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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 $Q/MQ \cong P$.

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If I'm not mistaken, the first problem is that it is a stack in categories, not a stack in groupoids. Are you throwing away non-invertible morphisms? – S. Carnahan Jun 18 '11 at 9:51
Either we just consider the invertible morphisms, or we extend the definition of a stack to categories fibered in categories. – Martin Brandenburg Jun 18 '11 at 10:30
If you look at quasi-coherent modules then the stack is certainly much too big to be algebraic, but for coherent modules you can have a look at Laumon & Moret-Bailly, Champs Algébriques, thm. – Matthieu Romagny Jun 18 '11 at 11:41
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. Matthieu is absolutely right. – Angelo Jun 18 '11 at 12:13
Thanks. Could you make this into an answer? – Martin Brandenburg Jun 18 '11 at 12:47
up vote 17 down vote accepted

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 $k[\epsilon]$.

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Thanks you for this clear answer :). – Martin Brandenburg Jun 18 '11 at 19:47
The statement in LMB is even a relative one: If $S$ is locally noetherian and $X/S$ is locally of finite type, then $Coh_{X/S}$, sending $U/S$ to the category of coherent sheaves on $U \times_S X$ which are flat over $U$, is an algebraic stack. But what happens when we drop the flatness hypothesis? – Martin Brandenburg Jun 18 '11 at 20:44
Dear Angelo, thank you for the further explanations! – Martin Brandenburg Jun 19 '11 at 8:39

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