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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

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  • $\begingroup$ 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. $\endgroup$ – Will Sawin Mar 30 '13 at 18:30
  • $\begingroup$ 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. $\endgroup$ – user32666 Mar 30 '13 at 19:13
  • $\begingroup$ MO seems to have collectively answered my question at this point: probably there are no such references, because these objects are not interesting. $\endgroup$ – user32666 Jul 11 '13 at 15:53
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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 C, in the natural way). For instance, is there any relationship between 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.

Well, it seems that the problem is you feel that there must be some functor inbetween the Category of Sequences and the Complexes. Complexes did arise as exact sequences, but when they do not arise as exact sequences, they're often of little interest. But, if you have to use the sequences, then you might want to define a strong boundary homo., which is often a vainless job. As@Will Sawin said, the two contra-directional inclusions hardly have relations.

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  • $\begingroup$ Why noy just make some quotient mapping to make an exact seq.? $\endgroup$ – Henry.L Apr 21 '13 at 10:01

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