By D. G. Northcott
In accordance with a sequence of lectures given at Sheffield in the course of 1971-72, this article is designed to introduce the coed to homological algebra fending off the frilly equipment often linked to the topic. This booklet offers a couple of very important themes and develops the mandatory instruments to deal with them on an advert hoc foundation. the ultimate bankruptcy comprises a few formerly unpublished fabric and may supply extra curiosity either for the willing scholar and his teach. a few simply confirmed effects and demonstrations are left as routines for the reader and extra routines are integrated to extend the most topics. suggestions are supplied to all of those. a quick bibliography presents references to different guides during which the reader may well keep on with up the topics handled within the booklet. Graduate scholars will locate this a useful direction textual content as will these undergraduates who come to this topic of their ultimate yr.
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Additional info for A First Course of Homological Algebra
Definition. Let A be a A-module. An 'injective envelope' of A is an injective, essential extension of A. Theorem 17. Let A be a A-module and E an injective extension module of A. Then E contains a submodule which is an injective envelope of A. Remark. In view of Theorem 14, this shows that every A-module has an injective envelope. We shall see later that injective envelopes are virtually unique. Proof. Let 2 consist of all the submodules of E that are essential extensions of A and let 2 be partially ordered by inclusion.
Accordingly rj is surjective and hence bijective. Since rj is a bijection and the centre of A is a subring of A, it only remains to be shown that rj preserves sums and products. This however is clear. PA Supplementary Exercises on Chapter 1 Exercise A. Let /i:A ->A'be a A-homomorphism. Show that the following statements are equivalent: (1) /i(ax) + ju>(a2) whenever ava2 are distinct elements of A; (2) pf\ — pf% (for A-homomorphisms f± andf2) always implies^ = f2. Solution. Assume (1). >)) a n d s °> b y (i)> Assume (2).
Hence y maps elements of the centre of A to natural transformations of the identity functor into itself. (y2)- T h e n V(7I)A = 7/(72)A' where A is considered as a left A-module, and in particular yx = 7i(y1)A (1) = 7)(y2)A (1) = y2. Hence 7/ is an injection. Next let [i\I->I be a natural transformation and put y = /iA(l). If now A eA, define a homomorphism / : A-> A by /(A') = A'A. The commutative diagram A f shows that yA = / ( y ) = / > A ( l ) = /iA(f(l)) =/(l)/* A (l) = Ay. Thus y is in the centre of A.
A First Course of Homological Algebra by D. G. Northcott