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The Five Lemma

 
 

Exact Sequences Of Homomorphisms

A sequence

~G_1 zDgrmRight{~f_1, _ } ~G_2 zDgrmRight{~f_2, _ } _ . _ . _ . _ . _ zDgrmRight{~f_{~n - 1}, _ } ~G_~n zDgrmRight{~f_~n, _ } ~G_{~n + 1}

of group homomorphisms between abelian groups is said to be ~#{exact} at ~G_~i _ if _ Im ~f_{~i - 1} = ker ~f_~i .

The sequence is ~#{exact} if it is exact at each ~i >= 2.

For the exact sequence

~O zDgrmRight{ &subset. , _ } ~G zDgrmRight{~g, _ } ~H zDgrmRight{~h, _ } ~K zDgrmRight{ _ , _ } ~O

where _ ~O _ is the trivial group \{ 0 \}_+ , we have

  1. ~g _ is injective since _ ker ~g _ = _ Im ( &subset. ) _ = _ \{ 0_~G \}
  2. ~h _ is surjective since _ Im ~h = ker (trivial map) = ~K
  3. ~H ./ Im ~g _ = _ ~H ./ ker ~h _ ~= _ Im ~h _ = _ ~K

The Five Lemma

~A_1 zDgrmRight{~f_1, _ } ~A_2 zDgrmRight{~f_2, _ } ~A_3 zDgrmRight{~f_3, _ } ~A_4 zDgrmRight{~f_4, _ } ~A_5
zDgrmDown{&alpha., _ }   zDgrmDown{&beta., _ }   zDgrmDown{&gamma., _ }   zDgrmDown{&delta., _ }   zDgrmDown{&epsilon., _ }
~B_1 zDgrmRight{ _ ,~g_1} ~B_2 zDgrmRight{ _ ,~g_2} ~B_3 zDgrmRight{ _ ,~g_3} ~B_4 zDgrmRight{ _ ,~g_4} ~B_5

Where all the ~A_~i and ~B_~i are abelian groups, the rows are exact sequences of homomorphisms, and all rectangles commute.

1) _ &alpha. surjective, &beta. and &delta. are injective _ => _ &gamma. is injective

Suppose _ ~x &in. ~A_3 _ such that _ &gamma. ~x = 0 ( &in. ~B_3 ) , _ then _ &delta. ~f_3 ~x = ~g_3 &gamma. ~x = 0 ( &in. ~B_4 ) , so since &delta. is injective , _ ~f_3 ~x = 0 ( &in. ~A_4 ) _ => _ ~x &in. ker ~f_3 = Im ~f_2 . _ So _ &exist. ~w &in. ~A_2 _ such that _ ~f_2 ~w = ~x . _ Now _ ~g_2 &beta. ~w = &gamma. ~f_2 ~w = &gamma. ~x = 0 ( &in. ~B_3 ) _ i.e. _ &beta. ~w &in. ker ~g_2 = Im ~g_1 , _ so _ &beta. ~w = ~g_1 ~u = ~g_1 &alpha. ~v , _ some ~v &in. ~A_1 _ since &alpha. is surjective. _ But _ &beta. ~f_1 ~v = ~g_1 &alpha. ~v = &beta. ~w , _ so _ ~w = ~f_1 ~v _ since &beta. is injective, _ i.e. _ ~w &in. Im ~f_1 = ker ~f_2 , _ so _ ~f_2 ~w = ~x = 0 (&in. ~A_3) , _ &therefore. ker &gamma. = \{0\} (&in. ~A_3) _ => _ &gamma. is injective.

2) _ &beta. and &delta. are surjective, &epsilon. injective _ => _ &gamma. is surjective

Suppose _ ~x &in. ~B_3 , _ then since &delta. is surjective _ &exist. ~y &in. ~A_4 _ such that _ &delta. ~y = ~g_3 ~x , _ further _ &epsilon. ~f_4 ~y = ~g_4 ~g_3 ~x = 0 (&in. ~B_5) , _ and since &epsilon. is injective _ ~f_4 ~y = 0 (&in. ~A_5) , _ so _ ~y &in. ker ~f_4 = Im ~f_3 , _ so _ ~y = ~f_3 ~z _ some ~z &in. ~A_3 . _ So _ &delta. ~f_3 = ~g_3 ~x , _ but _ &delta. ~f_3 ~z = ~g_3 &gamma. ~z , _ i.e. _ ~x - &gamma. ~z &in. ker ~g_3 = Im ~g_2 . _ I.e. _ &exist. ~u &in. ~B_2 , ~w &in. ~A_2 _ such that _ ~x - &gamma. ~z = ~g_2 ~u = ~g_2 &beta. ~w (&beta. surjective) = &gamma. ~f_2 ~w , _ so _ ~x = &gamma. ~f_2 ~w + &gamma. ~z = &gamma. ( ~f_2 ~w + ~z ) , _ any ~x &in. ~B_3 , _ so &gamma. is surjective.