Math 201a spring 2017

Announcements

Math 201a Spring 2017 will be a basic course in homological algebra. See the syllabus for more details.

Professor Rogalski: Office 5131 AP&M, e-mail drogalski@ucsd.edu.

4/3/17: Categories and examples. Preadditive categories. Ring of Endomorphisms of an object in a preadditive category. Category of chain complexes over a module category. Homology group of a complex. Functors, covariant and contravariant. Examples of functors.

4/5/17: Additive functors. Functors arising from Hom. Concrete categories. Exact sequences. Tensor products over noncommutative rings and their universal properties. Functors arising from tensor product.

4/7/17: Left exact and right exact functors. Hom functors are left exact (proof on homework) and tensor functors are right exact (proof later). Bimodules. Examples of bimodules. Extra structures on tensor and Hom arising from bimodules. Adjoint isomorphisms between tensor and Hom (two versions).

4/10/17: Adjoint functors. Hom and tensor give examples of adjoint functors. Other examples of adjoints. For functors between module categories, left adjoints are right exact and right adjoints are left exact. In particular, tensor products are left exact. Products. Examples of products in various categories. Products pull out of the second coordinate of Hom.

4/12/17: Coproducts. Examples of coproducts in various categories. Coproducts pull out of the first coordinate of Hom as products. Projective and injective modules. P is projective if and only if Hom(P, -) is exact, and E is injective if and only if Hom(-, E) is exact. Flat modules. Free modules are projective. Every module is a surjective image of a free module.

4/17/17: Split short exact sequences. A short exact sequence 0 to M to N to P to 0 with P projective is split. Projective modules are exactly the direct summands of free modules. Examples of projectives. Baer's criterion for injective modules.

4/19/17: Injective modules over a domain are divisible. Over a PID divisible modules are injective. Every Z-module is contained in an injective Z-module. Every R-module is contained in an injective R-module. Products of injectives are injective and direct sums of projectives are projective. Projective and injective resolutions and projective and injective dimension. Brief introduction to the idea of a derived functor.

4/24/17: Snake lemma. A short exact sequence of complexes produces a long exact sequence of homology groups. Definition of a morphism of complexes being homotopic to 0. A morphism of complexes which is homotopic to 0 induces the 0 map on homology. Lifting lemma for maps from a complex of projectives to an exact complex.

4/26/17: Proof of lifting lemma. Horseshoe lemma. Definition of left derived functors of a covariant additive functor. Proof that this is independent of choices made. The long exact sequence in left derived functors associated to a short exact sequence.

5/1/17: If F is right exact than L_0 F = F. Natural transformations of functors and natural isomorphisms. Above we really mean L_0 F is naturally isomorphic to F. When F = - \otimes_R M we write Tor_i^R(-, M) for L_i F. When F = N \otimes_R - we write Tor_i^R(N, -) for L_i F. Fact: Tor^i(N,M) is well defined whether it means Tor_i^R(-, M) applied to N or Tor_i^R(N, -) applied to M (proof later). F is an exact functor if and only if L_i F = 0 for all i \geq 1, if and only if L_1 F = 0. Example of calculating Tor. Right derived functors of a covariant functor. Right derived functors of a contravariant functor. The groups Ext^i_R(M,N).

5/3/17: Naturality of the long exact sequence. Extensions of modules. Equivalence of extensions. Theorem: extensions of N by M up to equivalence are in bijection with elements of Ext^1(N,M). Pushout squares. Beginning of proof of theorem.

5/5/17: Conclusion of proof of theorem. Abelian group structure on extensions is given by Baer sum. Scalar multiplication on extensions when R is a k-algebra. Examples.

5/8/17: Proof that Tor^R_i(M,N) can be calculated with projective resolutions in either coordinate. Mapping cones. Quasi-isomorphisms of complexes. Double complexes. The total complex of a double complex. Tensor product complexes. If a bounded below double complex has exact rows, its total complex is exact.

5/10/17: Global dimension. Schanuel's lemma and the generalized Schanuel lemma. Syzygies. Characterization of projective dimension in terms of Ext. Characterization of injective dimension in terms of Ext. left Global dimension is the sup of projective dimensions of all left modules, and the sup of injective dimensions of all left modules. Rings of global dimension 0 are semisimple rings.

5/15/17: Hereditary rings. Characterizations of hereditary rings. Left global dimension is the supremum of projective dimensions of cyclic left modules. Commutative hereditary domains are the same as Dedekind domains. More examples: matrix rings over hereditary rings. The Weyl algebra. Free associative algebras. Path algebras. An example of a ring which is hereditary on one side only.

5/24/17: Flat resolutions and flat dimension. Characterization of flat dimension in terms of Tor. Weak dimension. Left and right weak dimension are the same. Over a noetherian ring, finitely generated flat modules are projective. Noetherian rings have left and right global dimensions equal, and both are equal to the flat dimension. Some other results on global dimension (without proof): gldim R[x] = gldim R + 1. Commutative noetherian local rings are regular if and only if they have finite global dimension. Regular local rings are UFDs.

5/26/17: Exact couples. The derived couple of an exact couple. Filtered complexes. The exact couple associated to a filtered complex. Bicomplexes and total complexes. The two standard filtrations of a bicomplex.

5/31/17: A bounded filtered complex leads to a spectral sequence which converges. Proof of the convergence.

6/2/17: A first quadrant bicomplex gives two spectral sequences for each of the two filtrations of its total complex. Proof that the E^2 pages are formed by first taking vertical and then horizontal homology, or first horizontal and then vertical homology, respectively.

6/5/17: Applications of collapsing spectral seqeuences. Introduction to Grothendieck spectral sequences. Some examples of Tor-Tor spectral sequences.

6/7/17: Cartan-Eilenberg resolutions. Proof of the Grothendieck spectral sequence.