This question was prompted by the two following:

Constructible étale sheaves on X are étale algebraic spaces over X

Naive question about constructing constructible sheaves

If I have a constructible sheaf $F$ of $\mathbb{Z}$-modules (meaning the stalks are only required to be finite type) on a quasicompact quasiseparated scheme $X$ and a stratification $U_n=\emptyset \subset U_{n-1} \subset \cdots \subset U_0 = X$ of $X$ such that $F$ is locally constant on $X_i=U_i\backslash U_{i+1}$, how can I reconstruct a presentation of $F$ ? By that I mean an exact sequence $\mathbb{Z}_V \to \mathbb{Z}_U \to F \to 0$ where $U, V$ are étale schemes of finite presentation over $X$ and $\mathbb{Z}_V=j_!\underline{\mathbb{Z}}_V$ where $\underline{\mathbb{Z}}_V$ is the constant sheaf on $V$ and $j:V\to X$ is the structure map.

More precisely, denote $j_i$ and $k_i$ the inclusion of $U_{i+1}$ and $X_i$ in $U_i$, and suppose I am given a locally constant sheaf $F_i$ of $\mathbb{Z}$-modules on each $X_i$, together with a morphism $\phi_i : F_{i} \to k_i^\ast j_{i,\ast} (\mathrm{reconstruction}(F_{i-2},F_{i-1},\phi_{i-1}))$ where the $\mathrm{reconstruction}$ functor comes from Artin gluing (maybe this can be expressed without recourse to this functor, but let's leave it like that). Can I get an explicit presentation of $F$ ?

An inductive construction would need as heredity step the following : in an Artin gluing situation, compute the presentation of the sheaf $F$ corresponding to a triple $(F_U,F_Z,\phi:F_Z \to i^\ast j_\ast F_U)$ where $U \xrightarrow{j} X \xleftarrow{i} Z$ is an open-closed decomposition, $F_U$ is has a given presentation and $F_Z$ is a locally constant constructible sheaf of $\mathbb{Z}$-modules.

The first thing to do is to give a presentation of a locally constant sheaf; I don't know how to do that.

If we know how to compute presentations of locally constant sheaves, then in the most simple case $n=2$ (the initialization step), we are gluing two locally constant sheaves with given presentations. Can we say something in this case ? I feel like we can't say anything about this because the objects $\mathbb{Z}_U$ are not projective, so the standard argument of patching of projective resolutions does not apply.

If this is well-known/standard, I would gladly accept any reference.

EDIT: I had some confusion about what $\mathbb{Z}_U$ was. I'm not so sure now that a constructible sheaf of $\mathbb{Z}$-modules should come from an algebraic space. What is true is that it has a presentation by finite direct sums of sheaves of the form $\mathbb{Z}_U$ (SGA4, IX, 2.7)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.