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ALGEBRA: LECTURE NOTES §1. Categories and Functors 2 §1.1

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ALGEBRA: LECTURE NOTES §1. Categories and Functors 2 §1.1 ALGEBRA: LECTURE NOTES JENIA TEVELEV CONTENTS x1. Categories and Functors 2 x1.1. Categories 2 x1.2. Functors 5 x1.3. Equivalence of Categories 6 x1.4. Representable Functors 7 x1.5. Products and Coproducts 8 x1.6. Natural Transformations 9 x1.7. Exercises 10 x2. Tensor Products 12 x2.1. Tensor Product of Vector Spaces 12 x2.2. Tensor Product of R-modules 14 x2.3. Categorical aspects of tensors: Yoneda’s Lemma 16 x2.4. Hilbert’s 3d Problem 21 x2.5. Right-exactness of a tensor product 25 x2.6. Restriction of scalars 28 x2.7. Extension of scalars 29 x2.8. Exercises 30 x3. Algebraic Extensions 31 x3.1. Field Extensions 31 x3.2. Adjoining roots 33 x3.3. Algebraic Closure 35 x3.4. Finite Fields 37 x3.5. Exercises 37 x4. Galois Theory 39 x4.1. Separable Extensions 39 x4.2. Normal Extensions 40 x4.3. Main Theorem of Galois Theory 41 x4.4. Exercises 43 x5. Applications of Galois Theory - I 44 x5.1. Fundamental Theorem of Algebra 44 x5.2. Galois group of a finite field 45 x5.3. Cyclotomic fields 45 x5.4. Kronecker–Weber Theorem 46 x5.5. Cyclic Extensions 48 x5.6. Composition Series and Solvable Groups 50 x5.7. Exercises 52 x6. Applications of Galois Theory -II 54 x6.1. Solvable extensions: Galois Theorem. 54 1 2 JENIA TEVELEV x6.2. Norm and Trace 56 x6.3. Lagrange resolvents 57 x6.4. Solving solvable extensions 58 x6.5. Exercises 60 x7. Transcendental Extensions 61 x7.1. Transcendental Numbers: Liouville’s Theorem 61 x7.2. Hermite’s Theorem 62 x7.3. Transcendence Degree 64 x8. Algebraic Sets 66 x8.1. Noether’s Normalization Lemma 66 x8.2. Weak Nullstellensatz 67 x8.3. Affine Algebraic Sets. Strong Nullstellensatz 68 x8.4. Preview of Schemes: a double point. MaxSpec Z 69 x8.5. Exercises 71 x9. Geometry and Commutative Algebra 72 x9.1. Localization and Geometric Intuition Behind It 72 x9.2. Ideals in R and in S−1R 74 x9.3. Spectrum and Nilradical 75 x9.4. Going-up Theorem 76 x9.5. Exercises 78 x10. Geometry and Commutative Algebra - II 79 x10.1. Localization as a functor ModR ! ModS−1R. 79 x10.2. Nakayama’s Lemma 80 x10.3. Spec and MaxSpec. Irreducible Algebraic Sets. 81 x10.4. Morphisms of Algebraic Sets 83 x10.5. Dominant morphisms 85 x10.6. Finite Morphisms 85 x10.7. Exercises 86 x11. Representation Theory of Finite Groups 88 x11.1. Representations of Finite Groups 88 x11.2. Category of Representations 90 x11.3. Irreducible Representations of Abelian Groups 92 x11.4. Characters 94 x11.5. Schur Orthogonality Relations 95 x11.6. Decomposition of the Regular Representation 96 x11.7. Representation Theory of the Dihedral Group 96 x11.8. The Number of Irreducible Representations 96 x11.9. C[G] as an Associative Algebra 96 x11.10. dim Vi divides jGj 97 x11.11. Burnside’s Theorem 98 x11.12. Exercises 100 x1. CATEGORIES AND FUNCTORS x1.1. Categories. Most mathematical theories deal with situations when there are some maps between objects. The set of objects is usually some- what static (and so boring), and considering maps makes the theory more ALGEBRA: LECTURE NOTES 3 dynamic (and so more fun). Usually there are some natural restrictions on what kind of maps should be considered: for example, it is rarely interest- ing to consider any map from one group to another: usually we require this map to be a homomorphism. The notion of a category was introduced by Samuel Eilenberg and Saun- ders MacLane to capture situations when we have both objects and maps between objects (called morphisms). This notion is slightly abstract, but extremely useful. Before we give a rigorous definition, here are some ex- amples of categories: EXAMPLE 1.1.1. • The category Sets: objects are sets, morphisms are arbitrary func- tions between sets. • Groups: objects are groups, morphisms are homomorphisms. • Ab: objects are Abelian groups, morphisms are homomorphisms. • Rings: objects are rings, morphisms are homomorphisms of rings. Often (for example in this course) we only consider commutative rings with identity. • Top: topological spaces, morphisms are continuous functions. • Mflds: objects are smooth manifolds, morphisms are differentiable maps between manifolds. • Vectk: objects are k-vector spaces, morphisms are linear maps. Notice that in all these examples we can take compositions of morphisms and (even though we rarely think about this) composition of morphisms is associative (because in all these examples morphisms are functions with some restrictions, and composition of functions between sets is certainly associative). The associativity of composition is a sacred cow of mathemat- ics, and essentially the only axiom required to define a category: DEFINITION 1.1.2. A category C consists of the following data: • The set of objects Ob(C). Instead of writing “X is an object in C”, we can write X 2 Ob(C), or even X 2 C. • The set of morphisms Mor(C). Each morphism f is a morphism from an object X 2 C to an object Y 2 C. More formally, Mor(C) is a disjoint union of subsets Mor(X; Y ) over all X; Y 2 C. It is f common to denote a morphism by an arrow X −! Y . • There is a composition law for morphisms Mor(X; Y ) × Mor(Y; Z) ! Mor(X; Z); (f; g) 7! g ◦ f f g g◦f which takes X −! Y and Y −! Z to the morphism X −! Z. Id • For each object X 2 C, we have an identity morphism X −!X X. These data should satisfy the following basic axioms: • The composition law is associative. f Id • The composition of any morphism X −! Y with X −!X X (resp. with Id Y −!Y Y ) is equal to f. Here is another example. 4 JENIA TEVELEV EXAMPLE 1.1.3. Let G be a group. Then we can define a category C with just one object (let’s denote it by O) and with Mor(C) = Mor(O; O) = G: The composition law is just the composition law in the group and the iden- tity element IdO is just the identity element of G. f DEFINITION 1.1.4. A morphism X −! Y is called an isomorphism if there g exists a morphism Y −! X (called an inverse of f) such that f ◦ g = IdY and g ◦ f = IdX : In the example above, every morphism is an isomorphism. Namely, an inverse of any element of Mor(C) = G is its inverse in G. A category where any morphism is an isomorphism is called a groupoid, because any groupoid with one object can be obtained from a group G as above. Indeed, axioms of the group (associativity, existence of a unit, exis- tence of an inverse) easily translate into axioms of the groupoid (associativ- ity of the composition, existence of an identity morphism, existence of an inverse morphism). Of course not any category with one object is a groupoid and not any groupoid has one object. EXAMPLE 1.1.5. Fix a field k and a positive integer n. We can define a cate- gory C with just one object (let’s denote it by O) and with Mor(C) = Matn;n : The composition law is given by the multiplication of matrices. The iden- tity element IdO is just the identity matrix. In this category, a morphism is an isomorphism if and only if the corresponding matrix is invertible. Here is an example of a category with a different flavor: EXAMPLE 1.1.6. Recall that a partially ordered set, or a poset, is a set I with an order relation which is • reflexive: i i for any i 2 I, • transitive: i j and j k implies i k, and • anti-symmetric: i j and j i implies i = j. For example, we can take the usual order relation ≤ on real numbers, or divisibility relation ajb on natural numbers (ajb if a divides b). Note that in this last example not any pair of elements can be compared. Interestingly, we can view any poset as a category C. Namely, Ob(C) = I and for any i; j 2 I, Mor(i; j) is an empty set if i 6 j and Mor(i; j) is a set with one element if i j. The composition of morphisms is defined using transitivity of : if Mor(i; j) and Mor(j; k) is non-empty then i j and j k, in which case i k by transitivity, and therefore Mor(i; k) is non-empty. In this case Mor(i; j), Mor(j; k), and Mor(i; k) consist of one element each, and the composition law Mor(i; j) × Mor(j; k) ! Mor(i; k) is defined in a unique way. Notice also that, by reflexivity, i i for any i, hence Mor(i; i) contains a unique morphism: this will be our identity morphism Idi. ALGEBRA: LECTURE NOTES 5 Here is an interesting example of a poset: let X be a topological space. Let I be the set of open subsets of X. This is a poset, where the order re- lation is the inclusion of open subsets U ⊂ V . The corresponding category can be denoted by Top(X). x1.2. Functors. If we want to consider several categories at once, we need a way to relate them! This is done using functors. DEFINITION 1.2.1. A covariant (resp. contravariant) functor F from a cate- gory C to a category D is a rule that, for each object X 2 C, associates an f object F (X) 2 D, and for each morphism X −! Y , associates a morphism F (f) F (f) F (X) −! F (Y ) (resp.
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