Mathematics 683

Fall 2008

Syllabus

Lecture Notes 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 3.3 3.4 6.1 6.2 6.5 6.6 5.1 5.2 5.4 5.5 5.6 5.7 5.8 6.8

Course Handouts Review for Homological Algebra Homology of a Finite Graph Simplicial Homology of a Tetrahedron Products and Coproducts Some Examples of Categories Sheaf Cohomology The Category of Finite Groups does not have Coproducts The Snake Lemma Direct Limits Pushouts and Pullbacks Spectral Sequences

Suggested Homework Problems Section 1.1: Exercises 1.1.1, 1.1.3, 1.1.4 Section 1.2: What category theory properties hold for the category of projective modules? Section 1.2: What category theory properties hold for the category of sets, where morphisms are relations? Section 1.3: Exercise 1.3.3 Section 1.6: For an Abelian group A let F be the constant presheaf and G the constant sheaf. (1) Prove that there is a natural transformation f: F -> G such that f_U is an isomorphism for each connected open U, (2) Find an example to show that F can fail to be a sheaf. (3) Prove that F_P and G_P are isomorphic to A for each P in X. Section 1.6: Exercise 1.6.3. Section 2.1: Let F be a functor between Abelian categories. Let F_n = F if n = 0 and 0 if n > 0. Also, let F^n = F if n = 0 and 0 otherwise. Determine when {F_n} and {F^n} are homological and cohomological delta-functors, respectively. Section 2.2: Prove that the category of finite Abelian groups has no nonzero projective objects. Also prove that every object in the category of finite Abelian groups of exponent p (where p is a fixed prime) is projective. Section 2.2/2.3: Prove that in an arbitrary category, an arbitrary coproduct of projective objects is projective and an arbitrary product of injective objects is injective (provided that the coproduct/product does exist). Section 2.4: Exercise 2.4.1. Let R1 and R2 be two functors both right adjoint to a functor L. Prove that there is a natural isomorphism R1 -> R2. Let A be a commutative ring and let X = spec(A). For each open set U of X, define F(U) = S^{-1}A, where S is the intersection of A-P as the P range over elements of U. Prove that F is a presheaf on X, and its sheafification is isomorpic to O_X, the sheaf of regular functions on X. Let C (resp. C') : Ab -> Presheaves(X) (resp. Sheaves(X)) be defined by C(A) is the constant presheaf on X and C'(A) is the constant sheaf on X. Prove that C' is isomorphic to the sheafification of C. Prove that the global section functor Sheaves(X) -> Ab is right adjoint to the functor C' defined above. Suppose 0 -> A -> X -> B -> 0 is split exact. Show that it is equivalent to the "standard" split exact sequence 0 -> A -> A direct sum B -> B -> 0. Prove that each short exact sequence 0 -> Z/pZ -> Z/p^2Z -> Z/pZ -> 0 is equivalent to one we discussed in class (where the map Z/p^2Z -> Z/pZ sends a mod p^2 to a mod p). Let G be a group. Prove there is a natural isomorphism of functors hom_G(Z, -) -> ( )^G.