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