# What happens if you apply $\operatorname{QCoh}$ to $(X,\underline{\mathbb{C}}_X)$?

$$\DeclareMathOperator\QCoh{QCoh} \newcommand\numC{\mathbb{C}} \newcommand\numQ{\mathbb{Q}}$$Let $$X$$ be a topological space (possibly a scheme). What happens if you apply $$\QCoh$$ to the prestack $$(X,\underline{\numC}_{X})$$, where $$\underline{\numC}_{X}$$ is the constant sheaf at $$X$$? What about $$(X, \underline{\numQ}_{\ell})$$?

• You get locally constant sheaves of vector spaces. Jun 15 at 17:38
• @MarcHoyois I thought this, but I'm not sure this is true, since you could have a locally constant sheaf on an affine variety which is not constant and thus not quasicoherent in this definition. Jun 15 at 19:11
• @WillSawin That was also the answer I was hoping for, or possibly something related (perhaps providing a different approach to constructible sheaves). Unfortunately, a map $\operatorname{Spec}(A) \to (X,\underline{\mathbb{C}}_{X})$ is just any continuous map (seemingly without restrictions) and a map of sheaves of rings $\underline{\mathbb{C}}_{\operatorname{Spec}(A)} \to \mathcal{O}_{\operatorname{Spec}(A)}$. There can potentially be many such; if $A$ is a $\mathbb{C}$-algebra, any pathological automorphism of $\mathbb{C}$ provides you with one. Jun 16 at 11:49
• @WillSawin To clarify I was using the following definition, which I think is standard although rarely used: a sheaf of modules over a ringed space $(X,\mathcal O_X)$ is quasi-coherent if locally it is the cokernel of a map of free $\mathcal O_X$-modules (this is manifestly a local notion). For $\mathcal O_X=\underline{\mathbb C}_X$ we therefore get locally constant sheaves, at least if $X$ is locally connected or if we restrict to modules of finite presentation. Jun 16 at 12:32
• @MarcHoyois Sorry, I confused myself about what happens when you pull back a locally constant sheaf to an affine variety. But I don't think your definition makes sense for prestacks, so I see a map from the category of Zariski-locally constant sheaves to this category of quasicoherent sheaves but not yet vice versa. Jun 16 at 12:43