TITLE: Moduli of Extensions
QUESTION [7 upvotes]: Given two modules $M$ and $N$ there is a nice scheme parametrizing extensions
$0 \rightarrow M \rightarrow E \rightarrow N \rightarrow 0$
namely $Ext^1(N,M)$ or, leaving out the trivial extension, the projective space $P(Ext^1(N,M))$.
There are (at least) two natrual generalizations

n-step extensions
$M \rightarrow E_1 \rightarrow E_2 \rightarrow \dots \rightarrow E_n \rightarrow N$
between $N$ and $M$.
Filtered modules: Parametrize modules $E$ which admit a filtration 
$0 \subset F_1 \subset F_2 \dots F_n=E$
with fixed graded objects $E_i=F_i/F_{i-1}$.

I suppose in the first case one can use the group Ext^n(M,N), although I never
saw a construction of a universal family. Is there a good reference?
In the second case, I do not have a clue. So my main question is:
Is there a nice moduli space of filtered objects?

REPLY [3 votes]: Hi Heinrich,
in the situation you have in mind (sheaves on an algebraic variety),
such spaces are not too difficult to construct as Artin stacks. If you omit the condition that the i-th filtration quotient is isomorphic to a given one, then such a universal Artin stack is e.g. constructed in Bridgeland's introduction to Hall-algebras (arXiv:1002.4372, he calls them $\mathcal M^{(n)}$), but of course also in earlier articles by Joyce. Basically it follows from the existence of relative quot schemes.
These universal extension stacks have evaluation morphisms to $\mathcal M$, the stack of all sheaves, sending the filtration to its i-th quotients, so you can take a base change via the map from $\Spec k \to \mathcal M \times \dots \times \mathcal M$ given by your set of objects $E_i$, and the fiber product will be the Artin stack you are looking for.
If you want a scheme instead of an Artin stack - then I would ask back "why?" :) Nevertheless, it would be useful to understand this fiber product better when $n > 2$.