Schanuel's Lemma

The Schanuel's Lemma , named after Stephen Schanuel , is a simple and basic statement from the mathematical branch of homological algebra .

Formulation of the lemma

Let it be a ring and let it be ${\displaystyle R}$

${\displaystyle 0\rightarrow K_{1}\rightarrow P_{1}\rightarrow M\rightarrow 0}$

${\displaystyle 0\rightarrow K_{2}\rightarrow P_{2}\rightarrow M\rightarrow 0}$

short exact sequences in the category of left - modules with projective . Then it holds , that is, the two direct sums are isomorphic . ${\displaystyle R}$ ${\displaystyle P_{1},P_{2}}$ ${\displaystyle P_{1}\oplus K_{2}\cong P_{2}\oplus K_{1}}$

proof

We call the surjections , and consider their fiber product ${\displaystyle \pi _{i}:P_{i}\to M}$

${\displaystyle P_{1}\times _{M}P_{2}=\{(p,q)\in P_{1}\times P_{2}:\pi _{1}(p)=\pi _{2}(q)\}}$.

This is a module and there will be illustrations ${\displaystyle R}$

${\displaystyle {\hat {\pi }}_{i}:P_{1}\times _{M}P_{2}\to P_{i}}$

induced, which are also surjective, as general nonsense shows. It is easy to see that the kernel of isomorphic to also follows that the kernel of isomorphic to is. But since they are projective, these surjections disintegrate, which means that ${\displaystyle {\hat {\pi }}_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle {\hat {\pi }}_{2}}$ ${\displaystyle K_{1}}$ ${\displaystyle P_{i}}$

${\displaystyle P_{1}\oplus K_{2}\cong P_{1}\times _{M}P_{2}\cong P_{2}\oplus K_{1}\ .}$

application

If a projective resolution is such that is projective, then this is true for every projective resolution. ${\displaystyle \ldots \rightarrow P_{1}\rightarrow P_{0}{\xrightarrow {f_{0}}}M\rightarrow 0}$ ${\displaystyle \mathrm {ker} (f_{0})}$

If there is a further projective resolution, consider the short exact sequences ${\displaystyle \ldots \rightarrow Q_{1}\rightarrow Q_{0}{\xrightarrow {g_{0}}}M\rightarrow 0}$

${\displaystyle 0\rightarrow \mathrm {ker} (f_{0})\rightarrow P_{0}\rightarrow M\rightarrow 0}$

${\displaystyle 0\rightarrow \mathrm {ker} (g_{0})\rightarrow Q_{0}\rightarrow M\rightarrow 0}$

According to Schanuel's lemma, is , that is, is a direct summand of the module which is projective according to the assumption and therefore also projective. ${\displaystyle P_{0}\oplus \mathrm {ker} (g_{0})\cong Q_{0}\oplus \mathrm {ker} (f_{0})}$ ${\displaystyle \mathrm {ker} (g_{0})}$ ${\displaystyle Q_{0}\oplus \mathrm {ker} (f_{0})}$

Emergence

Stephen Schanuel discovered this lemma in 1958 during a lecture on homological algebra given by Irving Kaplansky . This essentially concerned the application mentioned above. Kaplansky reports:

During a lecture I carried out the first step of a projective resolution of a module and mentioned that when the core of a resolution is projective again, that also applies to everyone. I added that while this statement was simple, it would take some time to prove. Then Steve Schanuel took the floor and explained to me and the students that this was pretty easy and sketched what is now known as the "Schanuel Lemma".

Individual evidence

1. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-12-599841-4 ( Pure and Applied Mathematics 127), Proposition 2.8.26
2. ↑ Irving Kaplansky: Fields and Rings , University Of Chicago Press (1972), ISBN 0-226-42451-0 , p. 166