Notes on Tor and Ext
نویسنده
چکیده
1. Basic homological algebra 1 1.1. Chain complexes 2 1.2. Maps and homotopies of maps of chain complexes 2 1.3. Tensor products of chain complexes 3 1.4. Short and long exact sequences 3 1.5. Dual cochain complexes and Hom complexes 4 1.6. Relations between ⊗ and Hom 4 2. The universal coefficient and Künneth theorems 5 2.1. Universal coefficients in homology 5 2.2. The Künneth theorem 6 2.3. Universal coefficients in cohomology 7 2.4. Proof of the universal coefficient theorem 8 2.5. Künneth relations for cochain complexes 9 3. Torsion products 10 3.1. Projective resolutions 10 3.2. The definition and properties of Tor 11 3.3. Change of rings 13 3.4. Pairings and products 14 3.5. A sample computation 15 4. Ext groups 16 4.1. Two identities between Hom functors 16 4.2. Injective resolutions 17 4.3. The definition and properties of Ext 18 4.4. Change of rings 19 4.5. Extensions 20 4.6. Long extensions and the Yoneda product 20 4.7. Equivalences of long extensions 21 4.8. Pairings of Ext groups 23
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