Five lemma: Difference between revisions

Latest revision as of 19:44, 11 May 2008

This is a lemma involving commutative diagrams that can be proved by means of diagram chasing

Let $A_{1} \to A_{2} \to A_{3} \to A_{4} \to A_{5}$ and $B_{1} \to B_{2} \to B_{2} \to B_{4} \to B_{5}$ be exact sequences of homomorphisms. Suppose there are maps $f_{i} : A_{i} \to B_{i}$ such that the diagram of all these maps commutes. Then the following are true:

If $f_{2}$ and $f_{4}$ are injective and $f_{1}$ is surjective, then $f_{3}$ is injective
If $f_{2}$ and $f_{4}$ are surjective and $f_{5}$ is injective then $f_{3}$ is surjective
if $f_{1}, f_{2}, f_{4}, f_{5}$ are isomorphisms, then $f_{3}$ is also an isomorphism

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 * If <math>f_2</math> and <math>f_4</math> are injective and <math>f_1</math> is surjective, then <math>f_3</math> is injective
-* If <math>f_2</math> and <math>f_4</math/ are surjective and <math>f_5</math> is injective then <math>f_3</math> is surjective
+* If <math>f_2</math> and <math>f_4</math> are surjective and <math>f_5</math> is injective then <math>f_3</math> is surjective
 * if <math>f_1,f_2,f_4,f_5</math> are isomorphisms, then <math>f_3</math> is also an isomorphism