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

\[ I^{c + c'}E_ n \to I^{c'}E'_ n/E'_ n[I^{c}] \to E''_ n/E''_ n[I^{c + c'}] \]

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

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