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: http://math.stackexchange.com/questions/345173/sequences-of-maps-between-modules-such-that-kerd-n-subseteq-textimd-n

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

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: http://math.stackexchange.com/posts/345173/edit

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: http://math.stackexchange.com/questions/345173/sequences-of-maps-between-modules-such-that-kerd-n-subseteq-textimd-n