# Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$ for all $n$ (so the sequence is not necessarily a complex, and, if it is, it is an exact sequence). My question is this: does anyone know of any references where such objects have been studied?

The case I am interested in is where $M=\oplus_{n \in \mathbb{Z}} M_n$ is a curved dg-module over some curved dg-algebra; sequences of maps between modules that do not form complexes but which do have the above property come up naturally in this setting.

This question is also posted on MSE: https://math.stackexchange.com/questions/345173/sequences-of-maps-between-modules-such-that-kerd-n-subseteq-textimd-n

• Every module in the sequence has two filtrations - one into the kernels of $d^k$ for increasing $k$ and one into the images of $d^k$ for decreasing $k$. There are almost no relations between these two filtrations, meaning there are a lot of isomorphism classes of these objects. – Will Sawin Mar 30 '13 at 18:30
• Fair enough. But I still think perhaps there are interesting things one might be able to say about the category of such objects (if one were to form the category of these sequences of $R$ modules, call it $\mathcal{C}$, in the natural way). For instance, is there any relationship between $\mathcal{C}$ and the category of complexes of $R$-modules? This is the sort of thing I'd like to understand. I assume questions like these have been thought about before, but I can't seem to find any references for them. – user32666 Mar 30 '13 at 19:13
• MO seems to have collectively answered my question at this point: probably there are no such references, because these objects are not interesting. – user32666 Jul 11 '13 at 15:53