The Stacks project

Remark 15.100.6. Let $I$ be an ideal of a Noetherian ring $A$. Set $A_ n = A/I^ n$ for $n \geq 1$. Consider the following category:

1. An object is a sequence $\{ E_ n\} _{n \geq 1}$ where $E_ n$ is a finite $A_ n$-module.

2. A morphism $\{ E_ n\} \to \{ E'_ n\}$ is given by maps

   $$\varphi _ n : I^ cE_ n \longrightarrow E'_ n/E'_ n[I^ c] \quad \text{for }n \geq c$$

   where $E'_ n[I^ c]$ is the torsion submodule (Section 15.88) up to equivalence: we say $(c, \varphi _ n)$ is the same as $(c + 1, \overline{\varphi }_ n)$ where $\overline{\varphi }_ n : I^{c + 1}E_ n \longrightarrow E'_ n/E'_ n[I^{c + 1}]$ is the induced map.

Composition of $(c, \varphi _ n) : \{ E_ n\} \to \{ E'_ n\}$ and $(c', \varphi '_ n) : \{ E'_ n\} \to \{ E''_ n\}$ is defined by the obvious compositions

for $n \geq c + c'$. We omit the verification that this is a category.