Algorithmic Homological Algebra: WS 2016/17

Description

Vorwort Chapter 1. Modules 1.1. Rings 1.2. The category of modules 1.2.a. Objects 1.2.b. Morphisms 1.2.c. Subobjects and factor objects 1.2.d. Kernels and cokernels, images and co-images 1.2.e. Direct sums and direct products 1.2.f. Pull-backs and push-outs as kernels and cokernels Chapter 2. Abelian categories 2.1. Categories 2.2. Additive and Abelian categories Chapter 3. Categories of finite presentations 3.1. Computable rings 3.2. A constructively Abelian category of matrices Chapter 4. Generalized morphisms and functoriality 4.1. Generalized morphisms and their 3-arrow calculus 4.2. The category GA of generalized morphisms 4.3. The category G(A) is enriched over commutative inverse monoids 4.4. The computability of G(A) 4.5. The generalized inverse Chapter 5. Functors, natural transformations, and adjunctions 5.1. Functors 5.2. Natural transformations 5.3. Generalized natural transformations and functoriality 5.4. Adjoint functors Chapter 6. Resolutions and derived functors 6.1. Projectives, injectives, and resolutions 6.2. Homotopies of chain morphisms 6.3. Additive, left exact, and right exact functors 6.4. Derived functors of Abelian categories 6.5. The definitions of Ext and Tor 6.6. Very first steps in group (co)homology Appendix A. A generality on subobject lattices in Abelian categories 1.1. The posets of subobjects and factor objects in general categories 1.2. The Galois duality of the modular lattices Sub M and Fac M in Abelian categories 1.3. Hasse diagrams of specific configurations 1.3.a. One subobject K M 1.3.b. Two 2-chain L K M 1.3.c. Two subobjects L,K M in general position 1.3.d. A 2-chain B Z M and a third subject A M in general position 1.3.e. A pair of 2-chains B A M and D C M in general position 1.4. Hasse diagrams of morphisms in Abelian categories 1.4.a. One morphism 1.4.b. Two composable morphisms Appendix B. Inverse semigroups Bibliography Index