Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C\_n$ and $H\_n$ of $R$-modules, which are tame in every possible sense -- finite rank, zero for large enough $n$, etc. Here's the problem:



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> What is the set (space? category?) of all $R$-linear maps $d\_n:C\_n \to C\_{n-1}$ so that $(C\_\bullet,d\_\bullet)$ forms a chain complex with homology groups $H\_n$?
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I've tried to figure things out in baby cases without much luck in terms of seeing the bigger picture. You can easily create situations where no $d\_\bullet$s will suffice, but most fascinating are the cases where several different sequences of boundary operators will work!


A very general thing to do is fix a basis of all chain groups in sight and examine conditions that entries of various $d\_\bullet$-representing matrices have to satisfy. Just requiring the $d\_\bullet$s to produce a chain complex yields an algebraic variety generated by a system of multivariate quadratic equations (which sounds nightmarish). So unless forcing the homology to be prescribed by $H\_n$s greatly simplifies things, I should not expect a very explicit answer. However, I'd like to know what the obstructions to getting such an answer might be.


In the cases of interest, all my $d\_\bullet$s can be assumed to have an upper-triangular form as $R$-matrices, although I'd also be interested in a solution that ignores such structural constraints.


**Update**: Thanks to David Speyer's answer, which provides a wealth of information on the problem when $R$ is a field. I'm still seeking answers for more general choices of $R$.



Moduli space of boundary maps with prescribed chain and homology groups?
Over a field $k$, this is a special case of quiver loci. The general framework is that you are given a triangular array $(r\_{ij})\_{1 \leq i \leq j \leq n}$ of nonnegative integers. Consider a list of $n-1$ matrices $M\_1$, $M\_2$, …, $M\_{n-1}$ of size $r\_{ii} \times r\_{(i+1)(i+1)}$. The quiver locus is the set where $\mathrm{rank}(M\_i M\_{i+1} \cdots M\_{j-1}) \leq r\_{ij}$. Your case is where $r\_{ij}=0$ for $j-i \geq 2$. 


See [Knutson, Miller and Shimozono](http://arxiv.org/abs/math/0308142), and the many citations therein.


To be a bit more explicit, if you want $\dim V\_i = d\_i$ and $\dim H\_i(V\_{\bullet})=h\_i$, then take $r\_{ii} = d\_i$, compute $r\_{i(i+1)}$ by solving the linear equations $r\_{(i-1)i} + r\_{i(i+1)} + h\_i= d\_i$, and put all other $r\_{ij}=0$.


