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By Edwin Weiss
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2-1-7. Lemma. are G-homotopic. If we put A = @ A , then any two A’s 2-1. 51 HOMOMORPHISM OF PAIRS Proof: Let A and A' be two such A's and put E = A - A'. In all cases, E is an admissible map of differential graded groups (with operator group G ) ; in particular, an+,En+,= E,a:+,. ri) and A_, =0 where the notation is that of (2-1-5) and (2-1-6). v (since eEo = 0) This takes care of n = 0 in (*). For n > 0, we proceed by induction. Thus, assume that (*) holds for n, and put Then using the fact that so that (*) holds for n + 1.
E. a 2-module; right or left is irrelevant), then Horn (X, Y) may be made into a right R-module by cfr)(x) = f ( r x ) f~ Horn (X, Y) IE R 44 I. COHOMOLOGY GROUPS OF G IN A If X is a right R-module, then Hom (X, Y) may be made into a left R-module by f E Horn (X,Y ) ( r f ) ( x ) =f ( m ) YE R (In this connection, see (3-7-8)). (iii) If the Z-module Y is injective, then the R-module Hom (R, Y) is injective (here R may be viewed as either a left or a right R-module). Every R-module can be imbedded in an injective module; in fact, the injective R-module may be taken of form Hom (R, Y) where Y is an injective 2-module.
144. A ( 5 ) If f~ Horn (A, B) is an epimorphism, then morphism. f is a mono- (1) follows from (1-4-2)¶ and (3) follows from (1-4-3). As for (4), it too follows from the properties of the symbol (a, $)-for example, Proof: n g of = (gf, 1) = (f, 1) 0 (g,1) = 34. To prove (5), suppose that f ( t ) = O E A for some t ~ 8Thus . (f,l)t = t of = 0, and since f ( A ) = B this implies t(B) = 0hence t = 0. It remains to prove (2). ,n, u E G). , n xi,. xF=l define fi E A by putting u = 1, . 2'1 . otherwise and extend linearly from the Z-basis to all of A.
Cohomology of Groups by Edwin Weiss