TITLE: Objects of a category of homological dimension 1 is a smooth stack?
QUESTION [7 upvotes]: I heard a reference to a statement like:
Suppose $A$ is an (Abelian?) category of homological dimension one, then the stack of objects of $A$ is smooth. (I am not really sure what the stack of objects refers to exactly ... I guess it means that there is some stack which naturally equivalent to the underlying groupoid of $A$ (forgetting all non-isomorphisms), or something.)
Here homological dimension refers to the vanishing of Ext groups. So in this case, $Ext^i(M,N) = 0$ if $i \geq 2$.
Examples that were given were: quiver representations and coherent sheaves on a curve.
Could someone link me to a reference to this?
Edit: I heard it in this talk: Victor Ginzburg, Geometry of Quiver Varieties I
at about 9:50.

REPLY [2 votes]: It follows from general deformation theory. Infinitesimal automorphisms of an object $M$ are given by $Hom(M,M)$, first order infinitesimal deformations of $M$ are given by $Ext^1(M,M)$ and obstructions to lift some infinitesimal deformation to the next order live in $Ext^2(M,M)$. So if $Ext^2(M,M)=0$, these obstructions vanish and so the moduli space of objects is (formally) smooth at $M$, formal locally modeled on the quotient stack $Ext^1(M,M)/Hom(M,M)$.