The Stacks project

Lemma 12.31.1. Let $\mathcal{C}$ be a category.

  1. If $\mathcal{C}$ is an additive category, then the category of inverse systems with values in $\mathcal{C}$ is an additive category.

  2. If $\mathcal{C}$ is an abelian category, then the category of inverse systems with values in $\mathcal{C}$ is an abelian category. A sequence $(K_ i) \to (L_ i) \to (M_ i)$ of inverse systems is exact if and only if each $K_ i \to L_ i \to N_ i$ is exact.

Proof. Omitted. $\square$

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