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splicing together exact sequences
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This article proves a simple but very useful result about ``splicing'' together two exact sequences. Assume we are working in an abelian category such as groups, rings, or modules.
Proposition 1 Let $$ A \to B \xrightarrow{f} C $$ and $$ D \xrightarrow{g} E \to F $$ be exact, and assume that there is an isomorphism $\varphi : \coker f \to \ker g$ . Define $\psi : C\to D: c\mapsto \varphi(\bar{c})$ , where $\bar{c}$ is the image of $c$ in $\coker f$ . Then the following is exact: $$ A \to B \xrightarrow{f} C \xrightarrow{\psi} D \xrightarrow{g} E \to F $$
Proof. Exactness at $C$ : $$ c\in \ker \psi \iff \psi(c) = \varphi(\bar{c})=0 \iff \bar{c} = 0 \iff c\in\im f. $$ Exactness at $D$ : $$ d\in \ker g \iff d = \varphi(\bar{c}) \text{ for some } c\in C \iff d = \psi(c)\text{ for some } c\in C. $$
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"splicing together exact sequences" is owned by rm50.
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| Other names: |
splicing lemma |
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Cross-references: image, isomorphism, modules, rings, groups, abelian category, exact sequences
There is 1 reference to this entry.
This is version 1 of splicing together exact sequences, born on 2009-10-07.
Object id is 11943, canonical name is SplicingTogetherExactSequences.
Accessed 301 times total.
Classification:
| AMS MSC: | 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes) |
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Pending Errata and Addenda
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