A First Course of Homological Algebra by D. G. Northcott

By D. G. Northcott

In response to a sequence of lectures given at Sheffield in the course of 1971-72, this article is designed to introduce the coed to homological algebra averting the flowery equipment frequently linked to the topic. This booklet offers a few vital subject matters and develops the mandatory instruments to deal with them on an advert hoc foundation. the ultimate bankruptcy includes a few formerly unpublished fabric and may offer extra curiosity either for the prepared pupil and his show. a few simply confirmed effects and demonstrations are left as workouts for the reader and extra routines are incorporated to extend the most subject matters. suggestions are supplied to all of those. a brief bibliography presents references to different courses during which the reader may possibly stick to up the themes handled within the publication. 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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Aif. This will ensure that Xnc is a non-zero member of © Ai and complete the solution. iel Since Bix is an essential extension of Ati, there exists fixeA such that fJL1b1 is a non-zero member of Ai . Put Ax = /iv Assume next that we have constructed A1? , Ar for some r < n to meet our requirements. If Ar6r+1 = 0 we put Ar+1 = Ar. On the other hand, if AA + i * 0, then as Bir+i is an essential extension of Air+i there exists jur+1eA such that fir+1Xrbr+1 is a non-zero member of Air+1. In this case we put Ar+1 = /ir+1Ar.

For example, a module is always an essential extension of itself. An extension of a A-module A which is different from A will be said to be non-trivial. Lemma 4. Let A be a submodule of a A-module B and let {CJ ie/ be a non-empty family of submodules of B such that if iv i2£l, then either Cti c: Gx or Ci2 <= Gi . Denote by C the set-theoretic union of the Ct. Then C is a submodule of B. Further if each Gi is an essential extension of A, then C is an essential extension of A. Finally if C,L{\ A = 0 for all i, then C n A = 0.

Anqn= 1. Deduce that a projective ideal of R is necessarily finitely generated. If R is an integral domain and all its non-zero ideals are invertible, then R is called a Dedekind ring. f Exercises marked with an asterisk are likely to prove more difficult than those which are not. 4 Injective modules Let E belong to ^ A . Definition. E is said to be an ' injective A-module' if HomA (-, E) is an exact functor from tfA to the category of abelian groups. Clearly every A-module which is isomorphic to an injective module is also injective.

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