# Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules

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)