The Stacks project

Lemma 46.8.4. Let $A$ be a ring.

  1. Let $L \to M \to N$ be a universally exact sequence of $A$-modules. Let $K = \mathop{\mathrm{Im}}(M \to N)$. Then $K \to N$ is universally injective.

  2. Any universally exact complex can be split into universally exact short exact sequences.

Proof. Proof of (1). For any $A$-module $T$ the sequence $L \otimes _ A T \to M \otimes _ A T \to K \otimes _ A T \to 0$ is exact by right exactness of $\otimes $. By assumption the sequence $L \otimes _ A T \to M \otimes _ A T \to N \otimes _ A T$ is exact. Combined this shows that $K \otimes _ A T \to N \otimes _ A T$ is injective.

Part (2) means the following: Suppose that $M^\bullet $ is a universally exact complex of $A$-modules. Set $K^ i = \mathop{\mathrm{Ker}}(d^ i) \subset M^ i$. Then the short exact sequences $0 \to K^ i \to M^ i \to K^{i + 1} \to 0$ are universally exact. This follows immediately from part (1). $\square$

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