nLab Schanuel's lemma

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Overview

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

Statement

Let RR be a commutative ring.

Given two short exact sequences of RR-modules,

0P 0P 1M0 0 \rightarrow P_0 \rightarrow P_1 \rightarrow M \rightarrow 0
0P 0P 1M0 0 \rightarrow P_0' \rightarrow P_1' \rightarrow M \rightarrow 0

with P 0,P 1,P 0,P 1P_0,P_1,P_0',P_1' projective, there is an isomorphism of the direct sums

P 0P 1P 0P 1. P_0 \oplus P_1' \;\cong\; P_0'\oplus P_1 \,.

Proof

Observe the following commutative diagram

0 0 P 0 P 0 0 0 P 0 Q P 1 0 0 P 0 P 1 M 0 0 0 \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 QP 1MP 1Q-P_1'-M-P_1 is cartesian and the morphisms P 0QP_0\to Q and P 0QP_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 QQ. Thus rows and columns of the diagram are exact. Now, by the projectivity of P 1P_1' and P 1P_1 the upper and the left short exact sequences split, hence P 0P 1=Q=P 0P 1P_0\oplus P_1' = Q = P_0'\oplus P_1.

Literature

Textbook accounts:

It is Proposition 2.8.26 in

See also

category: algebra

Last revised on September 4, 2024 at 09:13:12. See the history of this page for a list of all contributions to it.