nLab Schanuel's lemma

Overview

Schanuel’s lemma is basic lemma in homological algebra, useful in the study of projective resolutions.

Statement

Let $R$ be a commutative ring.

Given two short exact sequences of $R$-modules,

$$0 \rightarrow P_0 \rightarrow P_1 \rightarrow M \rightarrow 0$$

$$0 \rightarrow P_0' \rightarrow P_1' \rightarrow M \rightarrow 0$$

with $P_0,P_1,P_0',P_1'$ projective, there is an isomorphism of the direct sums

$$P_0 \oplus P_1' \;\cong\; P_0'\oplus P_1 \,.$$

Proof

Observe the following commutative diagram

$$\array{ && && 0 && 0 && \\ && && \downarrow && \downarrow && \\ & & & & P_0' &\to& P_0'&\to& 0 \\ && && \downarrow && \downarrow && \\ 0 & \to & P_0 &\to& Q &\to& P_1' &\to& 0 \\ && \downarrow && \downarrow && \downarrow && \\ 0 & \to & P_0 &\to& P_1 &\to& M &\to& 0 \\ && && \downarrow && \downarrow && \\ && && 0 && 0 && }$$

where the square $Q-P_1'-M-P_1$ is cartesian and the morphisms $P_0\to Q$ and $P_0'\to Q$ are obtained by the universal property of cartesian square. These morphisms are then necessarily monic and the rows and columns are also exact at $Q$. Thus rows and columns of the diagram are exact. Now, by the projectivity of $P_1'$ and $P_1$ the upper and the left short exact sequences split, hence $P_0\oplus P_1' = Q = P_0'\oplus P_1$.

Literature

Textbook accounts:

• Carl Faith, Algebra: Rings, Modules, Categories, vol. 1, Grundlehren der mathematischen Wissenschaften 190, Springer (1973) [doi:10.1007/978-3-642-80634-6]

It is Proposition 2.8.26 in

• Louis Rowen, Ring theory, student edition.

See also

• Wikipedia, Schanuel’s lemma

category: algebra