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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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