stuff

algebra/ses.hlean

@@ -64,6 +64,39 @@ definition SES_of_surjective_map {B C : AbGroup} (g : B →g C) (Hg : is_surject
     intro b p, fapply tr, fapply fiber.mk, fapply sigma.mk, exact b, exact p, reflexivity,
   end
 
+definition SES_of_homomorphism {A B : AbGroup} (f : A →g B) : SES (ab_kernel f) A (ab_image f) :=
+  begin
+    fapply SES.mk,
+    exact ab_kernel_incl f,
+    exact image_lift f,
+    exact is_embedding_ab_kernel_incl f,
+    exact is_surjective_image_lift f,
+    fapply is_exact.mk,
+    intro a, induction a with a p, fapply subtype_eq, exact p,
+    intro a p, fapply tr, fapply fiber.mk, fapply sigma.mk, exact a, 
+    exact calc
+      f a = image_incl f (image_lift f a) : by exact homotopy_of_eq (ap group_fun (image_factor f)) a
+      ... = image_incl f 1 : ap (image_incl f) p
+      ... = 1 : by exact respect_one (image_incl f),
+    reflexivity
+  end
+
+definition SES_of_isomorphism_right {B C : AbGroup} (g : B ≃g C) : SES trivial_ab_group B C :=
+  begin
+    fapply SES.mk,
+    exact from_trivial_ab_group B,
+    exact g,
+    exact is_embedding_from_trivial_ab_group B,
+    fapply is_surjective_of_is_equiv,
+    fapply is_exact.mk,
+    intro a, induction a, fapply respect_one,
+    intro b p, 
+    have q : g b = g 1,
+    from p ⬝ (respect_one g)⁻¹, 
+    note r := eq_of_fn_eq_fn (equiv_of_isomorphism g) q, 
+    fapply tr, fapply fiber.mk, exact unit.star, rewrite r,
+  end
+
 structure hom_SES {A B C A' B' C' : AbGroup} (ses : SES A B C) (ses' : SES A' B' C') :=
   ( hA : A →g A')
   ( hB : B →g B')

move_to_lib.hlean

@@ -959,3 +959,36 @@ definition is_surjective_factor : is_surjective h → is_surjective g :=
   end
 
 end injective_surjective
+
+definition AbGroup_of_Group.{u} (G : Group.{u}) (H : Π x y : G, x * y = y * x) : AbGroup.{u} :=
+begin
+  induction G,
+  fapply AbGroup.mk,
+  assumption,
+  exact ⦃ab_group, struct, mul_comm := H⦄
+end
+
+definition trivial_ab_group : AbGroup.{0} := 
+begin
+  fapply AbGroup_of_Group Trivial_group, intro x y, reflexivity
+end
+
+definition trivial_homomorphism (A B : AbGroup) : A →g B :=
+begin
+  fapply homomorphism.mk, 
+  exact λ a, 1,
+  intros, symmetry, exact one_mul 1,
+end
+
+definition from_trivial_ab_group (A : AbGroup) :  trivial_ab_group →g A :=
+  trivial_homomorphism trivial_ab_group A
+
+definition is_embedding_from_trivial_ab_group (A : AbGroup) : is_embedding (from_trivial_ab_group A) :=
+  begin
+    fapply is_embedding_of_is_injective,
+    intro x y p,
+    induction x, induction y, reflexivity
+  end
+
+definition to_trivial_ab_group (A : AbGroup) : A →g trivial_ab_group :=
+  trivial_homomorphism A trivial_ab_group