constant functions. Let G be the sheafification of F in the ├ętale topology. Then G works. (For more details, look at the the section entitled “Sheaves and constructible functions” in the chapter “Examples” of the stacks project.)

Why did this come up? Consider the stack [Spec(Z)/G] classifying étale G-torsors. Then the morphism f : Spec(Z) —> [Spec(Z)/G] is an equivalence of categories of sections over the spectrum of any field, f is formally étale, and the stack [Spec(Z)/G] is limit preserving, but f is not an equivalence (as G is not zero). This answers a question posed by Dan Abramovich.

3 thoughts on “A sheaf

  1. Dear Johan,

    Could you explain a little more why the sheaf $G$ defined above is limit preserving? Consider the ring $R = k[x_1,x_2,…]$, a polynomial ring in countably many variables. This is the limit of the subrings $R_n = k[x_1,…,x_n]$. It is not the case that every constructible subset of $\text{Spec} R$ is the inverse image of a constructible subset of some $\text{Spec} R_n$, e.g., the (closed) singleton set with point $(x_1,x_2,…)$ is not the inverse image of any subset of any $R_n$. So why is the functor of constructible functions limit preserving?

    Best regards,

    • This is because a singleton in Spec(R) is not a constructible set for your R. The definition of a constructible set in a topological space X is that it is a finite union of subsets of the form U ∩ V^c with U, V retrocompact open in X, meaning that both U —>X and V —> X are quasi-compact maps of topological spaces. If X is the spectrum of a ring, then T ⊂ X is constructible if and only if T is the image of a finitely presented morphism Y —> X of affine schemes, see Lemma Tag 00F8 and Chevalley’s Theorem.

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