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Math 99R - Representations and Cohomology of Groups

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Math 99R - Representations and Cohomology of Groups Math 99r - Representations and Cohomology of Groups Taught by Meng Geuo Notes by Dongryul Kim Spring 2017 This course was taught by Meng Guo, a graduate student. The lectures were given at TF 4:30-6 in Science Center 232. There were no textbooks, and there were 7 undergraduates enrolled in our section. The grading was solely based on the final presentation, with no course assistants. Contents 1 February 1, 2017 3 1.1 Chain complexes . .3 1.2 Projective resolution . .4 2 February 8, 2017 6 2.1 Left derived functors . .6 2.2 Injective resolutions and right derived functors . .8 3 February 14, 2017 10 3.1 Long exact sequence . 10 4 February 17, 2017 13 4.1 Ext as extensions . 13 4.2 Low-dimensional group cohomology . 14 5 February 21, 2017 15 5.1 Computing the first cohomology and homology . 15 6 February 24, 2017 17 6.1 Bar resolution . 17 6.2 Computing cohomology using the bar resolution . 18 7 February 28, 2017 19 7.1 Computing the second cohomology . 19 1 Last Update: August 27, 2018 8 March 3, 2017 21 8.1 Group action on a CW-complex . 21 9 March 7, 2017 22 9.1 Induction and restriction . 22 10 March 21, 2017 24 10.1 Double coset formula . 24 10.2 Transfer maps . 25 11 March 24, 2017 27 11.1 Another way of defining transfer maps . 27 12 March 28, 2017 29 12.1 Spectral sequence from a filtration . 29 13 March 31, 2017 31 13.1 Spectral sequence from an exact couple . 32 13.2 Spectral sequence from a double complex . 32 14 April 4, 2017 34 14.1 Hochschild{Serre spectral sequence . 34 15 April 7, 2017 37 15.1 Homology and cohomology of cyclic groups . 37 16 April 11, 2017 39 16.1 Wreath product . 39 17 April 14, 2017 41 17.1 Cohomology of wreath products . 41 18 April 18, 2017 44 18.1 Atiyah completion theorem . 44 18.2 Skew-commutative graded differential algebras . 46 19 April 21, 2017 49 19.1 Cohomology of D2n ......................... 49 19.2 The third cohomology . 50 20 April 25, 2017 52 20.1 Swan's theorem . 52 20.2 Grothendieck spectral sequence . 53 20.3 Cechˇ cohomology . 56 2 Math 99r Notes 3 1 February 1, 2017 1.1 Chain complexes In the context of and abelian category, we can define a chain complex. Definition 1.1. An abelian category A is a category such that: • HomA (A; B) is an abelian group, • there exists a zero object 0 that is both initial and terminal, i.e., Hom(A; 0) and Hom(0;B) are always trivial, • composition Hom(A; B) × Hom(B; C) ! Hom(A; C) is bilinear, • it has finite products and coproducts (in this case they agree), • for any φ : A ! B there exists a kernel σ : K ! A with the required universal property, • for any φ : A ! B there exists a cokernel σ : B ! Q, • every monomorphism is the kernel of its cokernel, • every epimorphism is the cokernel of its kernel, • any morphism φ : A ! B factors like A ! C ! B so that A ! C is an epimorphism and C ! B is a monomorphism. Example 1.2. The category of R-modules is an abelian category. Definition 1.3. A functor F : A ! B is called additive if Hom(A; B) ! Hom(F (A);F (B)) is a group homomorphism. Definition 1.4. A chain complex fCngn2Z is a sequence of objects Cn 2 A with boundary maps dn : Cn ! Cn−1 such that dn−1 ◦ dn = 0. d2 d1 d0 d−1 d−2 ··· −! C1 −! C0 −! C−1 −−! C−2 −−!· · · Because we have the notion of a kernel, we can say that in a chain complex, there is a monomorphism im dn+1 ,! ker dn. We then define the homology as Hn(C•; d•) = coker(im dn+1 ! ker dn): Definition 1.5. A cochain complex is (C•; d•) has maps dn : Cn ! Cn+1 that satisfy dn+1 ◦ dn = 0. We likewise define the cohomology. A morphism f• :(C•; d•) ! (D•; δ•) is a collection of maps fn : Cn ! Dn such that the diagram dn Cn Cn−1 fn fn−1 δn Dn Dn−1 Math 99r Notes 4 commute for all n. In this case, the chain map gives rise to maps H•(f•): H•(C•; d•) ! H•(D•; δ•) on homology. Also there is an obvious equivalence between categories of chain complexes −n and cochain complexes given by Cn 7! C . Definition 1.6. A chain homotopy between two maps f•; g• :(C; d) ! (D; δ) is a collection of maps hn : Cn ! Dn+1 such that hn−1d + dhn = fn − gn: 1.2 Projective resolution Definition 1.7. An object P 2 C is called projective if you can always (maybe not uniquely) lift f to f~ making the diagram commute: X f~ f P Y A projective R-module is a projective object in the category of R-modules. Note that this is equivalent to saying that HomR(P; −) is an exact functor. This is because HomR(P; −) is automatically a left exact functor and this says projectivity says that it preserves epimorphisms. Definition 1.8. A projective resolution of A is a long exact sequence ···! Pn ! Pn−1 ! Pn−2 !···! P0 ! A ! 0; or equivalently a chain complex ···! P1 ! P0 with homology ( 0 if n > 0 Hn(P•) = A if n = 0: Definition 1.9. A positive complex is a complex (C•; d•) with Cn = 0 for all n < 0. Definition 1.10. An positive acyclic chain complex is a complex (C; d) with Cn = 0 for n < 0 and Hn(C) = 0 for n > 0. So a projective resolution is positive chain complex that is both acyclic and projective (i.e., consisting of projective objects). Proposition 1.11. If (C; d) and (D; δ) are two positive chain complexes, (C; d) is projective, and (D; d0) is acyclic, then ' H0(') (C• −! D•) 7! H0(C•) −−−−! H0(D•) Math 99r Notes 5 induces a bijection homotopy classes of ! fH0(C•) ! H0(D•)g: chain maps C• ! D• Proof. So we have to prove two things: first that we can get maps on chains from H0(C) ! H0(D), and that if two maps give the same maps then they are chain homotopic. We are inductively going to lift maps. Cn Cn−1 Cn−1 dn dn−1 Dn Dn−1 Dn−2 Note that the composition Cn ! Cn−1 ! Dn−1 has image lying in the ker dn−1 = im dn. Then using that Dn ! im dn is an epimorphism and that Cn is projec- tive, we can lift this map. Now we show that two chain maps giving the same maps on homology is homotopic. We may assume that f• gives the zero map H0(C) ! H0(D) and show that f• is null-homotopic. You can check this by lifting maps similarly. Math 99r Notes 6 2 February 8, 2017 Last time we defined abelian categories, chain complexes and cochain complexes, associated homology and cohomology. Also we defined a projective resolution of some object A 2 A , which is basically a long exact sequence ending with ···! A ! 0 consisting of projective objects. For two objects A; B 2 A , we saw that the set of morphisms A ! B is the homotopy classes of a chain map from a projective complex to an acyclic complex. Corollary 2.1. Any two projective resolutions P•;Q• of A are homotopic to each other. In other words, there are chain maps φ : P• ! Q• and : Q• ! P• such that φ ◦ ∼ idQ and ◦ φ ∼ idP . Definition 2.2. We say that A has enough projectives if for any A 2 A there exists a projective P 2 A and an epimorphism P ! A. This is equivalent to saying that any A 2 A has a projective resolution, because we can inductively choose the projective that surjects onto the kernel. Theorem 2.3. The category R−Mod has enough projectives. 2.1 Left derived functors Suppose we have an additive functor F : A ! B, i.e., functor that preserves the additive group structure of Hom. If we apply F to a projective resolution ···! P2 ! P1 ! P0 ! A ! 0; we get a chain complex ···! F (P2) ! F (P1) ! F (P0) ! 0: This is not in general exact, but it is a chain complex and so we can compute its homology. We define the n-th left derived functor associated to P• as P• Ln F (A) = Hn(F (P•)): Why is it a functor? If there is a map A ! B then we get a map between projective resolutions, and then we get a map between homology. Theorem 2.4. Suppose we have an additive functor F : A ! B. Let P; Q be two assignment of projective resolutions to objects of A . There exists a P Q canonical natural isomorphism Ln F ' Ln F between the two functors. The main idea is that there is a chain map PA ! QA given by the identity idA and then this induces a Hn(PA) ! Hn(QA). Lemma 2.5. An additive functor F : A ! B induces an additive functor F : ChA ! ChB, given by F (C•)n = F (Cn) and F ( •)n = F ( n). This satisfies the following: Math 99r Notes 7 (i) If Σ• : • ! '• is a homotopy in two maps •;'• 2 HomChA (C•;D•), then F (Σ•): F ( •) ! F ('•) is also a homotopy between F ( •);F ('•) 2 HomChB(F (C•);F (D•)). (ii) If C• ' D• are homotopic in ChA , then F (C•) ' F (D•) in ChB.
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