Qcoh(-) algebraic stack?

Question

The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, how far is it from being algebraic? For example, which fragment of Artin's criteria is satisfied? I expect that this should be well-known, but currently I don't know where this example is treated in the literature.

For example I can show that for every pair of surjective ring homomorphisms $A \to B, C \to B$, the canonical functor $\text{Mod}(A \times_B C) \to \text{Mod}(A) \times_{\text{Mod}(B)} \text{Mod}(C)$ is an equivalence of categories. In particular, the functor has a deformation theory (in the sense of Artin), sending $(A_0,M,P \in \text{Mod}(A_0))$ to the category of $A_0[M]$-modules $Q$ with an isomorphism $Q/MQ \cong P$.

Answer 1

Artin's axioms do not apply in this case, because the stack is not limit-preserving. They only work with stacks that are locally finitely presented.

In any case, it is easy to give examples of quasi-coherent sheaves whose functor of automorphisms is not representable (for example, an infinite dimensional vector space), and this of course prevents the stack from being algebraic.

As Matthieu says, one should consider coherent sheaves.

[Edit] The question is: what about the stack of coherent sheaves, without flatness hypothesis? First of all, one should interpret "coherent" as meaning "quasi-coherent of finite presentation". The notion of coherent sheaf, as defined in EGA, is not functorial, that is, pullbacks of coherent sheaves are not necessarily coherent. Hartshorne's book defines "coherent" as "quasi-coherent and finitely generated", but this is a useless notion when working with non-noetherian schemes.

The stack of quasi-coherent finitely presented sheaves is not algebraic either. For example, let $k$ be an algebraically closed field, $k[\epsilon] = k[t]/(t^2)$ the ring of dual numbers. If $S = \mathop{\rm Spec} k[\epsilon]$, consider the coherent sheaf $F$ corresponding to the $k[\epsilon]$-module $k = k[\epsilon]/(\epsilon)$. Then I claim that the functor of automorphisms of $F$ over $S$ is not represented by an algebraic space.

Suppose it is represented by an algebraic space $G \to S$. Denote by $p$ the unique rational point of $S$ over $k$; the tangent space of $S$ at $p$ has a canonical generator $v_0$. Furthermore, if $X$ is a $k$-scheme with a rational point $x_0$ and $v$ is a tangent vector of $X$ at $x_0$, then there exists a unique $k$-morphism $S \to X$ sending $p$ to $x_0$ and $v_0$ to $v$. The inverse image of $S_{\rm red} = \mathop{\rm Spec} k$ in $G$ is isomorphic to $\mathbb G_{\mathrm m, k}$; so $G_{\rm red}$ is an affine scheme, hence $G$ is an affine scheme. The differential of the projection $G \to S$ at the origin of $G$ has a $1$-dimensional kernel, the tangent space of $\mathbb G_{\mathrm m, k}$ at the origin. On the other hand there is a unique section $S \to G$ sending $p$ to the origin, corresponding to $1 \in k^* = \mathrm{Aut}_{k[\epsilon]}k$; this means that there is a unique tangent vector of $G$ at the origin mapping to $v_0$. These two facts give a contradiction.

Here is another way to look at this. Given a contravariant functor $F$ on $k$-schemes and an element $p$ of $F(\mathop{\rm Spec}k)$, one can define the tangent space of $F$ at $p$ as the set of element of $F(\mathop{\rm Spec}k[\epsilon])$ that restrict to $p$. However, in order for this tangent space to be a $k$-vector space, one needs a Schlessinger-like gluing condition on $F$ (this is standard in deformation theory). The analysis above shows that this condition is not satisfied for the functor of automorphisms of $k$ over $k[\epsilon]$.