8
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ I think in this context "stack of objects" means someone has in mind an implicit way to upgrade your category to a stack of categories, and "stack of objects" refers to the corresponding stack of groupoids given by forgetting non-isomorphisms. Quiver representations and coherent sheaves on a curve can both be upgraded to a stack of categories in this way: to a commutative ring $R$ they assign quiver representations over $R$ resp. coherent sheaves on the base change of the curve to $R$. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 11, 2016 at 18:45

1 Answer 1

Reset to default
3
$\begingroup$

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 $Ext^1(M,M)/Hom(M,M)$.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks. I only vaguely understand your answer, however. Do you know of a reference where this might be explained in more detail? $\endgroup$
    – Elle Najt
    Commented Sep 11, 2016 at 19:34
  • 5
    $\begingroup$ I would be very interested to see an elaboration of the statement "infinitesimal deformations of $M$ are given by $Ext^1(M,M)$". For example, in the category of $\mathbb Z$-modules, we have $Ext^1(\mathbb Z/2,\mathbb Z/2)=\mathbb Z/2$". What are the corresponding infinitesimal deformations of $\mathbb Z/2$? If I'm not allowed to take the category of $\mathbb Z$-modules as an example, then I would like to understand which categories I'm allowed to consider. $\endgroup$
    – André Henriques
    Commented Sep 11, 2016 at 21:15

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .