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Unformatted text preview: HOMOLOGICAL ALGEBRA Contents Part 0. Announcements 3 0. What does the Homological algebra do? 3 0.1. Some examples of applications 3 0.2. Topics 4 0.3. The texts 4 Part 1. Intro 5 1. Algebraic topology: Homology from triangulations 5 1.1. Linear Simplices 5 1.2. Topological simplices 7 1.3. Triangulations 7 1.4. Combinatorial Topology of Simplicial Complexes 8 1.5. Complex C * ( X, T ; k ) 10 1.6. Examples 13 2. Duality for modules over rings 15 2.1. Capturing a class of objects in terms of a smaller and better behaved subclass 15 2.2. Rings 15 2.3. Duality and biduality of kmodules 16 2.4. What is the dual of the abelian group Z n ? 17 2.5. Resolutions 19 2.6. Free resolutions 21 2.7. Derived category of kmodules 22 2.8. Derived versions of constructions 24 2.9. A geometric example: duality for the ring of polynomials 25 2.10. Comments 26 Date : ? 1 2 3. Sheaves 27 3.1. Definition 27 3.2. Global sections functor Γ : S heaves ( X )→S ets 28 3.3. Projective line P 1 over C 29 3.4. ˇ Cech cohomology of a sheaf A with respect to a cover U 29 3.5. Vector bundles 30 3.6. ˇ Cech cohomology of line bundles on P 1 31 3.7. Geometric representation theory 32 3.8. Relation to topology 33 So Okada has suggested a number of corrections. 3 Part 0. Announcements 0. What does the Homological algebra do? Homological algebra is a general tool useful in various areas of mathematics. One tries to apply it to constructions that morally should contain more information then meets the eye. Homological algebra, when it applies, produces “derived” versions of the construction (“the higher cohomology”), which contain the “hidden” information. In a number of areas, the fact that that with addition of homological algebra one is not missing the less obvious information allows a development of superior techniques of calculation. The goal of this course is to understand the usefulness of homological ideas in applica tions. and as usual, to use this process as an excuse to visit various interesting topics in mathematics. 0.1. Some examples of applications. Some basic applications : (1) Algebraic topology. It can loosely be described as a “systematic way of counting holes in manifolds”. While we can agree that a circle has a 1dimensional hole (in the sense of “a hole that can be made by a one dimensional object”) and a sphere has a 2dimensional hole, algebraic topology finds that the surface of a pretzel has one 2dimensional hole and four 1dimensional holes. These “holes” or “cycles” turn out to be essential in problems in geometry and analysis. (2) Cohomology of sheaves. It deals with an omnipresent problem of relating local and global information on a manifold....
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 Winter '11
 PhanThuongCang
 Algebraic Topology, Vector Space, Topological space, Homological algebra, simplices

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