Given two modules $M$ and $N$ there is a nice scheme parametrizing extensions


$0 \rightarrow M \rightarrow E \rightarrow N \rightarrow 0$


namely $\operatorname{Ext}^1(N,M)$ or, leaving out the trivial extension, the projective space $P(\operatorname{Ext}^1(N,M))$.


There are (at least) two natural generalizations


1. $n$-step extensions


$M \rightarrow E\_1 \rightarrow E\_2 \rightarrow \dots \rightarrow E\_n \rightarrow N$


between $N$ and $M$.
2. 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 $\operatorname{Ext}^n(M,N)$, 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?**



Moduli of extensions of modules
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 $\operatorname{Spec} k \to \mathcal M \times \dots \times \mathcal M$ 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 $n > 2$.


