An Elementary Approach to Homological Algebra Book

1	Modules	1
1.1	Modules	1
1.2	Free Modules	4
1.3	Exact Sequences	10
1.4	Homomorphisms	18
1.5	Tensor Product of Modules	21
1.6	Direct and Inverse Limits	28
1.7	Pull Back and Push Out	35
2	Categories and Functors	41
2.1	Categories	41
2.2	Functors	54
2.3	The Functors Hom and Tensor	63
3	Projective and Injective Modules	73
3.1	Projective Modules	73
3.2	Injective Modules	80
3.3	Baer's Criterion	83
3.4	An Embedding Theorem	88
4	Homology of Complexes	97
4.1	Ker - Coker Sequence	97
4.2	Connecting Homomorphism - the General Case	105
4.3	Homotopy	112
5	Derived Functors	117
5.1	Projective Resolutions	117
5.2	Injective Resolutions	122
5.3	Derived Functors	125
6	Torsion and Extension Functors	145
6.1	Derived Functors - Revisited	145
6.2	Torsion and Extension Functors	147
6.3	Some Further Properties of Tor[subscript n][superscript R]	155
6.4	Tor and Direct Limits	161
7	The Functor Ext[subscript R][superscript n]	167
7.1	Ext[superscript 1] and Extensions	167
7.2	Baer Sum of Extensions	178
7.3	Some Further Properties of Ext[subscript R][superscript n]	184
8	Hereditary and Semihereditary Rings	189
8.1	Hereditary Rings and Dedekind Domains	189
8.2	Invertible Ideals and Dedekind Rings	196
8.3	Semihereditary and Prufer Rings	200
9	Universal Coefficient Theorem	203
9.1	Universal Coefficient Theorem for Homology	203
9.2	Universal Coefficient Theorem for Cohomology	206
9.3	The Kunneth Formula - A Special Case	209
10	Dimensions of Modules and Rings	215
10.1	Projectively and Injectively Equivalent Modules	215
10.2	Dimensions of Modules and Rings	220
10.3	Global Dimension of Rings	224
10.4	Global Dimension of Noetherian Rings	227
10.5	Global Dimension of Artin Rings	234
11	Cohomology of Groups	239
11.1	Homology and Cohomology Groups	239
11.2	Some Examples	242
11.3	The Groups H[superscript 0](G,A) and H[subscript 0](G,A)	245
11.4	The Groups H[superscript 1](G,A) and H[subscript 1](G,A)	246
11.5	Homology and Cohomology of Direct Sums	250
11.6	The Bar Resolution	256
11.7	Second Cohomology Group and Extensions	262
11.8	Some Homomorphisms	267
11.9	Some Exact Sequences	278
12	Some Applications	293
12.1	An Exact Sequence	293
12.2	Outer Automorphisms of p-Groups	298
12.3	A Theorem of Magnus	303
	Bibliography	309
	Index	313