0¬ABC÷0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s : C → B so that ps = idc. 3. B AC, and there is an isomorphism from the given sequence to the sequence 0→A→A@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.
0¬ABC÷0 be a short exact sequence of abelian groups. Show that the following are equivalent. 1. There is a homomorphism r: B → A such that ri = id; 2. There is a homomorphism s : C → B so that ps = idc. 3. B AC, and there is an isomorphism from the given sequence to the sequence 0→A→A@C→C→0 with the obvious inclusion and projection maps. We call such exact sequences split.
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Exercise 3.4. Let
0¬ABC 0
be a short exact sequence of abelian groups. Show that the following are equivalent.
1. There is a homomorphism r: B → A such that ri = id;
2. There is a homomorphism s: C → B so that ps = idc.
3. BAC, and there is an isomorphism from the given sequence to the sequence
0→AA@C→C→0 with the obvious inclusion and projection maps.
We call such exact sequences split.
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