Splitting Lemma
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent. Given a short exact sequence with maps q and r: 0 rightarrow A overset{q}{longrightarrow} B overset{r}{longrightarrow} C rightarrow 0 one writes the additional arrows t and u for maps that may not exist: 0 rightarrow A {{q atop longrightarrow} atop {longleftarrow atop t}} B {{r atop lon...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent. Given a short exact sequence with maps q and r: 0 rightarrow A overset{q}{longrightarrow} B overset{r}{longrightarrow} C rightarrow 0 one writes the additional arrows t and u for maps that may not exist: 0 rightarrow A {{q atop longrightarrow} atop {longleftarrow atop t}} B {{r atop longrightarrow} atop {longleftarrow atop u}} C rightarrow 0 Then the following are equivalent: 1. left split there exists a map t: B A such that tq is the identity on A, 2. right split there exists a map u: C B such that ru is the identity on C, 3. direct sum B is isomorphic to the direct sum of A and C, with q being the natural injection of A and r being the natural projection onto C.
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