# exact sequences for modules with finite projective dimension

Let $R$ be a ring and $M$ be a (left) $R$-module, such that $\mbox{proj dim}(M)=n<\infty$. If

 $0\rightarrow K\rightarrow P_{n}\rightarrow\cdots\rightarrow P_{0}\rightarrow M\rightarrow 0$

is an exact sequence of $R$-modules, such that each $P_{i}$ is projective, then $K$ is projective.

Proof. Since $\mbox{proj dim}(M)=n<\infty$, then there exists exact sequence of $R$-modules

 $0\rightarrow P^{\prime}_{n}\rightarrow\cdots\rightarrow P^{\prime}_{0}% \rightarrow M\rightarrow 0,$

Note that sequences

 $P_{n}\rightarrow\cdots\rightarrow P_{0}\rightarrow M\rightarrow 0;$
 $P^{\prime}_{n}\rightarrow\cdots\rightarrow P_{0}\rightarrow M\rightarrow 0,$

are projective resolutions of $M$. Let $\delta:P_{n}\rightarrow P_{n-1}$ and $\beta:P^{\prime}_{n}\rightarrow P^{\prime}_{n-1}$ be maps take from these resolutions. Then generalized Schanuel’s lemma implies that $\mathrm{ker}\delta$ and $\mathrm{ker}\beta$ are projectively equivalent. But $\mathrm{ker}\delta\simeq K$ and $\mathrm{ker}\beta=0$. This means, that there are projective modules $P,Q$ such that

 $K\oplus P\simeq Q.$

Therefore $K$ is a direct summand of a free module (since $Q$ is), which completes the proof. $\square$

Title exact sequences for modules with finite projective dimension ExactSequencesForModulesWithFiniteProjectiveDimension 2013-03-22 19:04:55 2013-03-22 19:04:55 joking (16130) joking (16130) 4 joking (16130) Corollary msc 16E10 msc 18G20 msc 18G10