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Math 8030 Introduction to Homological Algebra

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Math 8030 Introduction to Homological Algebra MATH 8030 INTRODUCTION TO HOMOLOGICAL ALGEBRA PETE L. CLARK Contents 1. First Steps in Homological Algebra 3 2. R-Modules 4 2.1. Some Constructions 4 2.2. Tensor and Hom 9 2.3. Splitting 11 2.4. Free Modules 11 2.5. Projective Modules 14 2.6. Injective Modules 15 2.7. Flat Modules 19 3. The Calculus of Chain Complexes 21 3.1. Additive Functors 21 3.2. Chain Complexes as a Category 21 3.3. Chain Homotopies 21 3.4. The Comparison Theorem 23 3.5. The Horseshoe Lemma 24 3.6. Some Diagram Chases 24 3.7. The Fundamental Theorem on Chain Complexes 27 4. Derived Functors 28 4.1. (Well)-definedness of the derived functors 28 4.2. The Long Exact Co/homology Sequence 29 4.3. Delta Functors 30 4.4. Dimension Shifting and Acyclic Resolutions 32 5. Abelian Categories 34 5.1. Definition and First Examples 36 5.2. The Duality Principle 37 5.3. Elements Regained 39 5.4. Limits in Abelian Categories 39 6. Tor and Ext 41 6.1. Balancing Tor 41 6.2. More on Tor 41 6.3. Balancing Ext 44 6.4. More on Ext 44 6.5. Ext and Extensions 50 6.6. The Universal Coefficient Theorems 52 7. Homological Dimension Theory 55 7.1. Syzygies, Cosyzygies, Dimension Shifting 55 7.2. Finite Free Resolutions and Serre's Theorem on Projective Modules 58 7.3. Projective, Injective and Global Dimensions 64 1 2 PETE L. CLARK 7.4. Kaplansky Pairs 67 7.5. The Weak Dimension 69 7.6. The Change of Rings Theorems 70 7.7. The Hilbert Syzygy Theorem 73 8. Dimension Theory of Local Rings 74 8.1. Basics on Local Rings 75 8.2. Some Results From Commutative Algebra 75 8.3. Regular Local Rings 77 8.4. The Auslander-Buchsbaum Theorem 81 8.5. Some \Global" Consequences 82 9. Cohomology (and Homology) of Groups 82 9.1. G-modules 82 9.2. Introducing Group Co/homology 83 9.3. More on the Group Ring 84 9.4. First Examples 85 9.5. Cohomological Dimension 87 9.6. Products 89 10. Functorialities in Group Co/homology 90 10.1. Induction, Coinduction and Eckmann-Shapiro 90 10.2. The Standard Resolutions; Cocycles and Coboundaries 93 10.3. Group co/homology as bifunctors 94 10.4. Inflation-Restriction 95 10.5. Corestriction 95 11. Galois Cohomology 95 11.1. Hilbert's Satz 90 95 11.2. Topological group cohomology 97 11.3. Profinite Groups 97 11.4. The Krull Topology on Aut(K=F ) 99 11.5. Galois Cohomology 100 12. Applications to Topology 102 12.1. Some Reminders 102 12.2. Introducing Eilenberg-MacLane Spaces 104 12.3. Recognizing Eilenberg-MacLane Spaces 106 12.4. Examples of Eilenberg-MacLane Spaces 106 12.5. Eilenberg-MacLane spaces and Group Co/homology 108 References 109 Global notation: In this course \a ring" is not necessarily commutative, but it is associative with a multiplicative identity. For a ring R, R• denotes R n f0g, a monoid under multiplication, and R× denotes the group of units of R, i.e., the set of elements x 2 R such that there is y 2 R with xy = yx = 1. (R× is the group of units of the monoid R•.) In general my terminology follows Bourbaki. For instance compact and locally compact imply Hausdorff. The condition that every open covering has a finite subcovering is called quasi-compact. MATH 8030 INTRODUCTION TO HOMOLOGICAL ALGEBRA 3 1. First Steps in Homological Algebra Exercise 1.1) Let k be a field. a) Let 0 ! V 0 ! V ! V 00 ! 0 be a short exact sequence of k-vector spaces (not assumed to be finite-dimensional). 0 00 Show that dimk V = dimk V + dimk V . b) Let ! ! ! ! ! ! 0 Vn Vn−1 ::: V1 V0 0 P n − i be an exact sequence of finite-dimensional k-vector spaces. Show that i=0( 1) dimk Vi = 0.1 c) Let 0 ! A0 ! A ! A00 ! 0 be a short exact sequence of finite Z-modules. Show #A = #A0 · #A00. d) Let 0 ! A ! A − ! :::A ! A ! 0 n n 1 1Q 0 Z n (−1)i be an exact sequence of finite -modules. Show i=0(#Ai) = 1. e) State and prove a result which simultaneously generalizes parts b) and d). Exercise 1.2) LetL us say that a chain complex M• of left R-modules is finitely generated if n2Z Mn is a finitely generated R-module. Show that this holds iff each Mn is a finitely generated R-module and Mn = 0 for all but finitely many n. Exercise 1.3) Let M• be a finite-dimensional complex of k-vector spaces. Define its Euler characteristic X n χ(M) = (−1) dimk Mn: n Show that X n χ(M) = (−1) dimk Hn(M•): n We say that the Euler characteristic is a homological invariant of M•. Exercise 1.4) Let R be an integral domain (that is, a commutative ring without nonzero divisors of zero) with fraction field K. For a finitely generated complex M• of R-modules, we define the Euler characteristic χ(M•) as χ(M• ⊗R K). Show that X n χ(M) = (−1) dimK Hn(M•) ⊗R K: n Remark: If necessary, feel free to use that K is a flat R-module. Exercise 1.5) Let k be a field, V a fixed K-vector space, and consider the fol- lowing functor from the category of k-vector spaces to itself: FM = M ⊗k V . a) Show that F is an exact functor: that is, if 0 ! M1 ! M2 ! M3 ! 0 1We need to assume finite-dimensionality even for the statement to make sense: subtraction is not a well-defined operation on infinite cardinals. However, if we rewrote the alternating sum as \the sum of the dimensions of the even-indexed vector spaces equals the sum of the dimensoins of the odd-indexed vector spaces, then this is true without the hypothesis of finite dimensionality. 4 PETE L. CLARK is a short exact sequence of k-vector spaces, then so is 0 ! F (M1) ! F (M2) ! F (M3) ! 0: b) Show that if M• is a complex of K-vector spaces, for all n 2 Z there is an isomorphism ∼ Hn(FM•) = FHn(M•): Exercise 1.6) Let 0 ! M1 ! M2 ! M3 ! 0 be a short exact sequence of Z-modules. We tensor the sequence with Z=2013Z. a) Show that M1 ⊗ Z=2013Z ! M2 ⊗ Z=2013Z ! M3 ⊗ Z=2013Z ! 0 is exact. b) Show by example that M1 ⊗ Z=2013Z ! M2 ⊗ Z=2013Z need not be injective. Exercise 1.7) Let X be the Cantor set (X is characterized up to homeomorphism by being compact, second countable, totally disconnected and without isolated points). a) Show that there is no CW-complex with underlying topological space X. b) Show that X is not even homotopy equivalent to the underlying topological space of a CW-complex. Remark: Spaces homeomorphic to the Cantor set arise frequently in number the- ory: e.g. the ring Zp of p-adic integers. (However, admittedly such spaces are not interesting from the perspective of topological homology.) Exercise 1.8) Let F be an exact functor from the category of R-modules to the category of S-modules. Let M• be a chain complex of R-modules. Show: for all n 2 Z, ∼ Hn(FM•) = FHnM•: 2. R-Modules 2.1. Some Constructions. In this section we discuss various basic module-theoretic constructions. Most of them are special cases of vastly more general category-theoretic constructions, and this more general perspective will be taken later: for now, we stay concrete. There is a common theme running through all of these constructions: namely, they all satisfy universal mapping properties. In general, universal mapping properties make for good definitions because the uniqueness of the associated object is assured, and they are often (though not always) useful for giving clean proofs of various properties satisfied by these objects. One still has the task of proving existence of the objects in question, which in all these cases can be done relatively straightforwardly. For our first two examples the constructions are so simple and familiar that they are given first, followed by the universal mapping properties that they sat- isfy. Thereafter the approach is inverted: we give the universal mapping property, followed by the (messier) explicit construction. MATH 8030 INTRODUCTION TO HOMOLOGICAL ALGEBRA 5 2.1.1. Direct Products. f g Let Mi i2I be an indexed family of modulesQ (here I denotes an arbitrary set). We define an R-module, the direct product i2I Mi, as follows: as a set, it is the 2 usual Cartesian product, i.e., the collection of all families of elements fxi j xi 2 Ig. We endow it with the structure of an R-module by fxig + fyig = fxi + yig; f g f g r xi = rxi Q: 2 ! f g 7! For each i I there is a projection map πi : i Mi Mi, xi xi. Proposition 2.1. (Universal Property of the Direct Product) Let fMigi2I be a family of R-modules, let A be an R-module, and suppose that for each i 2 I we ! are givenQ an R-module map 'i : A Mi. Then there is a unique R-module map ! 2 ◦ Φ: A i2I Mi such that for all i I, 'i = πi Φ: namely Φ: a 2 A 7! f'i(a)g: Exercise: Prove it. 2.1.2. Direct Sums. f g LFor a family Mi i2I of R-modules, we define an R-module, the direct sum i2I Mi as follows: it is the R-submodule spanned by theL elements ei, which is 1 in the ith coordinate and otherwise 0.
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