An Introduction to Categories and Homological Algebra: A Tale of Four Functors By Caedmon Ragland July, 2024 Director of Thesis: Heather Ries, Ph.D. Major Department: Mathematics Abstract We assume familiarity with group theory and discuss the technical background necessary to construct and understand the functors Hom,Ext, Tor and tensor on the category of abelian groups, with special attention paid to Hom and Ext. In doing so, we hope to provide an introduction to the study of category theory and homological algebra understandable to an undergraduate with 1-2 semesters of abstract algebra. An Introduction to Categories and Homological Algebra: A Tale of Four Functors A Thesis Presented to the Faculty of the Department of Mathematics East Carolina University In Partial Fulfillment of the Requirements for the Degree Master’s of Art in Mathematics By Caedmon Ragland July, 2024 Director of Thesis: Heather Ries, Ph.D. Thesis Committee Members: Heather Ries, Ph.D. Chris Jantzen, Ph.D. Michael Spurr, Ph.D. ©2024, Caedmon Ragland Acknowledgements This work, just like all others, is not the product of a single person but of many. Thanks go out to my partner Sig Webb, who kept me company during many long nights spent on this thesis; to my family for their love and support through this process; to Dr. Susan Howard, Mr. John McDonald, Ms. Lara Smith, and Dr. Manuel Salazar, each of whom inspired and pushed me to pursue education in mathematics; to Dr. Chris Carolan for helping me grow as a mathematician during my undergraduate years; and to my committee. Thank you to to Dr. Chris Jantzen for your advice and assistance during my graduate education; to Dr. Michael Spurr for the same, and for advice which lead me to the school I will begin my PhD at in the fall; to Dr. Jungmin Choi who, despite many challenges, has adamantly helped me keep my degree on track. Finally, with the utmost gratitude, I’d like to thank my thesis director Dr. Heather Ries. Her help, guidance, and willingness to put up with me subjecting her to adjoint functors over and over has no doubt made this thesis ten times better than it would’ve been. For a master’s thesis, a 100-page elementary introduction to category theory and homological algebra was an ambitious choice that proved to take much longer than expected. Despite the size of this task, my insistence on category theory, and hundreds of typos and mismatched variables, she has steadfastly seen me through this process. For that, I cannot thank her enough. Contents Introduction 1 1 Category Theory 3 1.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Universals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Adjoint Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.7 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Exact Sequences 35 2.1 Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Equivalent Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Split Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Exact Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 Hom(A,B) 53 3.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Induced Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Viewing Hom(A,B) as a Functor . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 The Exactness of Hom, part I . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 The Exactness of Hom, part II: Projectives and Injectives . . . . . . . . . . 69 4 Extensions and Ext(A,B) 74 4.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Viewing E(A,B) as a Functor . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Ext(A,B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 The Equivalence of E, Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5 Computing Ext(A,B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5 An Adjoint Relationship and Tor(A,B) 99 5.1 The Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Tensor and Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 Tor(A,B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References 105 Introduction In 1941 Dr. Saunders Mac Lane calculated the group of extensions of Z by the Prüfer p- group during a lecture at the University of Michigan. Afterwards, he discussed the calculation with Dr. Samuel Eilenberg, who recognized the computation as the homology of a particular topological space. The discussion of this connection later lead to the publication of the paper “General Theory of Natural Equivalences” [EM45] by the two, which birthed the field of mathematics we now know as category theory [Rie17]. In this work, we follow the reverse path, beginning with category theory and eventually developing the group of extensions of A by B. Our aim is to provide an introduction to homological algebra and category theory through the lens of abelian groups. We assume familiarity with group theory, and utilize classic results and reference foundational theorems without citation. Should the reader be unfamiliar with these results, they are encouraged to consult the texts referenced in the creation of this work. Particularly, [Art11], [Rot12], and [Pin10]. In particular, [HW89] provided a singular resource of almost all results necessary to construct this work. For the final two chapters however, we prefer [HS12] as it provides more elementary exposition. The choice to use abelian groups is one made out of an effort to keep results more elementary. It is more general and thus more correct in a sense to use modules as our primary lens, as via the full embedding theorem [HS12], any small abelian category may be fully embedded in the category of modules over an appropriate ring. Choosing our ring to be the integers however, and thus our modules to be abelian groups, allows an exploration unburdened by the additional prerequisite knowledge required to understand general modules. In the first and second chapters we present necessary background to construct and under- stand the functors Hom,Ext, Tor, and tensor. The first chapter focuses entirely on category theory, establishing the categorical lens on our work. This serves not only to allow the proof of desired results, but provides a pathway to generalize our results to the arbitrary abelian category case. The second chapter defines exact sequences and establishes the definition of an exact functor, which is necessary to construct Ext and Tor. The final three chapters examine the namesake four functors, with the final chapter among them covering tensor and Tor, serving as a short epilogue to previous work. A reader interested in our results but unfamiliar with category theory and unprepared for the level of abstraction it necessitates is encouraged to begin with chapter 3, returning to chapters 1 and 2 when we refer to results found therein. We, however, assert that beginning with a full dive into category theory is not only the correct choice, but is more enlightening than only referring to it when absolutely necessary. 2 Chapter 1 Category Theory Dr. Eugenia Cheng, an educator and category theorist, is often quoted as stating “category theory is the mathematics of mathematics” [Che22]. Dr. Emily Riehl states in the preface to her textbook Category Theory in Context, that if mathematics is the science of analogy, “the purview of category theory is mathematical analogy” [Rie17]. The truth behind these statements lies in the ability of category theory to translate results across different areas of mathematics. Perhaps the most famous area of study to utilize this is algebraic topology. By taking a topological space and associating it with a group, algebraic topologists form a functor from the category of topological spaces1 to the category of groups. In doing so, they are permitted to apply the techniques of group theory to problems in topology, and turn complex problems into simple solutions. Riehl presents the representative example of the Brouwer Fixed Point Theorem in two dimensions. By assuming there exists a function from the closed disk in two dimensions to itself which does not have a fixed point, one may utilize the aforementioned functor to derive homomorphisms f : Z → 0 and g : 0 → Z such that gf = 1Z. However, by way of passing through the trivial group 0, this composition must be the trivial homomorphism 0, and we have our contradiction. Per Riehl, we have reduced “a seemingly intractable problem in topology to a trivial one (0 ̸= 1) in algebra” [Rie17]. 1To be clear, the domain of the functor we speak of is not the category Top of topological spaces, but the category Top∗ of topological spaces with a specifed basepoint, for reasons we do not delve into here. This power to translate results and techniques across fields, category theory derives from the primary kind of structure it studies: relationships between mathematical structures. This study of relationships positions category theory as the field to call upon when attempting to relate two previously unrelated ideas across areas of mathematics. However, category theory may also be utilized to deepen the study of a chosen area of mathematics. In this work we choose the area of abelian groups, and seek to use category theory to examine and construct new ideas out of familiar ideas. We use this first chapter to cover all that is necessary to generalize the properties inherent to the category of abelian groups, and discuss the categorical ideas we later use in our proofs. Along the way we discuss related examples of categories and functors, allude to the broader study of category theory, and assemble more than our fair share of categories. 1.1 Categories Definition 1.1.1 (Category [Rie17] [HW89]). A category C consists of the following (i) A collection of objects X, Y, Z, . . . (ii) A collection of morphisms f, g, h, . . . , each of which having a specified domain object and codomain object (usually written as f : X → Y , for the morphism f which has domain X and codomain Y ) (iii) An associative operation of composition, such that any two morphisms f : X → Y, g : Y → Z , where the domain of g is equal to the codomain of f , may be com- posed to form the morphism g ◦ f : X → Z (iv) For each object X, an “identity morphism” 1X : X → X, which serves as the identity with respect to the operation of composition. That is, for any morphism f : X → Y , 4 we have 1Y ◦ f = f = f ◦ 1X With such an abstract definition, it is necessary to relate the concept of a category to structures that are already familiar, such as 1. The category Set, whose objects are sets, and whose morphisms are functions. Since each set has the identity function to itself, and function composition is associative, Set is indeed a category. 2. The category Ab whose objects are abelian groups, and whose morphisms are group homomorphisms. Since each group has the identity homomorphism to itself, and ho- momorphism composition is associative, Ab is a category.2 Example 1.1.2 (The single object category G [Rie17]). We may consider a group itself as a category. Let G be a group, and consider a category G having only one object, which we refer to as G. We then let there be a morphism g : G → G for each element g ∈ G. We compose these morphisms following the rules of G’s group operation, which ensures associativity and existence of identity. Particularly, we may consider an individual morphism g as “multiplication by g” using the group operation of G. The identity morphism 1G is then simply the identity element of G. Then G has an identity morphism for each object (namely, for its only object G), and its morphisms may be composed associatively, giving that G is a category. Example 1.1.3 (Z under ≤ [Rie17]). We may consider a partially ordered set as a category. We consider Z and the usual ≤, letting the objects be elements of Z and declaring that there exists a unique morphism x → y iff y ≤ x. Reflexivity ensures there exists an identity morphism x → x, since x ≤ x. Transitivity ensures composition is associative. 2While we rarely utilize it in this work, groups in general do indeed form a category in the same manner. In fact, Ab may be considered a subcategory of Grp, in a sense similar to the idea of a subgroup. 5 1.2 Functors If category theory derives its strength from its study of relationships between mathematical structures, it is no surprise that immediately after defining a new structure, the category, we examine the relationships between them. These relationships take the form of functors, which are described as “essentially morphisms of categories” by Hilton and Wu [HW89]. While we may indeed assemble a category Cat whose objects are themselves categories, and whose morphisms are functors, our primary focus is on the ability of functors to preserve morphism structure. Definition 1.2.1 (Functor [HW89]). If C ,D are categories, a functor F : C → D consists of two mappings. The first, gives for each object X in C an object FX in D . The second, gives for each morphism f : X → Y in C , a morphism Ff : FX → FY such that F (f ◦ g) = Ff ◦ Fg for any two composable morphisms f, g, and F (1X) = 1FX That is, functors preserve composition of morphisms, and preserve identities. Examples of functors abound, we present a selection below. 1. The “forgetful” functor U : Ab → Set, which sends abelian groups to their under- lying sets, and homomorphisms to their underlying functions. This functor is called “forgetful”, as it may be summarized as “forgetting” the group structure of a group A. 2. The identity functor 1C : C → C , for which 1C (X) = X for all objects X, and 1C (f) = f for all morphisms f . 6 3. The aforementioned functor from the category Top∗ of topological spaces (with a specified basepoint) to the category Grp of groups, sending topological spaces to their fundamental group, and continuous functions to their induced homomorphisms. We omit further detail, as were we to include it, it would fill an entire chapter. Example 1.2.2 (Group Homomorphisms as Functors [Rie17]). We return to considering a group as a single-object category as in Example 1.1.2. Let G,K be groups, each considered as a category. Let ϕ : G → K be a homomorphism. Then ϕ(eG) = eK , and ϕ(gg ′) = ϕ(g)ϕ(g′) for any g, g′ ∈ G. Since for G considered as a category, its elements are its morphisms, we have that ϕ preserves composition of morphisms. Since eG is the only identity morphism in G, we further have that ϕ preserves identity morphisms. Thus, ϕ satisfies the definition of a functor. Example 1.2.3 (The Free Abelian Functor [HW89]). Let S, T be sets in Set, and f : S → T be a function between them. The free abelian functor Fr : Set→ Ab sends S to the free abelian group on the set S as defined in §3.6, and f : S → T to the homomorphism Fr(f) : Fr(S)→ Fr(T ) such that Fr(f)(s) = f(s) for all s.3 By this definition, the identity function is mapped to the identity homomorphism, and for any composable functions f, g, Fr(fg)(s) = fg(s) and Fr(f)Fr(g)(s) = fg(s). Thus, Fr satisfies the definition of a functor. 1.3 Universals In the study of category theory, it is often handy to take familiar notions, such as the Cartesian product of sets, and reformulate their definitions in the language of objects and morphisms. This reformulation makes obvious the commonalities across different areas of mathematics, and facilitates the transfer of results that we have claimed is the strength of category theory. 3The existence of a homomorphism satisfying this property follows from the fact that the free abelian group on S is equivalent to a direct sum of |S| copies of Z, a fact we discuss in §3.6 7 Remark 1.3.1. The collection of reformulated definitions we present in this section, bar a choice few, will each follow a useful pattern. 1. There exists an object X equipped with a collection of morphisms {fi} such that a certain diagram “commutes”. 2. For any other object Y equipped with a collection of morphisms {gi} allowing the same diagram to commute, there exists a unique morphism h : Y → X such that gi = fih for each i. This pattern is common because they are each an example of a limit of a functor [Rie17]. We decline to explore the details of this fact, and instead focus on how we may exploit this pattern. The second part of this definition, particularly the unique morphism h, may be exploited in a nearly identical way across each of these definitions to prove that these objects are unique up to isomorphism. Rather than prove this for each object we define, instead we establish what is meant by isomorphism in categorical terms, what it means for a diagram to commute, provide a representative example of this proof of uniqueness, and leave the rest to the reader. In the spirit of reformulating familiar ideas, to define an isomorphism in categorical terms we look to the definition of isomorphism with respect to groups. In a usual abstract algebra course, an isomorphism is defined as a homomorphism that is surjective and injective. While we will find anologues of these terms, the fact that objects do not necessarily have “elements” as is shown in Example 1.1.3 means we currently lack ideas of surjectivity and injectivity. Rather, we look to what we can say about an isomorphism through its composition. Particularly, that isomorphisms come in pairs. If ϕ : G→ K is an isomorphism, there exists an isomorphism ϕ−1 : K → G. Even more precisely, if ϕ is an isomorphism, there exists ϕ−1 such that the following compositional identities hold ϕϕ−1 = 1G ϕ−1ϕ = 1K 8 This generalizes intuitively to the definition Definition 1.3.2 (Isomorphism [Rie17]). A morphism f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that fg = 1X gf = 1Y We allow ourselves a small detour to define the related terms of monomorphism and epimor- phism, by noting that if f is an isomorphism, and we have fϕ = fϕ′ or ψf = ψ′f for any morphisms ϕ, ϕ′ having codomain X, ψ, ψ′ having domain Y , then we necessarily have ϕ = ϕ′, ψ = ψ′. For the left-hand equality we may postcompose4 by g to receive gfϕ = gfϕ′, which simplifies to 1ϕ = 1ϕ′, giving the desired result. Proof for the right-hand equality follows similarly. These two equalities should not be viewed as one property unique to the isomorphism, but as two distinct properties for which the isomorphism satisfies both, as the following definitions suggest. Definition 1.3.3 (Monomorphism [Rie17]). Let f : X → Y be a morphism such that fϕ = fϕ′ =⇒ ϕ = ϕ′. Then f is referred to as a monomorphism and is denoted as f : X ↣ Y . Definition 1.3.4 (Epimorphism [Rie17]). Let g : A→ B be a morphism such that ψg = ψ′g =⇒ ψ = ψ′. Then g is referred to as an epimorphism, and is denoted as g : A↠ B. 4The meaning of the term “postcompose” is hopefully intuitive. In case it is not, we note that to “postcompose” a morphism f by another morphism g is to perform the composition gf , read as “g after f”. In a similar fashion, to “precompose” f by g is to perform the composition fg. Both of these terms are adopted from Categories in Context by Dr. Emily Riehl, whose work greatly influenced this thesis. 9 Note that the remark of analogues to surjective and injective apply here. For the cate- gory Set, monomorphisms are precisely injective functions, and epimorphisms are precisely surjective functions. The same holds true for Ab, which is shown in Corollary 1.7.7. We now address what is meant by commutativity by presenting the most basic example of a commutative diagram: The commutative square. Example 1.3.5 (Commutative Square). In a category C , let A,B,C,D be objects, and each arrow depicted below represent a morphism whose domain is the object it points from, and whose codomain is the object it points to. A B C D f h g k To say this diagram “commutes” is to say that the composites gf and kh are equal. More broadly, if a diagram commutes, we say that any path of morphisms beginning at the same object and leading to the same object is equal. We now have sufficient information to give our representative example. Definition 1.3.6 (Product of X and Y [HW89]). Let C be a category having the objects X, Y . Then the object X × Y is called the product of X, Y , if it satisfies the universal property described by the commutative diagram P X X × Y Y pX pY h πX πY In words, X × Y is the object having morphisms πX : X × Y → X and πY : X × Y → Y such that for any other object P having morphisms pX : P → X and pY : P → Y , there exists a unique morphism h : P → X × Y such that πY h = pY , πXh = pX . That is to say, there exists a unique h such that the diagram commutes.5 5Occasionally we may prefer to refer to h by the names of the morphisms it arises from. In this case for the product, we write h = ⟨x1, x2⟩. 10 Here, X×Y is a reformulation of the familiar idea of the cartesian product of sets. In the language of Remark 1.3.1, the object in question is X × Y and the collection of morphisms allowing commutativity is {πX , πY }. Commutativity of the diagram X πX←− X × Y πY−→ X is rather trivial, as there are no paths of morphisms beginning at the same object and leading to the same object. Nonetheless, for any other object P having morphisms {pX , pY } allowing this vacuous commutativity, we may utilize the unique morphism h : P → X × Y to prove X × Y is unique up to isomorphism. Theorem 1.3.7. The finite product is unique up to isomorphism. Proof. Suppose that for two objects X, Y , there exist two objects X ×Y, (X ×Y )′ such that both are the product of X, Y as defined above. As such, we may form the diagrams (X × Y )′ X × Y X X × Y Y X (X × Y )′ Y π′ X π′ Yh h′ πYπX πYπX π′ X π′ Y Thus through the universal property of the product, we have unique morphisms h : X × Y → (X × Y )′ and h′ : (X × Y )′ → X × Y allowing the diagrams to commute. We claim that their composite, h′h : X × Y → X × Y , allows the following diagram to commute X × Y X X × Y Y πX πY h′h πX πY (1.1) To show as much, we may stack our diagrams like so X × Y X (X × Y )′ Y X X × Y Y h πX πY h′ π′ X π′ Y π′ X π′ Y πYπX to see that π′ Y h = πY and πY h ′ = π′ Y together give that πY h ′h = π′ Y h = πY , and similarly so for πX , giving that h′h allows 1.1 to commute. 11 However, the identity morphism 1X×Y : X × Y → X × Y would also allow the above diagram to commute. Since the morphism X×Y → X×Y allowing the diagram to commute must be unique by definition, we therefore have h′h = 1X×Y . Similarly, hh′ = 1(X×Y )′ . Therefore, h, h′ are isomorphisms between X × Y, (X × Y )′, and as such the product is unique up to isomorphism. As stated previously, this proof of uniqueness up to isomorphism is representative of the proof of uniqueness for every other definition in this section. Moving forward, we omit the proofs and simply state the results where applicable. A further comment on the product as described above is that while we define it with respect to two objects, it extends inductively to any finite number of objects, as we may take the product of X, Y , then take the product of said product with some other object Z, and continue. In fact, we may extend the definition of product to an arbitrary collection of objects, as we now show. Definition 1.3.8 (Product [HW89]). Let {Xi}i∈J be a collection of objects in some category C . Then an object ∏ i∈J Xi is called the product of this collection if it satisfies the universal property described by the diagram P {Xi}i∈J ∏ i∈J Xi {pi}i∈J h {πi}i∈J We began our definitions with the product, as it is very common in the study of elemen- tary set and group theory, and thus is likely to be familiar to the reader. We now turn a construction of a particular commutative square whose analogues outside category theory are less obvious: the pullback. Hilton and Wu give the following example to illustrate the essence of a pullback. Example 1.3.9 ([HW89]). Let A1, A2, A be sets in Set such that A1 ⊂ A, A2 ⊂ A. We then have the inclusion function 12 mapping x ∈ A1 to itself in A. Similarly so for A2. We may illustrate this relationship via the diagram A1 A2 A To turn this relationship into a commutative square, we require an object to fill the upper left corner, and morphisms from said object to A1, A2 allowing the diagram to commute. The pullback is, in essence, the “best” object that fills this role. For this specific example, the pullback is the set A1 ∩ A2, together with the inclusion morphisms to A1, A2. Clearly for any x ∈ A1 ∩ A2, we may map it to itself in A1, then to itself in A. The result is the same as if we map it to itself in A2, then itself in A. Note that any subset of A1 ∩ A2 could fulfill the same role, but that for A0 ⊂ A1 ∩ A2, we have the following commutative diagram A0 A1 ∩ A2 A1 A2 A where all morphisms are inclusions, and the dashed morphism is the inclusion of the subset into A1 ∩ A2. This idea is generalized by the following definition. Definition 1.3.10 (Pullback [HW89]). For C a Category, A,A1, A2 objects in C having Morphisms ϕ1 : A1 → A, ϕ2 : A2 → A, the commutative square B A1 A2 A b1 b2 ϕ1 ϕ2 13 is the pullback of these morphisms if it satisfies the universal property described by the diagram X B A1 A2 A x1 x2 h b2 b1 ϕ1 ϕ2 That is, there exist morphisms b1 : B → A1, b2 : B → A2 allowing the diagram to com- mute, such that for any other object X, having morphisms x1, x2 allowing the diagram to commute, there exists a unique morphism h : X → B such that b2h = x2, b1h = x1. Again in the language of Remark 1.3.1, the object in question is B, the collection of morphisms is {b1, b2}, and the diagram they allow to commute is the square. Thus, the pullback is unique up to isomorphism. The final definition we wish to provide is a reformulation of the idea of the kernel of a homomorphism in group theory. In order to do so, we must first introduce a few ancillary definitions. Definition 1.3.11 (Terminal Object [HW89]). An object T is called terminal if for any other object C, there exists a unique morphism C → T . Definition 1.3.12 (Initial Object [HW89]). An object I in a category C is called initial, if for any other object C, there exists a unique morphism I → C. We claim that the terminal object is yet another example, if a particularly vacuous one, of the pattern described in Remark 1.3.1. We omit the details of why so as not to confuse our point, but the details may be found on page 78 of [Rie17]. The initial object is an example of a different pattern, discussed in the next section. 14 Example 1.3.13 (The Singleton Set). In Set, a terminal object T is any singleton set. For any other set S, there is exactly one function S → T , mapping every element of S to the single element of T .6 Notice that while T is not unique, being any singleton set, it is unique up to isomorphism as we claimed earlier. This, since for {a}, {b} singleton sets, the only function {a} → {b} sends a to b, and likewise for {b} → {a}. Their composition is of course the identity. Example 1.3.14 (The Trivial Group). In Grp, the initial object is the trivial group {e}. Since all other groups have an identity element by definition, and homomorphisms must preserve identities, we have that to any other group G, there exists the unique homomorphism mapping {e} to the identity in G. However, there is also a unique homomorphism G → {e}, defined as mapping all elements in G to the identity in the trivial group. Thus, {e} is both an initial and terminal object. This observation leads to the following definition. Definition 1.3.15 (Zero Object [HW89]). If an object 0 in a category C is both an initial and terminal object, we refer to it as a zero object. The combination of the universal properties of the initial and terminal objects lead to the following definition. Definition 1.3.16 (Zero Morphism [HW89]). Let C be a category with a zero object. Let A,B be any two objects in C . Then via the universal properties of the zero object, there exists a unique morphism A → B passing through the zero object A→ 0→ B We call this morphism the zero morphism, and denote it 0AB, or simply 0. 6If S = ∅, this morphism is vacuously defined, but does exist. 15 Since the defining property of the zero morphism is that it is unique and passes through the zero object, for A,B,C objects and f : A → B, g : B → C the zero morphism has the following compositional properties g0AB = 0AC 0BCf = 0AC We take the notational convention of dropping the domain and codomain. Thus, we have that 0 composed with any other morphism is once again 0. We now turn to reformulating the kernel. Definition 1.3.17 (Kernel [HW89]). For a category C having a zero object, and f : G→ H a morphism between the objects G, H, a morphism κ is called the kernel of f if it satisfies the universal property described by the diagram K G H L κ 0 f h ℓ 0 That is, K has a monomorphism κ : K ↣ G, such that the composition fκ is equal to the zero morphism, and for any other object L having a morphism ℓ : L → G satisfying fℓ = 0, there exists a unique morphism h : L → K allowing the diagram to commute, i.e: such that κh = ℓ. 1.4 Duality In the study of category theory, it is often true that for the work of proving one result, two results are proven. This is due to the principle of duality. This principle may be summarized as noting that for any theorem applying to an arbitrary category, we may for every argument “reverse the direction” of our morphisms, and receive a “dual theorem”[Rie17]. This process 16 of reversing direction of our morphisms appears arbitrary at first glance, but is founded in an important construction: that of the opposite category. Definition 1.4.1 (Opposite Category [Rie17]). For C a category, the opposite category C op is the category whose objects are the objects of C , and whose morphisms are such that for each morphism f : X → Y in C , there exists a morphism f op : Y → X in C op. C op as described does indeed form a category. Whenever f : X → Y , g : Y → Z are composable, gop : Z → Y , f op : Y → X are composable. Additionally, 1opX : X → X serves the role of the identity morphism in C op. Thus, if a result holds in arbitrary categories, it holds in the opposite category. Therefore, the dual result holds in arbitrary categories as well. This is true precisely because the axioms of the definition of category are self-dual, meaning that the dual of an axiom is itself an axiom. We later see that the axioms describing the properties of Ab are themselves self- dual, which allows us to obtain dual results of our proofs when necessary. We now examine the relationship of a dual to its original result through the dual of the product introduced in the previous section. We confine ourselves to the case of only two objects for the sake of exposition, though what is said translates to the arbitrary case as well. Example 1.4.2 (The Dual of the Product). Let C ,C op be a category and its opposite. Suppose each have products, and examine a product of X, Y in C op. We then have the following diagram P X X ⊕ Y Y popX popY hop fopX fopY Where X ⊕ Y is the product in C op, and P is another object with morphisms to X, Y . Since this structure exists within C op, there must be a corresponding structure in C . To construct it, we reverse the direction of our morphisms to receive the diagram 17 P X X ⊕ Y Y fX pX h fY pY We call this structure the dual of the product, and name it the coproduct. Definition 1.4.3 (Coproduct [HW89]). Let {Xi}i∈J be a collection of objects in some category C . Then an object ∐ i∈J Xi is called the coproduct of this collection if it satisfies the universal property described by the diagram P {Xi}i∈J ∐ i∈J Xi {f ′i}i∈J {fi}i∈J h That is, ∐ i∈J Xi is the object having morphisms fi : Xi → ∐ i∈J Xi for each Xi such that for any other object P having morphisms f ′ i : Xi → P , there exists a unique morphism h : ∐ i∈J Xi → P such that hfi = f ′ i for all i. 7 We may dualize other definitions as well. Some duals already introduced are the terminal object and its dual, the initial object. The dual of a monomorphism is an epimorphism. In fact, all definitions given in the previous section have duals that we discuss shortly. Before providing the remainder of the dual definitions, we dualize the proof that products are each unique up to isomorphism. Theorem 1.4.4. The coproduct is unique up to isomorphism Proof. Suppose for the collection of objects {Xi}i∈J , there exist two objects ∐ i∈J Xi, (∐ i∈J Xi )′ such that both are the coproduct of said collection. As such, we may form the diagrams 7Occasionally we may prefer to refer to h by the names of the morphisms it arises from. In this case for the coproduct, we write h = {f ′ 1, f ′ 2} in the finite case, or h = {f ′ i}i∈J in the arbitrary case. 18 (∐ i∈J Xi )′ ∐ i∈J Xi {Xi}i∈J ∐ i∈J Xi {Xi}i∈J (∐ i∈J Xi )′ {f ′i}i∈J {fi}i∈J h {fi}i∈J {f ′i}i∈J h′ Giving unique morphisms h : ∐ i∈J Xi → (∐ i∈J Xi )′ , and h′ : (∐ i∈J Xi )′ → ∐ i∈J Xi. We may of course form a similar diagram with ∐ i∈J Xi and itself, where the composition h′h is the unique morphism ∐ i∈J Xi → ∐ i∈J Xi. We may of course also let the identity 1∐ i∈J Xi be the unique morphism in said diagram. Since this morphism must be unique, we therefore have h′h = 1∐ i∈J Xi , and similarly so for hh′ = 1( ∐ i∈J Xi) ′ , giving that h, h′ are isomorphisms as desired. In the previous section a similar proof was given to show that the product of two objects is unique up to isomorphism. The above prove is deliberately similar to show how little must change to dualize a proof. We again claim this proof to be representative of the proofs for following definitions; just as the previous definitions were all examples of limits, their duals are each examples of colimits. We return to introducing the duals of previously defined objects, beginning with the kernel. Definition 1.4.5 (Cokernel [HW89]). For a category C having a zero, and f : H → G a morphism between objects H, G, a morphism κ : G↠ K is called the cokernel of f if it satisfies the universal property described by the diagram K G H L k g κ f 0 0 19 That is, K has an epimorphism κ : G ↠ K such that the composition κf = 0, and for any other object L having a morphism g : G → L satisfying gf = 0, there exists a unique morphism k : K → L allowing the diagram to commute. For the pullback, we break from our pattern of naming dual objects with the prefix “co”. Rather than co-pullback, the dual of the pullback is referred to as the pushout. Definition 1.4.6 (Pushout [HW89]). For a Category C , A,A1, A2 objects in C having morphisms ψ1 : A→ A1, ψ2 : A→ A2, the commutative square A A1 A2 B ψ2 ψ1 b1 b2 is the pushout of these morphisms if it satisfies the universal property described by the diagram A A1 A2 B X ψ2 ψ1 b1 x1 b2 x2 h That is, there exist morphisms b1 : A1 → B, b2 : A2 → B such that for any other object X having morphisms x1 : A1 → X, x2 : A2 → X, there exists a unique morphism h : B → X allowing the diagram to commute. We have now applied the principle of duality to categories with the opposite category, to morphisms with monomorphism being the dual of epimorphism, and our constructed definitions as well via above. We now turn to applying duality to functors. Definition 1.4.7 (Contravariant Functor [Rie17]). A contravariant functor F from C to D is a functor F : C op → D . 20 Just as previously, we have two mappings making up F . An object mapping, giving an object FX in D for every object X in C , and a morphism mapping, which gives a morphism Ff : FY → FX in D for every morphism f : X → Y in C . If we interpret F as a functor C op → D , it is intuitive why Ff has domain FY and codomain FX. This, because Ff is truly Ff op, so since f op has domain Y and codomain X, so too must Ff op. However, interpreting F as a functor C → D , as we often do, shows F to have the effect of reversing our morphisms. Thus, the composition property is not F (fg) = Ff ◦ Fg as before, but F (fg) = Fg ◦ Ff . To differentiate between these functors, which we have called contravariant, and our original definition of functors, we now call the original definition “covariant”. We will see more examples of contravariant functors later in chapters 3 and 4, but allow ourselves to be satisfied with the following example for now. Example 1.4.8 (The Contravariant Power Set Functor [HW89]). The functor Q : Set → Set which sends sets to their power sets, and sends a function f : X → Y to the function Qf : P (Y ) → P (X) defined by Qf(B) = f−1(B) where B ⊂ Y . Since the identity function sends each element to itself, and therefore each subset to itself, Q preserves identities. Then letting g : Y → Z, we have Qgf(C) = (gf)−1(C) = f−1g−1(C) = QfQg(C) for C ⊂ Z, giving that Q reverses composition, and is a contravariant functor as desired. 1.5 Natural Transformations We continue our study of the relationships between structures. Categories scaffolded the study of relationships between like structures, functors allowed study of relationships between categories, and now natural transformations facilitate the study of relationships between functors. Definition 1.5.1 (Natural Transformation [Rie17]). 21 Let F : C → D , G : C → D be functors sharing a source category and target category. A natural transformation τ : F → G is a collection of morphisms τX : FX → GX in D for each object X in C such that for any morphism f : X → Y in C , the following diagram commutes FX GX FY GY τX Ff Gf τY If each τX is an isomorphism as in Definition 1.3.2, we call τ a natural equivalence. Natural transformations are in fact, the original motivator of the development of Category Theory as a whole. In truth, the construction of category theory followed precisely the reverse path we have. First came natural transformations, then Functors to ensure natural transformations had a defined domain and codomain. Finally, categories were constructed to ensure functors had a defined domain and codomain [Rie17]. Natural transformations will prove to be exactly as important and powerful as this developmental path implies. We omit examples for but a moment, as we give a detailed one in the very next section. 1.6 Adjoint Functors Natural transformations pave the way to speak of another way functors may relate to each other. We first define a functor to assist in examining these relationships. Definition 1.6.1 (The Functors C (A,−) and C (−, B) [Rie17]). For a category C , objects A,B in C , we have the functors C (A,−) : C → Set and C (−, B) : C → Set. C (A,−) sends an object X to the set C (A,X) of morphisms in C from A → X, and sends a morphism f : X → Y to the function f∗ : C (A,X) → C (A, Y ) defined as f∗(g) = fg for g ∈ C (A,X). 22 The functor C (−, B) is defined similarly, with the one exception being that C (−, B) is contravariant, with its morphism mapping being f : X → Y is sent to the function f ∗ : C (Y,B)→ C (X,B) defined as f ∗(h) = hf for h ∈ C (Y,B). We then use the above definition to define the notion of adjoint functors. Definition 1.6.2 (Adjoint Functors [HW89]). For F : C → D , G : D → C functors, we say F,G are adjoint if there exists a natural equivalence η : D(F (−), B) ∼= C (A,G(−)) F is then referred to as left adjoint to G, while G is labeled right adjoint to F . The fact that this is a natural equivalence requires a bit of pause and explanation. For a natural equivalence to make sense, we must have that these functors share a domain and codomain. Sharing a codomain follows from the definition, their shared codomain is the category Set. Shared domain on the other hand requires a short detour. We may assemble a category Cat whose objects are themselves categories, and whose morphisms are functors [Rie17]. This implicitly tells us that composition of functors produces another functor. We utilize this fact, as well as the product in Cat to construct the following diagram C ×G(D) C ×D Set F (C )×D C (−,−)1C×G F×1D D(−,−) which when condensed, gives C ×D Set C (−,G(−)) D(F (−),−) 23 showing that our two functors are parallel as desired, and their shared domain is the product category C ×D . In practice, this natural equivalence presents itself as showing that for any X in C , Y in D , we have an isomorphism (which in Set, we may simply regard as a bijection) between the sets C (X,GY ),D(FX, Y ) of morphisms X → GY in C , FX → Y in D respectively. This, along with checking the commutativity of the naturality square given in §1.5, is sufficient to show existence of the equivalence. We wish to provide a representative example of adjoint functors that is related to what we study. In order to do so, we first provide more detail on the previously introduced free abelian group. Definition 1.6.3 (Free Abelian Group [Rot12]). An abelian group Fr(S) is free abelian if it is a direct sum of infinite cyclic groups. More precisely, there is a subset S ⊂ F (S) of elements of infinite order which serves as a basis of Fr(S). That is, Fr(S) = ∑ s∈S ⟨s⟩ ∼= ∑ Z. We may use the fact that in Ab, the direct sum is precisely the coproduct, to prove the following theorem. Theorem 1.6.4 ([Rot12]). Let Fr(S) be a free abelian group with basis S, A be any abelian group, and f : S → A be a function. Then there exists a unique homomorphism ϕ : Fr(S) → A extending f ; that is, ϕ(s) = f(s) for all s ∈ S. This is expressed via the diagram below, where ι is the inclusion of the set of generators. Fr(S) S A ϕ ι f 24 Proof. We begin with a lemma. Lemma 1.6.5. If C is a cyclic group, then ϕ : C → K is uniquely determined by ϕ(c), where ⟨c⟩ = C. Proof. To prove this assertion, recall that since C is cyclic any element c′ ∈ C is expressible as a sum consisting only of c. Then if c′ = c+ c+ · · ·+ c︸ ︷︷ ︸ n times = nc, for any homomorphism ϕ, we have ϕ(c′) = ϕ(nc) = ϕ(c) + ϕ(c) + · · ·+ ϕ(c)︸ ︷︷ ︸ n times = nϕ(c) Thus, for any c′ ∈ C, we have that ϕ(c′) may be expressed only in terms of ϕ(c), where ⟨c⟩ = C. Then let Fr(S) be free abelian, thus equal to ∑ s∈S ⟨s⟩. For each ⟨s⟩ we may define a homomorphism ϕs : ⟨s⟩ → A by ϕ(s) = f(s) and ϕ(s′) = nϕ(s′) for s′ = ns. By lemma, these homomorphisms are uniquely determined by f(s). Then by the universal property of the coproduct, there exists a unique homomorphism ϕ : Fr(S)→ A such that the following digram commutes for each s, where ιs is the embedding of ⟨s⟩ into the direct sum Fr(s). A Fr(S) ⟨s⟩ ϕ ϕs ιs In particular, ϕ(s) = ϕs(s) = f(s) for each s, completing the proof. Above we provide a proof illustrating how uniqueness arises from the direct sum’s status as the coproduct, and the universal property it has as a result. A more common proof would define the homomorphism as we have, but note that it is unique by way of any two ϕ’s extending f being forced to agree on the generators of Fr(S). The property described in Theorem 1.6.4 is precisely what gives the maps allowing Fr to be a functor. If there exists a function f : S → T for S, T sets, then there exists a function ιTf : S → Fr(T ) formed by composition with the function ιT embedding T into Fr(T ). We then use the above property to find the homomorphism Fr(f) : Fr(S)→ Fr(T ). 25 Fr(S) S Fr(T ) Fr(f) ιS ιT f In the upcoming example, we freely associate the homomorphism ϕ found through the free abelian group property and the homomorphism Fr(f) found as the image of the free functor. Example 1.6.6 (The Adjoint Functors Fr and U [HW89]). In §1.2 we defined the functor U : Ab→ Set as sending groups to their underlying sets and homomorphisms to their underlying functions. We additionally defined Fr : Set → Ab as sending each set to the free abelian group on that set, and functions between those sets to the homomorphisms guaranteed by the universal property of the free group. We claim Fr is left adjoint to U , and consequently U is right adjoint to Fr. Proof. To show natural equivalence, we must define a family of isomorphisms Set(S, U(A))→ Ab(Fr(S), A) and show it is natural in S and in A. As stated above, since these iso- morphisms are in Set, it is sufficient to show they are bijections. We define a function fSA : Set(S, U(A)) → Ab(Fr(S), A) and show it to be a bijection. Let fSA(g), where g : S → U(A) is a function, be defined as Fr(g), the homomorphism Fr(g) : Fr(S) → A guaranteed by the universal property of the free group Fr(S). Fr(S) S A Fr(g) ι g To prove fSA is a bijection, we need only show it is injective and surjective. For injectivity, we simply refer to Theorem 1.6.4, which ensures Fr(g) is uniquely deter- mined by the value of g on each s ∈ S. As such, Fr(g) = Fr(g′) =⇒ g(s) = g′(s) for each s ∈ S, therefore giving that g = g′, and fSA is injective. For surjectivity, we merely reverse the process. Letting ϕ : Fr(S) → A be a homo- morphism, we define a function gϕ : S → U(A) by the rule gϕ(s) = ϕ(s). Then via the 26 universal property of the free group, there is a homomorphism Fr(gϕ)ι = gϕ. We then have Fr(gϕ)ι(s) = gϕ(s) = ϕ(s) for all s ∈ S. Since these elements s ∈ S are the generators of Fr(S), we have fSA(gϕ) = Fr(gϕ) = ϕ. Thus, fSA is surjective. Therefore for any S,A, we may form a bijection fSA : Set(S, U(A))→ Ab(Fr(S), A) We verify that these bijections satisfy the naturality condition in both variables. First, in A, we consider Set(S, U(A)) Ab(Fr(S), A) Set(S, U(B)) Ab(Fr(S), B) fSA U(ϕ)∗ ϕ∗ fSB Where ϕ : A → B and for g : S → U(A) we have U(ϕ)∗(g) = U(ϕ)g , ϕ∗(ψ) = ϕψ as in Definition 1.6.1. Noting that for g : S → U(A), fSBU(ϕ)∗(g) = Fr(U(ϕ)g) is a homomor- phism such that Fr(U(ϕ)g)(s) = ϕg(s), and ϕ∗fSA(g) = ϕFr(g) is a homomorphism such that ϕFr(g)(s) = ϕg(s). We have that the diagram commutes, and fSA is natural in A. For S we proceed similarly, considering the diagram Set(T, U(A)) Ab(Fr(T ), A) Set(S, U(A)) Ab(Fr(S), A) fTA h∗ Fr(h)∗ fSA Where h : S → T , and for g : T → U(A) we have h∗(g) = gh and Fr(h)∗(ψ) = ψFr(h) as in Definition 1.6.1. Then since fSAh ∗(g) = Fr(gh) and Fr(h)∗fTA(g) = Fr(g)Fr(h)), we have that fSAh ∗(g) sends s ∈ S to gh(s), and Fr(h)∗fTA(g) sends s to gh(s). Thus the diagram commutes, and fSA is natural in S. Thus, we have a collection of isomorphisms fSA : Set(S, U(A))→ Ab(Fr(S), A) 27 natural in S and A, giving the desired natural equivalence, and therefore the adjoint rela- tionship. 1.7 Abelian Categories To conclude this chapter, we present a definition describing the particular kind of category we study in the remainder of the work. In doing so, we prove a few additional results we make use of later, and show that all previously introduced reformulated definitions exist in our category of interest. Definition 1.7.1 (Abelian Category [HW89] [HS12]). 8 An abelian category A is a category satisfying the following: 1. A has a zero object. 2. A has finite products and coproducts. 3. Every morphism has a kernel and a cokernel. 4. Every monomorphism is the kernel of its cokernel. 5. Every epimorphism is the cokernel of its kernel. 6. Every morphism is expressible as an epimorphism followed by a monomorphism. The final condition is of particular importance: 7.a The set of morphisms A (X, Y ) has a unique abelian group structure for any X, Y . 7.b The zero morphism 0: X → Y serves as the identity for this structure. 7.c For any f, g, h ∈ A (X, Y ) we have (f + g)h = fh+ gh and f(g + h) = fg + fh. 8Note that axioms 1, 2, 7.a-7.c are borrowed from the more general notion of an additive category, which we have elected to exclude from this work for sake of streamlining. 28 The axioms above are observably self-dual, i.e. Axiom 4 is the dual of Axiom 5, the dual of Axiom 1 is itself, the dual of Axiom 2 is itself, and similarly so for Axioms 3, 6, and 7. As mentioned in §1.5, this fact allows us to obtain a dual result for each result we prove. The archetypal example of an abelian category, and the one we work in for the majority of this thesis, is the category Ab. Further examples include the category ModR of left R-modules and the category Vectk of vector spaces over a field k [Rie17]. The fact that abelian categories have products and coproducts, as well as zero morphisms by existence of a zero object, allow us to show the following Theorem 1.7.2 ([HW89]). If A1 ⊕A2 is the product of A1, A2, there exist morphisms such that A1 ⊕ A2 is the coproduct of A1, A2. Proof. For A1 we have the morphisms 1A1 : A1 → A1 and 0: A1 → A2. Similarly so for A2. We then utilize the universal property of the product as shown A1 A2 A1 A1 ⊕ A2 A2 A1 A2 ⊕ A2 A2 h1 01 h2 0 1 π2π1 π1 π2 to find morphisms h1 : A1 → A1 ⊕ A2, h2 : A2 → A1 ⊕ A2 such that π1h1 = 1A1 , π2h1 = 0: A1 → A2, and similarly so for h2. We claim h1, h2 are the morphisms such that A1 ⊕ A2 is the coproduct. To show this is the case, we examine the diagram X A1 A1 ⊕ A2 A2h1 α1 α h2 α2 and show there exists a unique α such that the diagram commutes. Define α = α1π1 + α2π2. Then (α1π1 + α2π2)h1 = α1π1h1 + α2π2h1 = α11A1 + 0 = α1. Similarly, (α1π1 + α2π2)h2 = α1π1h2 + α2π2h2 = 0 + α21A2 = α2. Thus, α as defined allows the diagram to commute. To show uniqueness, we first prove that h1π1 + h2π2 : A1⊕A2 → A1⊕A2 is the identity. Since π1(h1π1 + h2π2) = (π1h1)π1 + (π1h2)π2 = 1A1π1 + 0π2, we have π1(h1π1 + h2π2) = π1. 29 Similarly, π2(h1π1+h2π2) = (π2h1)π1+(π2h2)π2 = 0π2+1A2π2, giving that π2(h1π1+h2π2) = π2. Then, the following diagram commutes. A1 ⊕ A2 A1 A1 ⊕ A2 A2 π1 π2h1π1+h2π2 π1 π2 But then as we have noted for previous proofs, the identity also allows this diagram to commute. Since the morphism allowing this diagram to commute must be unique, we then have that h1π1 + h2π2 is the identity. Then to finish uniqueness, we assume there exists some α′ allowing the coproduct diagram to commute. That is, such that α′h1 = α1, α ′h2 = α2. Then α(h1π1 + h2π2) = (αh1)π1 + (αh2)π2 = (α′h1)π1 + (α′h2)π2 giving α′h1π1 + α′h2π2 = α′(h1π1 + h2π2) Since h1π1+h2π2 is the identity, we then have α1 = α′1 and thus α = α′, showing uniqueness. This proof extends inductively for any finite collection of objects Ai, and shows that the product and coproduct coincide in the finite case. Then, the only topic not touched by Definition 1.7.1 is that of pullbacks and pushouts. This, however, is solved through the note that pullbacks and pushouts may be re-expressed as kernels and cokernels of specific morphisms, making use of the abelian group structure of morphisms in abelian categories. Theorem 1.7.3 ([HS12]). In an abelian category A , the diagram B A1 A2 A b1 b2 ϕ1 ϕ2 is a pullback if and only if for the sequence 30 B A1 ⊕ A2 A {b1,b2} ⟨ϕ1,−ϕ2⟩ {b1, b2} is the kernel of ⟨ϕ1,−ϕ2⟩. Proof. Suppose the diagram X B A1 A2 A x1 x2 h b2 b1 ϕ1 ϕ2 is a pullback. Thus, we have x1 = b1h, x2 = b2h. We then utilize b1, b2, x1, x2 in the following two product diagrams B X A1 A1 × A2 A2 A1 A1 × A2 A2 b1 b2{b1,b2} x1 x2{x1,x2} π1 π2 π1 π2 calling the unique morphisms guaranteed by the universal property of the product by the names {b1, b2} and {x1, x2} respectively. Then {x1, x2} = {b1, b2}h, as we may form the diagram X B A1 A1 × A2 A2 h x1 x2 {b1,b2} b1 b2 π1 π2 showing that {b1, b2}h fulfills the role of {x1, x2}, which must be unique, giving {x1, x2} = {b1, b2}h as desired. Then, since ϕ2 ∈ A (A2, A), a set of morphisms in an abelian category, which has a canonical abelian group structure, we can find −ϕ2 such that ϕ2 + (−ϕ2) = 0. We then utilize ϕ1,−ϕ2 in the following coproduct diagram A A1 A1 ⊕ A2 A2 ϕ1 ι1 ⟨ϕ1,−ϕ2⟩ −ϕ2 ι2 31 calling the unique morphism guaranteed by the coproduct by ⟨ϕ1,−ϕ2⟩. Since in an abelian category the product and coproduct of two objects coincide, we may call them both by A1 ⊕ A2 and form the following diagram, which we claim is a kernel diagram. B A1 ⊕ A2 A X {b1,b2} ⟨ϕ1,−ϕ2⟩ h {x1,x2} We have already established that h allows commutativity. Since h is found by the universal property of the pullback, it is unique and exists for any X, giving the universal property of the kernel. We now need only show that the composition ⟨ϕ1,−ϕ2⟩{b1, b2} is zero. By the pullback diagram, we have ϕ1b1 = ϕ2b2, which by the abelian group structure of morphisms gives that ϕ1b1 − ϕ2b2 = 0. We then stack the previously shown product and coproduct diagrams as such A A1 A1 ⊕ A2 A2 B ι1 ϕ1 π1 π2 ⟨ϕ1,−ϕ2⟩ ι2 −ϕ2 b2 b1 {b1,b2} and present the following sequence of equalities. 0 = ϕ1b1 − ϕ2b2 = ϕ1π1{b1, b2} − ϕ2π2{b1, b2} = ⟨ϕ1,−ϕ2⟩ι1π1{b1, b2}+ ⟨ϕ1,−ϕ2⟩ι2π2{b1, b2} = ⟨ϕ1,−ϕ2⟩(ι1π1 + ι2π2){b1, b2} Since in the proof of Theorem 1.7.2, we showed that ι1π1 + ι2π2 is the identity on A1 ⊕ A2, we have 0 = ϕ1b1 − ϕ2b2 = ⟨ϕ1,−ϕ2⟩(ι1π1 + ι2π2){b1, b2} = ⟨ϕ1,−ϕ2⟩{b1, b2} Thus, 0 = ⟨ϕ1,−ϕ2⟩{b1, b2} as desired. Of course, since {x1, x2} = {b1, b2}h, we have 0 = 0h = ⟨ϕ1,−ϕ2⟩{b1, b2}h = ⟨ϕ1,−ϕ2⟩{x1, x2} as well. 32 Finally, the uniqueness of the morphism h gives that {b1, b2} is monomorphic. If {b1, b2}f = {b1, b2}g for f, g : Y → B, Y is such that it has b1f = b2g and b2f = b2g such that the pull- back diagram Y B A1 A2 A f g b1f=b1g b2f=b2g b1 b1 ϕ1 ϕ2 commutes. Then the universal property gives that there is a unique h such that b1f = b1g = b1h and b2f = b2g = b2h. Since h is unique, and both f, g satisfy its role, we therefore must have f = h = g, and {b1, b2} is monomorphic. Thus the pullback implies the kernel. For the converse, we simply reverse the line of reasoning to form the pullback out of the kernel. Dually, we have the following Theorem 1.7.4 ([HS12]). In an abelian category A , the diagram A A1 A2 B ψ1 ψ2 b1 b2 is a pushout if and only if for the sequence A A1 ⊕ A2 B {ψ1,ψ2} ⟨b1,−b2⟩ ⟨b1, b2⟩ is the cokernel of {ψ1, ψ2}. Together, these theorems allow the construction of pullbacks and pushouts so long as we have kernels, cokernels, and a coinciding product/coproduct. These are all present in abelian categories. We present one final theorem. 33 Theorem 1.7.5 ([HW89]). In an abelian category, the kernel of every monomorphism is 0. Proof. Let f : X ↣ Y be a monomorphism. Then we have 0↣X f ↣Y where f0 = 0 by the compositional properties of the zero morphism. Then assume there exists ℓ : L→ X such that fℓ = 0. Then fℓ = f0, and we have ℓ = 0 since f is monomorphic. Thus 0 is the unique map such that 00 = ℓ, and 0 has the universal property of the kernel. Finally, 0 : 0 ↣ X is vacuously monomorphism, as there is exactly one unique morphism into the zero object from any other object. Dually, we have the following Theorem 1.7.6 ([HW89]). In an abelian category, the cokernel of every epimorphism is 0. This gives the following corollary for Ab. Corollary 1.7.7. In Ab, monomorphisms are precisely injective homomorphisms, epimor- phisms are precisely surjective homomorphisms. Proof. The kernel in Ab is the embedding of the usual understanding of the kernel, that is for ϕ : A → B, it is the embedding of the subgroup K = {a ∈ A|ϕ(a) = 0} into A. The cokernel in Ab is the projection onto the usual understanding of the cokernel, that is for ϕ : A → B, it is the projection from B onto B im(ϕ) . Since injective homomorphisms are those with trivial kernel and surjective homomorphisms are those with trivial cokernel, the corollary is immediate. From here forward, we work entirely within abelian categories. The existence of objects previously defined, as well as the fact morphisms “factor” as an epimorphism followed by a monomorphism, is be of great import, and serves as the foundation for the category-theoretic definition of exactness in the following chapter. 34 Chapter 2 Exact Sequences The concept of an exact sequence is a foundational one for this work. Of the four functors we study, a vital property we examine is that of exactness, that is, the property of preserving exact sequences. As a whole, this leads to the development of Ext in the 4th chapter. We therefore spend the entirety of this chapter on building the necessary foundation to support this development. 2.1 Exact Sequences Definition 2.1.1 (Exact Sequence [HS12]). Let . . . An−1 An An+1 . . . fn−1 fn εn µn be a sequence of objects in an abelian category A , where each fn = µnεn with εn epimorphic, µn monomorphic. This sequence is said to be exact at An if µn−1 is the kernel of εn and εn is the cokernel of µn−1. We then refer to the full sequence as exact if it is exact at An for every n. In practice, we work almost entirely with the far more manageable short exact sequence. Definition 2.1.2 (Short Exact Sequence [HS12]). A short exact sequence in an abelian category A is an exact sequence of the form 0→ A′ µ ↣A ε ↠A′′ → 0 Since in abelian categories we have by Theorem 1.7.5 and its dual that the kernel of every monomorphism is 0, and the cokernel of every epimorphism is 0, we may invoke the uniqueness of kernels and cokernels to declare that the zeroes on each side of our sequence are superfluous, and we may write our sequence as A′ µ ↣A ε ↠A′′ Further note that, being an isomorphism, the identity is both monomorphic and epimorphic and we may write µ = µ1A′ and ε = 1A′′ε as the previously spoken of monomorphism followed by epimorphism decompositions of µ, ε. Thus, the above sequence is exact at A (and thus exact overall) if µ is the kernel of ε, and ε is the cokernel of µ. A shift towards short exact sequences rather than general exact sequences then allows us to simplify our notation and conditions for what makes the sequence exact. We have thus established almost all purely categorical definitions needed for our further work. For now, we move from using the language of category theory to using the language of groups. Know that underlying our discussion is an understanding and eye towards categories, but that the structures we study have interest solely from the perspective of group theory. When examining exact sequences in Ab we may simplify our conditions for exactness even further through the observations given in Corollary 1.7.7. The fact that the usual understanding of both the kernel and cokernel align with our definitions allow us to simply examine images of subsets rather than busy ourselves with proving universal properties, as the following definition suggests. Definition 2.1.3 (Exact Sequence (in Ab) [HW89]). 36 A sequence of abelian groups · · · → An−1 fn−1−−→ An fn−→ An+1 → . . . where each fn is a homomorphism fn : An → An+1, is called exact at An if im(fn−1) = ker(fn). We then refer to the full sequence as exact if it is exact at each An. For a short exact sequence A′ µ ↣A ε ↠A′′ this simplifies even further to only three conditions. im(µ) = ker(ε), µ is injective, and ε is surjective. We now move to examples. Given two abelian groups A,B, we may utilize the canonical injection ιA : A↣A ⊕ B defined as a 7→ (a, 0) and the canonical projection πB : A ⊕ B↠B defined as (a, b) 7→ b to form the sequence A ιA ↣A⊕B πB ↠B The kernel of πB is all elements of the form (a, 0), which is precisely the image of ιA, giving that the sequence is exact. While this sequence always exists for any A,B it is not necessarily the only sequence of the form A↣E↠B that does. Example 2.1.4. Consider the case of A = B = Z2. We may form both the sequence Z2 ι ↣Z2 ⊕ Z2 π ↠Z2 and the sequence Z2 µ ↣Z4 ε ↠Z2 with ι, π the canonical injection and projection, µ defined as 1 7→ 2 (a sufficient description by Lemma 1.6.5), and ε defined as {0, 2} 7→ 0 and {1, 3} 7→ 1. Clearly im(µ) = ker(ε), µ is injective, and ε is surjective, so the sequence is exact. While we have yet to establish any 37 sort of equivalence between these sequences, since Z2 ⊕ Z2 ̸∼= Z4, we would certainly expect these sequences to be distinct. However for A = Z2, B = Z3, we may form the sequences Z2 ι ↣Z2 ⊕ Z3 π ↠Z3 and Z2 µ ↣Z6 ε ↠Z3 with ι, π the canonical injection and projection, µ defined as 1 7→ 3 and ε defined as {0, 3} 7→ 0, {1, 4} 7→ 1, and {2, 5} 7→ 2. We again have injectivity, surjectivity, and that im(µ) = ker(ε), giving exactness. Since Z2⊕Z3 ∼= Z6, we are no longer certain what, if any distinction exists between the two. This curiosity leads us to define when exactly we may say two sequences are equivalent. 2.2 Equivalent Exact Sequences Definition 2.2.1 (Equivalent Exact Sequences [HS12]). Let each of B µ ↣E1 ε ↠A, B µ′ ↣E2 ε′ ↠A be short exact sequences such that the following diagram commutes B E1 A B E2 A µ ε ϕ µ′ ε′ Then ϕ is an isomorphism, and the two sequences are said to be equivalent. Here, the fact that ϕ is an isomorphism given that the diagram commutes is not proven directly. Rather, we give a stronger result in order to illustrate the power of exact sequences, and showcase the utility of the diagram chase. Should the reader seek a direct proof, they are encouraged to read case (iii) of the proof below, and let ϕ′, ϕ′′ be the identity. A direct proof may also be found on page 29 of [HW89]. 38 Theorem 2.2.2 ([HS12]). Given the commutative diagram with exact rows A′ A A′′ B′ B B′′ ϕ′ µ ϕ ε ϕ′′ µ′ ε′ if any two of the three homomorphisms ϕ′, ϕ, ϕ′′ are isomorphisms, then the third is an isomorphism as well. Proof. We prove each of the three possible cases. For brevity, we explicitly note two equalities that are of vital importance. Since the above diagram commutes, we have µ′ϕ′ = ϕµ and ϕ′′ε = ε′ϕ Then for each case, we show the homomorphism in question is injective and surjective. Our method of showing this is choosing an element in the kernel, and showing it to be 0 for injectivity, and choosing an element in the codomain and finding an element in the domain that maps to it for surjectivity. (i) ϕ′, ϕ are isomorphisms. Let a′′ ∈ ker(ϕ′′). Since ε is surjective via exactness, we may find a ∈ A s.t. ε(a) = a′′. Then ϕ′′ε(a) = 0, so ε′ϕ(a) = 0. Thus, ϕ(a) ∈ ker(ε′), and is therefore of the form µ′(b′) for some b′ ∈ B′. Then since ϕ′ is an isomorphism, there exists a′ ∈ A′ such that ϕ′(a′) = b′, giving that µ′ϕ′(a′) = ϕ(a). Thus, ϕ(a) = ϕµ(a′), so a = µ(a′) since ϕ′ is injective and a ∈ ker(ε) by exactness, giving that a′′ = ε(a) = εµ(a′) = 0. Thus a′′ ∈ ker(ϕ′′) =⇒ a′′ = 0, and ϕ′′ is injective. Then let b′′ ∈ B′′. Since ε′ surjective, we may find b ∈ B such that ε′(b) = b′′. Since ϕ is an isomorphism, there exists a ∈ A such that ϕ(a) = b. Then ε′ϕ(a) = b′′ so ϕ′′ε(a) = b′′, and thus ε(a) ∈ A′′ has ϕ′′(ε(a)) = b′′, giving that ϕ′′ is surjective. 39 Thus, ϕ′′ is surjective and injective, thus an isomorphism. (ii) ϕ, ϕ′′ are isomorphisms. Let a′ be in ker(ϕ′). Then µ′ϕ′(a′) = µ′(0) = 0, so ϕµ(a′) = 0. But then ϕ is an isomorphism, so µ(a′) = 0, and µ is injective, so a′ = 0. Thus a′ ∈ ker(ϕ′) =⇒ a′ = 0, and ϕ′ is injective. Then let b′ ∈ B′. Consider µ′(b′). Since µ′(b′) ∈ B, and ϕ an isomorphism, we may find a ∈ A such that ϕ(a) = µ′(b′). Then since ε′ϕ(a) = ε′µ′(b′) = 0 via exactness, we have ϕ′′ε(a) = 0, giving that a ∈ ker(ε) since ϕ′′ is an isomorphism. Then a = µ(a′) for some a′ ∈ A′ via exactness, and ϕ(a) = ϕµ(a′) = µ′(b′) giving that µ′ϕ′(a′) = µ′(b′), and thus ϕ′(a′) = b′ since µ′ injective. Thus, ϕ′ is surjective. Now having that ϕ′ injective and surjective, we conclude it is an isomorphism. (iii) ϕ′, ϕ′′ are isomorphisms. Let a ∈ ker(ϕ). Then ϕ(a) = 0, so ε′ϕ(a) = 0, and ϕ′′ε(a) = 0. Since ϕ′′ is an isomorphism, we then have ε(a) = 0, so a = µ(a′) for a′ ∈ A′ via exactness. Then ϕ(a) = ϕ(µ(a′)) = 0, so µ′ϕ′(a′) = 0. Since both ϕ′, µ′ injective µ′ϕ′(a′) = 0 =⇒ a′ = 0. Then µ(a′) = µ(0) = a, so we have a = 0, which gives ker(ϕ) = 0, and therefore ϕ injective. Then let b ∈ B. Consider ε′(b). Since ϕ′′ an isomorphism, we may find a′′ such that ϕ′′(a′′) = ε′(b). Then since ε surjective, we may find a such that ε(a) = a′′. Then ϕ′′ε(a) = ε′(b), so ε′ϕ(a) = ε′(b). Then since ε′(b) − ε′ϕ(a) = 0, ε′(b − ϕ(a)) = 0, and therefore b− ϕ(a) = µ′(b′) for some b′ ∈ B′ by exactness. Since ϕ′ is an isomorphism, 40 we may find a′ ∈ A′ such that ϕ′(a′) = b′, and thus µ′ϕ′(a′) = ϕµ(a′) = µ′(b′), which equals b− ϕ(a). Finally, we have ϕ(µ(a′) + a) = ϕµ(a′) + ϕ(a) = b− ϕ(a) + ϕ(a) = b Thus, ϕ(µ(a′) + a) = b, and ϕ is surjective. Since ϕ is injective and surjective, we conclude it is an isomorphism, completing the proof of all three cases. In the light of the above theorem, we see that our earlier statement is simply a spe- cial case of the stronger result. Since equality (or identity) is clearly an isomorphism, any homomorphism ϕ : E1 → E2 allowing commutativity must be an isomorphism. This of course does not mean that any two exact sequences B ↣ E1 ↠ A and B ↣ E2 ↠ A are equivalent, as we saw earlier with Example 2.1.4. Equivalence, in this case, stems precisely from the fact that ϕ allows commutativity. Even if E1, E2 are isomorphic, it is not a guarantee that the sequences are equivalent, as the following example shows. Example 2.2.3. Consider the exact sequences Z 3 ↣Z ε ↠Z3 and Z 3 ↣Z ε′ ↠Z3 where 3 denotes multiplication by 3, and ε(1) = 1 mod 3, ε′(1) = 2 mod 3. We prove by contradiction that they are not equivalent. Consider the diagram Z Z Z3 Z Z Z3 3 ϕ ε 3 ε′ 41 We assume there exists a homomorphism ϕ : Z → Z allowing the diagram to commute, and begin by examining the right hand square. Since the diagram commutes, we have 1Z3ε = ε′ϕ =⇒ 1Zε(1) = ε′ϕ(1). We evaluate to obtain 1 mod 3 = 2n mod 3, where ϕ(1) = n. Utilizing the fact that 2 is the multiplicative inverse of itself modulo 3, we solve the congruence as n ∼= 2 mod 3. Thus, n = 3k + 2 for some k ∈ Z. We then turn to the left hand square, which commutes via assumption, giving that ϕ ◦ 3 = 3 ◦ 1Z, and thus ϕ ◦ 3(1) = 3 ◦ 1Z(1). Then ϕ(3) = 3 =⇒ 3n = 3. As such, we must have n = 1. Thus n = 1 = 3k + 2 via our earlier work, giving that k = −1 3 , a contradiction since k ∈ Z. Thus, there does not exist ϕ : Z → Z allowing the diagram to commute, and the sequences are not equivalent. Example 2.2.4. We return to Example 2.1.4 of A = B = Z2, and the sequences Z2 ι ↣Z2 ⊕ Z2 π ↠Z2 and Z2 ι ↣Z4 π ↠Z2 By our definition of equivalence, there must exist an isomorphism Z2 ⊕ Z2 ∼= Z4 for the sequences to be equivalent. Since they are not isomorphic, they cannot be equivalent. Continuing, we let A = Z2, B = Z3, claim that the sequences as constructed in 2.1.4 are equivalent, construct the diagram Z2 Z2 ⊕ Z3 Z3 Z2 Z6 Z3 ι π ϕ µ ε and define ϕ such that the diagram commutes. Define ϕ as (1, 1) 7→ 1 and therefore (0, 2) 7→ 2, (1, 0) 7→ 3, (0, 1) 7→ 4, and (1, 2) 7→ 5. Then ϕ({(0, 0), (1, 0)}) = {0, 3}, so ϕι = µ1Z2 and the left square commutes. For the right square, we have εϕ((0, 0)) = 0 εϕ((1, 1)) = 1 εϕ((0, 2)) = 2 εϕ((1, 0)) = 0 εϕ((0, 1)) = 1 εϕ((1, 2)) = 2 42 matching the values for 1Z3π, giving that the right square commutes. Together, this gives that the diagram commutes, ϕ is an isomorphism by Theorem 2.2.2, and the sequences are equivalent. 2.3 Split Exact Sequences We return to the example of the sequence B ιB ↣B ⊕ A πA ↠A Recall that we previously stated that this sequence exists for any two abelian groups. Thus, the sequence A ιA ↣B ⊕ A πB ↠B is exact as well. Together, we then have the sequence B πB ⇆ ιB B ⊕ A ιA ⇆ πA A (2.1) with πBιB = 1B and πAιA = 1A. We generalize this notion as follows Definition 2.3.1 (Split Exact Sequence [HS12]). A short exact sequence B µ ↣E ε ↠A (2.2) is referred to as split if there exists η : E ↠ B and ν : A ↣ E such that ηµ = 1B and εν = 1A. Just as with the isomorphism, the two defining equalities of the split exact sequence should not be viewed as one property, but as two distinct properties for which the split sequence satisfies both. This is suggested by the following definitions. 43 Definition 2.3.2 (Left Split Exact Sequence [HW89]). A short exact sequence 2.2 is referred to as left split if there exists η : E ↠ B such that ηµ = 1B. We then call η a left splitting of 2.2. Definition 2.3.3 (Right Split Exact Sequence[HW89]). A short exact sequence 2.2 is referred to as right split if there exists ν : A ↣ E such that εν = 1A. We then call ν a right splitting of 2.2. We assert through the upcoming theorem that in the category Ab, a sequence being left split, right split, or both gives that it is equivalent to the sequence 2.1. The proof of this fact and some subsequent comments make up the remainder of this section. Theorem 2.3.4 ([HW89]). Let A′ µ ↣A ε ↠A′′ be a short exact sequence of abelian groups. Then the following statements are equivalent. (i) There exists η : A→ A′ with ηµ = 1A′. (ii) There exists η : A→ A′ so that (A; η, ε) is the product (direct product) of A′, A′′. (iii) There exists ν : A′′ → A with εν = 1A′′. (iv) There exists ν : A′′ → A so that (A;µ, ν) is the coproduct (direct sum) of A′, A′′. (v) There exist η : A→ A′ and ν : A′′ → A with µη + νε = 1A. We begin with a lemma. Lemma 2.3.5 ([HW89]). Let A′ µ ↣A ε ↠A′′ be a short exact sequence of groups. Then it is equivalent to a product if and only if there exists η : A→ A′ such that ηµ = 1A′. 44 Proof. Consider the diagram A′ A A′′ A′ A′ × A′′ A′′ µ ϕ ε ιA′ πA′′ πA′ ιA′′ Assume A is equivalent to a direct product. Then the above diagram commutes. As such, ϕµ = ιA′1′A. We may then define η = πA′ϕ and see that ηµ = πA′ϕµ = πA′ιA′ = 1A′ , showing the existence of η satisfying ηµ = 1A′ . Conversely, suppose there exists a homomorphism η such that ηµ = 1A′ . In light of 2.2.2, we need only show a ϕ that ensures commutativity to have that it is an isomorphism, and the sequences are equivalent. Let ϕ(a) = (η(a), ε(a)). We of course have that ϕ is a homomorphism, as ϕ(a)+ϕ(a′) = (η(a), ε(a))+(η(a′), ε(a′)) = (η(a)+η(a′), ε(a)+ε(a′)) = (η(a+a′), ε(a+a′)) = ϕ(a+a′) For the diagram to commute, we then need that ιA′ = ϕµ and πA′′ϕ = ε. Let a′ ∈ A′. Then ιA′(a′) = (a′, 0), and ϕµ(a′) = (ηµ(a′), εµ(a′)) = (a′, 0) via ηµ = 1, εµ = 0. Similarly for a ∈ A, we have πA′′ϕ(a) = πA′′(η(a), ε(a)) = ε(a). This of course is equal to ε(a), so the diagram commutes, ϕ is an isomorphism, and the sequences are equivalent. We now prove the theorem. The above lemma gives (i) ⇔ (ii). We then use the fact that the direct sum and direct product coincide for finite collections of abelian groups to obtain (iv) =⇒ (ii). From the coinciding product/coproduct, we obtain (iii) and (v), as in the introduction to this section. The manner of obtaining (iii) in particular is near identical to showing (ii) =⇒ (i), defining ν as ϕ−1ιA′′ . (v) then follows from the fact that ιA′πA′ + ιA′′πA′′ = 1A′⊕A′′ , and ϕ is an isomorphism. We thus have the following paths of implication (i) (ii) (iv) (iii) (v) 45 All that remains is to give (iii) =⇒ (v) =⇒ (i). For this, we require that for any two homomorphisms ϕ, ψ of abelian groups, the mapping (ϕ + ψ)(a) = ϕ(a) + ψ(a) always defines a homomorphism. Similarly so for −ϕ(a) defined by mapping each ϕ(a) to its inverse in the codomain. In the next chapter, we discuss the details of this fact, but for now we simply reference the fact that Ab is an abelian category, and as such its morphisms (homomorphisms) have a canonical abelian group structure. Lemma 2.3.6. (iii) =⇒ (v) =⇒ (i) Proof. Let A′ µ ↣A ε ↠A′′ be a short exact sequence and assume that there exists ν : A′′ → A such that εν = 1′′A. Consider the mapping 1 − νε : A → A. Via the referenced result above, this mapping is a homomorphism. Applying ε, we receive ε(1− νε) = ε− ενε = ε− ε = 0 Thus since ker(ε) = im(µ) via exactness, we must have that the image of 1 − νε is in the image of µ. But then since 1− νε : A → A, and µ : A′ → A, we must have 1− νε = µη for some η : A → A′. By the fundamental homomorphism theorem, µ is an isomorphism from A′ to im(µ), so µ−1 (after the projection onto the image of µ) is the η in question. Solving, we then have 1 = µη + νε, giving (v) as desired. For (v) =⇒ (i) we then precompose by µ, obtaining µ = (µη + νε)µ = µηµ+ νεµ = µ(ηµ) + ν0 = µ(ηµ) Thus µ = µ(ηµ), giving that ηµ must be 1A′ , since µ injective via exactness. Thus (v) =⇒ (i) and as such (iii) =⇒ (v) =⇒ (i), completing the proof of our larger theorem. While only proven with respect to abelian groups, Theorem 2.3.4 in fact applies in any 46 abelian category. In that context, it is commonly referred to as the splitting lemma [Hat02]. We allow a small aside to address an important fact: Theorem 2.3.4 does not hold without the assumption that all groups are abelian. However, much of it does. A review of Lemma 2.3.5 shows that we did not specify our groups to be abelian. It is then true that in the larger category Grp we have (i) ⇔ (ii), and even that (ii) =⇒ (iii) and (ii) =⇒ (v). We nonetheless fail to have equivalence of all statements. Particularly, we lack equivalence because the existence of a right splitting does not guarantee the existence of a left splitting, giving that (iii) ≠⇒ (i), which we show via the following example. Example 2.3.7 (An, Sn,Z2). Consider the sequence 0→ An µ ↣Sn ε ↠Z2 → 0 where µ : An → Sn is the inclusion of the alternating group An into the permutation group Sn, and ε : Sn → Z2 is the sign function, sending even permutations to 0, and odd permutations to 1. Then let ν : Z2 → Sn be defined as ν(0) = e, ν(1) = (1 2). Since the identity is an even permutation, and (1 2) is an odd permutation, ν satisfies εν = 1Z2 . Thus, ν is a right splitting of the sequence. If the prior result held, we would expect to be able to find a left splitting of the sequence as well. That is, some consequently surjective homomorphism η : Sn → An s.t. ηµ = 1. Assume by way of contradiction that we can, and that η is this homomorphism. By the fundamental homomorphism theorem, since η defines a map from Sn to An, we must have Sn/ker(η) ∼= An. Since η is surjective, and An has index 2 in Sn, it then must be the case that ker(η) has order 2. We now show that no subgroups of order 2 in Sn are normal, giving that such a homomorphism cannot exist. The only subgroups of order 2 in Sn are those of the form {e, T}, where T is some transposition, or some product of disjoint transpositions. Consider then a product T of disjoint transpositions (a1 b1)(a2 b2)(a3 b3) · · · (ak bk). Then choose a transposition τ = (a1 c), 47 with the value of c being chosen according to the following cases dependent on the value of n for Sn. Case 1: n is odd. Since our product consists of exclusively transpositions, it will permute 2k elements. Since n is odd, we therefore have that there is at least one element c not permuted by our product. Then (a1 c) is disjoint with, and as such commutes with, all but the first transposition in our product. We conjugate our product by (a1 c), to obtain (a1 c)(a1 b1)(a2 b2)(a3 b3) · · · (ak bk)(a1 c) = (a1 c)(a1 b1)(a1 c)(a2 b2) · · · (ak bk) Since (a1 c)(a1 b1)(a1 c) = (b1 c), we then have that (a1 c)(a1 b1)(a2 b2)(a3 b3) · · · (ak bk)(a1 c) = (b1 c)(a2 b2) . . . (ak bk) an element not in the subgroup generated by our product. Case 2: n is even. With no guarantee we may chose c such that it is not permuted by our product, we instead choose c = b2. Then (a1 b2) is disjoint with all but the first two transpositions in our product, and we have (a1 b2)(a1 b1)(a2 b2)(a3 b3) · · · (ak bk)(a1 b2) = (a1 b2)(a1 b1)(a2 b2)(a1 b2) · · · (ak bk) Then since (a1 b2)(a1 b1)(a2 b2)(a1 b2) = (a1 a2)(b1 b2), we have (a1 b2)(a1 b1)(a2 b2)(a3 b3) · · · (ak bk)(a1 b2) = (a1 a2)(b1 b2) · · · (ak bk) an element not in the subgroup generated by our product. Thus for any product of disjoint transpositions T , we may find a transposition τ such 48 that τTτ−1 /∈ ⟨T ⟩. As such, no subgroups of order 2 are normal in Sn. Therefore, we cannot have Sn/ker(η) ∼= An, giving that there are no surjective homomorphisms η : Sn → An, guaranteeing that Sn has no left splitting, and thus: Sn cannot be isomorphic to a direct product. 2.4 Exact Functors Definition 2.4.1 (Exact Functor [HW89]). Let F : A → B be a functor between abelian categories A ,B. Then F is exact if it preserves short exact sequences. For F covariant and exact, the exactness of the sequence A′ f ↣A g ↠A′′ ensures that its image under F FA′ Ff↣FA Fg ↠FA′′ is exact as well. If F is contravariant and exact, the exactness of the sequence A′ f ↣A g ↠A′′ ensures that its image under F FA′′ Fg↣FA Ff ↠FA′ is exact. This definition is limited to precisely what is needed for our further work, but is somewhat cumbersome for showing whether or not a functor is exact. To simplify this process, we introduce additional variants of exact sequences below. 49 Definition 2.4.2 (Left and right exact sequences [HW89]). Let a sequence of the form 0→ A′ µ ↣A ε→A′′ be exact. We then refer to it as a left exact sequence. Similarly, if a sequence of the form A′ µ→A ε ↠A′′ → 0 is exact, we refer to it as a right exact sequence. Note again, that here the zeroes are superfluous. Definition 2.4.3 (Left and Right Exact Functors [HW89]). A functor F : A → B between abelian categories A ,B is considered left exact if it preserves left exact sequences, and right exact if it preserves right exact sequences. Note that if F contravariant and left/right exact, then the image of a left exact sequence is a right exact sequence, and the image of a right exact sequence is a left exact sequence. Theorem 2.4.4. A functor on abelian categories is exact if and only if it is left and right exact. Proof. Assume F exact. Without loss of generality, we let F be covariant. Let A′ f ↣A g−→ A′′ be a left exact sequence. Utilizing the fact that each morphism in an abelian category factors as an epimorphism followed by a monomorphism, we write g = µgεg. Then for the following sequence, the first two morphisms form an exact sequence. A′ f ↣A εg ↠ im(g) µg ↣A′′ Here, we borrow the notation im(g) fromAb for the domain of µg. Since F exact, it preserves 50 the exact sequence. We thus have the following in the image of F . F (A′) Ff ↣F (A) Fεg ↠ F (im(g)) Fµg−−→ F (A′′) Here, the first two morphisms form an exact sequence. By our definition of exactness, we then have if f = µfεf , Fεg is the cokernel of Fµf . This fact does not change when we compose Fεg and Fµg to obtain F (µgεg) = Fg. Thus, the sequence F (A′) Ff ↣F (A) Fg−→ F (A′′) is exact at F (A′), F (A), and is therefore left exact. As such, F preserves left exact sequences. Showing that F preserves right exact sequences follows similarly. Thus, F is both left and right exact. Assume F both left and right exact. Consider the short exact sequence A′ µ ↣A ε ↠A′′ Then the sequence ker(ε) ↣ A ε−→ A′′ is left exact, and preserved by F . Thus F preserves kernels. Similarly, the sequence A µ−→ A′′ ↠ coker(µ) is right exact, and preserved by F . Thus F preserves cokernels. Since µ is the kernel of ε, and ε is the cokernel of µ by exactness, F preserves each, and thus preserves the full sequence. Thus, rather than approaching exactness directly, we may show left and right exactness, or perhaps only achieve partial exactness in the functors we consider. In particular, one of 51 the final results of the following chapter is determining whether the functor Hom is exact. It is the result of this inquiry that allows the development of Ext as promised at the outset of this chapter. 52 Chapter 3 Hom(A,B) 3.1 Definition and Examples Definition 3.1.1 (The Abelian Group Hom(A,B) [HW89]). Let A,B be abelian groups. Then Hom(A,B) is the abelian group whose underlying set is the set of homomorphisms A→ B, and whose operation is “elementwise” addition, defined as (ϕ1 + ϕ2)(a) = ϕ1(a) + ϕ2(a) So that the image of ϕ1 + ϕ2 is the sum of their images in B. We must ensure closure. That is, that ϕ1 + ϕ2 is a homomorphism A → B. Since the domain of each ϕi is A, and ϕ1(a), ϕ2(a) ∈ B, closure of B ensures that ϕ1(a) + ϕ2(b) ∈ B and thus ϕ1 + ϕ2 is a mapping A→ B. Then (ϕ1 + ϕ2)(a+ a′) = ϕ1(a+ a′) + ϕ2(a+ a′) = ϕ1(a) + ϕ1(a ′) + ϕ2(a) + ϕ2(a ′) since ϕ1, ϕ2 are homomorphisms. Utilizing the commutativity of B’s operation, we then have ϕ1(a)+ϕ1(a ′)+ϕ2(a)+ϕ2(a ′) = ϕ1(a)+ϕ2(a)+ϕ1(a ′)+ϕ2(a ′) = (ϕ1+ϕ2)(a)+(ϕ1+ϕ2)(a ′) showing that ϕ1 + ϕ2 is a homomorphism, and thus Hom(A,B) is closed under the operation. To ensure this is then an abelian group is claimed, we first note that since we borrow the operation of B, the axiom of associativity is satisfied, and the commutativity of the operation is inherited. It is quick to show that the identity element of Hom(A,B) is the zero homomorphism, as (0 + ϕ)(a) = 0(a) + ϕ(a) = eB + ϕ(a) = ϕ(a) However, the fact that for any ϕ ∈ Hom(A,B), there exists a −ϕ such that (−ϕ)(a) = −(ϕ(a)), and therefore that Hom(A,B) contains inverses, requires some special considera- tion. We have for ϕ ∈ Hom(A,B), −ϕ is a homomorphism, as −ϕ(a+ a′) = −(ϕ(a+ a′)) = −(ϕ(a) + ϕ(a′)) = −(ϕ(a′)) + (−(ϕ(a))) Above we utilize the fact that in general, (ab)−1 = b−1a−1. However, we may once again utilize commutativity of B to rewrite as −(ϕ(a′)) + (−(ϕ(a))) = −(ϕ(a)) + (−(ϕ(a′))) = (−ϕ(a)) + (−ϕ(a′)) giving that −ϕ is a homomorphism from A→ B. Since ϕ+ (−ϕ) = 0, Hom(A,B) satisfies every group axiom, and is an abelian group as desired. Special conditions of Hom(A,B) are of particular importance for later proofs, so we consider a few here. First among these, are groups of the form Hom(Z, K), with K any abelian group. Theorem 3.1.2 ([HW89]). For any abelian group K, we have Hom(Z, K) ∼= K 54 Proof. We define a mapping ψ : Hom(Z, K) → K by ψ(ϕ) = ϕ(1), and show that it is an isomorphism. Note if ϕk is such that ϕk(1) = k, we have (ϕk1 + ϕk2)(1) = ϕk1(1) + ϕk2(1) = k1 + k2 = ϕk1+k2(1). We now show ψ is a homomorphism, as ψ(ϕk1 + ϕk2) = (ϕk1 + ϕk2)(1) = k1 + k2 = ϕk1(1) + ϕk2(1) = ψ(ϕk1) + ψ(ϕk2) For injectivity, since Z cyclic, we have by Lemma 1.6.5 that any homomorphism ϕ : Z→ K is uniquely determined by ϕ(1), since ⟨1⟩ = Z. Thus if ψ(ϕ) = ψ(ϕ′), we have ϕ(1) = ϕ′(1), and thus by Lemma 1.6.5 we have ϕ = ϕ′, giving that ψ is injective. For surjectivity, we require that for any k ∈ K, there exists ϕk ∈ Hom(Z, K) such that ϕk(1) = k. Let k ∈ K. We then let ϕk : Z→ K be such that ϕk(1) = k. We claim this func- tion is a homomorphism. Letting ϕk(1) = k, and therefore ϕ(n) = ϕ(1) + ϕ(1) + · · ·+ ϕ(1)︸ ︷︷ ︸ n times = k + k + · · ·+ k︸ ︷︷ ︸ n times = nk, we have that for n,m ∈ Z: ϕk(n) + ϕk(m) = nk +mk = (n+m)k = ϕk(n+m) Giving that ϕk as defined above is a homomorphism. Thus ψ : Hom(Z, K) → K is injective by lemma, and surjective by above. Thus, Hom(Z, K) ∼= K as desired.1 Following the same idea, we have the theorem Theorem 3.1.3 ([HW89]). Hom(Zn,Zm) ∼= Zgcd(n,m). Proof. Let ϕ ∈ Hom(Zn,Zm). Since Zn is a cyclic group, by Lemma 1.6.5 we may uniquely determine each homomorphism on Zn by where it sends some generator of Zn. Without loss 1An alternate proof of this theorem utilizes the fact that Z is the free abelian group on any singleton set. Theorem 1.6.3 then guarantees for each function from a singleton set to A, we have a unique homomorphism from Z sending 1 to the element which is the image of the function, giving that Hom(Z, A) has an element for each element of A. From there, defining the isomorphism is as above. 55 of generality, choose 1 as the generator in question. Since ϕ is a homomorphism, we must have ϕ(0) = 0. Additionally, we must have ϕ(1) = a for some a ∈ Zm. Then, 1 + 1 + 1 + · · ·+ 1︸ ︷︷ ︸ n times = 0 in Zn =⇒ ϕ(1) + · · ·+ ϕ(1)︸ ︷︷ ︸ n times = nϕ(1) = na = 0 in Zm, so it must be the case that na ∼= 0 mod m. Thus, na = mk for some k ∈ Z. If we let d = gcd(n,m), then n = n1d,m = m1d, and we may simplify further through na = n1da = m1dk = mk =⇒ n1a = m1k Using this simplification, we may solve for a as a = m1k n1 . It is a well known result from number theory that if d = gcd(n,m), then n d , m d are coprime. Thus since amust be an integer, and n1 ̸ |m1, we must have n1|k, so k = n1k ′ for some k′ ∈ Z+. Then a = m1 n1k′ n1 = m1k ′. Consider k′ = 1. Then a = m1, so ϕa(1) = m1. Then for any other choice of k′, we have that a = m1k ′ = m1 +m1 + · · ·+m1︸ ︷︷ ︸ k′ times . Thus, ϕm1k′ = ϕm1 + ϕm1 + · · ·+ ϕm1︸ ︷︷ ︸ k′times , and each ϕm1k′ ∈ Hom(Zn,Zm) is expressible as a sum of ϕm1 . Thus, Hom(Zn,Zm) = ⟨ϕm1⟩. Then since dm1 = m = 0 in Zm, we have ϕdm1 = 0, and thus |ϕm1| = d, so Hom(Zn,Zm) is a cyclic group generated by ϕm1 , whose order is d. As such, Hom(Zn,Zm) ∼= Zd. 3.2 Sums and Products Given that the elements of Hom(A,B) are homomorphisms, it is natural to ask how it interacts with previously discussed universal properties. For the Direct Product (Product) and Direct Sum (Coproduct) we examine these questions, and find the following results: Theorem 3.2.1 ([HW89]). For A an abelian group and {Bi}i∈I a collection of abelian groups, 56 we have ∏ i∈I Hom(A,Bi) ∼= Hom(A, ∏ i∈I Bi) The reader should note that here, this theorem is hopefully intuitive. Interpreted, this theorem essentially tells us that for every collection of homomorphisms A → Bi for each Bi, there is a corresponding homomorphism from A → ∏ i∈I Bi, that is “the same”. This is precisely the universal property of the product object, and is vital for the proof of the theorem. Proof. Let ϕ : Hom(A, ∏ i∈I Bi) → ∏ i∈I Hom(A,Bi) be defined as ϕ(ψ) = (ρ1ψ, ρ2ψ, . . . ) with ρi : ∏ i∈I Bi → Bi the canonical projection. We seek to show it is an isomorphism. We have that ϕ is a homomorphism as ϕ(ψ) + ϕ(ψ′) = (ρ1ψ, . . . ) + (ρ1ψ ′, . . . ) = (ρ1(ψ + ψ′), . . . ) = ϕ(ψ + ψ′) It is injective, as if ϕ(ψ) = (0, 0, . . . ), then ρiψ = 0 for each i. Since none of ρi are 0, ψ must be. Next, it is surjective as for any (f1, f2, . . . ) ∈ ∏ i∈I Hom(A,Bi), we may find ψ ∈ Hom(A, ∏ i∈I Bi) such that ψ = (f1, f2, . . . ) via the universal property of the product, as shown below. {Bi}i∈I ∏ i∈I Bi A ρi fi ψ Thus ϕ is a bijective homomorphism, and is therefore an isomorphism, giving that Hom(A, ∏ i∈I Bi) ∼= ∏ i∈I Hom(A,Bi). Theorem 3.2.2 ([HW89]). For B an abelian group, and {Ai}i∈I a collection of abelian groups, we have Hom(⊕ i∈I Ai, B) ∼= ∏ i∈I Hom(Ai, B) 57 Similar to the previous theorem, this theorem follows from the universal property of the coproduct object. That is, for every collection of homomorphisms Ai → B from each Ai we may find a unique homomorphism from ⊕ i∈I Ai → B that is “the same”. Again, this is vital for the following proof. Proof. Let ϕ : Hom(⊕ i∈I Ai, B) → ∏ i∈I Hom(Ai, B) be defined as ϕ(ψ) = (ψι1, ψι2, . . . ) with ιi : Ai → ⊕ i∈I Ai the canonical injections. We seek to show it is an isomorphism. We have that ϕ is a homomorphism, as ϕ(ψ) + ϕ(ψ′) = (ψι1, . . . ) + (ψ′ι1, . . . ) = ((ψ + ψ′)ι1, . . . ) = ϕ(ψ + ψ′) It is injective, as ϕ(ψ) = (0, 0, . . . ) =⇒ ψιi = 0 for each i, which implies ψ is 0 for all Ai since ιi is never 0. Thus, ψ = 0. Then for surjectivity, we utilize the universal property of the coproduct. Let (f1, f2, . . . ) be an element in ∏ i∈I Hom(Ai, B) consisting of homomorphisms from each Ai → B. Then via the universal property of the coproduct, we may find ψ : ⊕ Ai → B such that ψιi = fi via the diagram below {Ai}i∈I ⊕ i∈I Ai B fi ιi ψ Then since ϕ(ψ) = (ψι1, ψι2, . . . ) = (f1, f2, . . . ), we have that ϕ is a bijective homomor- phism, and therefore an isomorphism. Thereby givingHom(⊕ i∈I Ai, B) ∼= ∏ i∈I Hom(Ai, B). 3.3 Induced Maps An important thing to note about Hom(A,B), is that we may let the groups therein vary. We may for example, consider all groups Hom(A,−), where the second position may be treated as in input for an abelian group. Since for any two abelian groups A,X, there exists 58 the zero homomorphism 0: A→ X, the set of homomorphisms A→ X is always nonempty. Since it is nonempty, by our previous proof, it forms an abelian group Hom(A,X). Letting X, Y be abelian groups, and f : X → Y be a homomorphism between them, we consider Hom(A,X), Hom(A, Y ), and ask the question: Can we form a corresponding homomorphism f∗ : Hom(A,X)→ Hom(A, Y )? That is, assuming we have an element ϕ of Hom(A,X), which is a homomorphism ϕ : A → X. Can we then use f : X → Y to find an element of Hom(A, Y )? The answer is yes. We define f∗ : Hom(A,X) → Hom(A, Y ) as f∗(ϕ) = fϕ. By simply postcomposing ϕ by f , we find a homomorphism fϕ : A→ Y via the diagram X Y A f ϕ fϕ Thus postcomposition with f gives us a method of turning any homomorphism A→ X into a homomorphism A→ Y . Further, since f is a homomorphism, we have f∗(ϕ1 + ϕ2) = f(ϕ1 + ϕ2) = fϕ1 + fϕ2 = f∗(ϕ1) + f∗(ϕ2) giving that f∗ is a homomorphism! We then call f∗ the homomorphism induced by f . We may perform a similar process with Hom(−, B), with an important distinction. Uti- lizing the same f : X → Y , an illustration of the scenario we now have is the following X B Y ϕ f There is now no clear way to, given a homomorphism ϕ : X → B, find a homomorphism Y → B using f . We would hope that, similar to above, composing f with ϕ somehow would give us what we desire. But since ϕ has domain X and codomain B, f has domain X and codomain Y , neither can be composed with the other. 59 Instead, as the diagram may have given away, we begin with a homomorphism ψ : Y → B, and precompose with f to receive ψf : X → B, as illustrated by the diagram. Y X B ψ f ψf Just as before, this method gives us a homomorphism, but rather than a homomorphism Hom(X,B)→ Hom(Y,B), it is a homomorphism f ∗ : Hom(Y,B)→ Hom(X,B) defined as f ∗(ψ) = ψf . To summarize, we have Definition 3.3.1 (The Hom-induced maps f ∗ and f∗ [HW89]). Let X, Y be abelian groups such that there exists a homomorphism f : X → Y . There are then induced homomorphisms f∗ : Hom(A,X)→ Hom(A, Y ) and f ∗ : Hom(Y,B)→ Hom(X,B) defined as f∗(ϕ) = fϕ and f ∗(ψ) = ψf , respectively. 3.4 Viewing Hom(A,B) as a Functor The observant reader may notice that, in fixing one of A or B in Hom(A,B) and allowing the other to vary, we have found a mapping of abelian groups to abelian groups. Simi- larly with the induced homomorphisms, we have found a mapping of homomorphisms to homomorphisms. This is precisely what is necessary to define a functor! In particular, we may define the covariant functor Hom(A,−) : Ab → Ab and the contravariant functor Hom(−, B) : Ab→ Ab. 60 Definition 3.4.1 (The functor Hom(A,−) : Ab→ Ab [HW89]). Define the object map of Hom(A,−) as G 7→ Hom(A,G). Then the morphism map as (ϕ : G→ K) 7→ (ϕ∗ : Hom(A,G)→ Hom(A,K)) defined by ϕ∗(f) = fϕ for f ∈ Hom(A,G). We verify that Hom(A,−) defined this way is indeed a functor by letting H,G,K ∈ Ab, ϕ : G→ K, ψ : K → H, and considering the diagram G K H Hom(A,G) Hom(A,K) Hom(A,H) Hom(A,−) ϕ Hom(A,−) ψ Hom(A,−) ϕ∗ ψ∗ Wemay compose ϕ, ψ as ψϕ, whose image underHom(A,−) is (ψϕ)∗ defined as (ψϕ)∗(f) = (ψϕ)f . Similarly, we may compose ϕ∗, ψ∗ as ψ∗ϕ∗, defined as ψ∗ϕ∗(f) = ψ∗(ϕf) = ψ(ϕf) for f ∈ Hom(A,G). Thus, Hom(A,−)(ψϕ) = Hom(A,−)(ψ)Hom(A,−)(ϕ) since the compo- sition of homomorphisms is associative. Then we need only check that Hom(A,−) preserves identities. As such, let 1G : G→ G. Then (1G)∗(f) = 1Gf = f for f ∈ Hom(A,G), so (1G)∗ is the identity on Hom(A,G). As such, Hom(A,−) is a functor as desired. Definition 3.4.2 (The contravariant functor Hom(−, B) : Ab→ Ab [HW89]). Define the object map as G 7→ Hom(G,B). Then the morphism map as (ϕ : G 7→ K) 7→ (ϕ∗ : Hom(K,B)→ Hom(G,B)) defined as ϕ∗(f) = ϕf for f ∈ Hom(K,B). We verify that Hom(−, B) defined this way is indeed a functor, by letting H,G,K ∈ Ab, ϕ : G→ K, ψ : K → H, and considering the diagram G K H Hom(G,B) Hom(K,B) Hom(H,B) ϕ Hom(−,B) ψ Hom(−,B) Hom(−,B) ϕ∗ ψ∗ 61 Wemay compose ϕ, ψ as ψϕ, whose image underHom(−, B) is (ψϕ)∗ defined as (ψϕ)∗(β) = β(ψϕ) for β ∈ Hom(H,B). Similarly, we may compose ψ∗, ϕ∗ as ϕ∗ψ∗, defined as ϕ∗ψ∗(β) = ϕ∗(βψ) = (βψ)ϕ. Thus, Hom(−, B)(ψϕ) = Hom(−, B)(ϕ)Hom(−, B)(ψ) since the compo- sition of homomorphisms is associative. For the preservation of identity, we have that if 1H : H → H is the identity homomor- phism, then (1H) ∗(β) = β1H = β for β ∈ Hom(H,B), and is therefore the identity on Hom(H,B). Thus, Hom(−, B) is indeed a functor. We may consider these functors at the same time to receive the bifunctorHom(−,−) : Ab× Ab→ Ab, contravariant in the first argument, covariant in the second. It is perhaps more “correct” to do so, but is often more illuminating to fix an argument and examine from there. 3.5 The Exactness of Hom, part I We now examine the exactness of each of the Hom functors, first showing that they are both left exact. Theorem 3.5.1 ([HW89]). Hom(A,−) is a left exact functor. Proof. Let the sequence B′ ϕ ↣B ψ−→ B′′ be left exact. For Hom(A,−) to be left exact, we seek to show that its image under Hom(A,−) Hom(A,B′) ϕ∗ ↣Hom(A,B) ψ∗−→ Hom(A,B′′) is also left exact. To do so, we need only show that ϕ∗ is injective, and that ker(ψ∗) = im(ϕ∗). For injectivity let ϕ∗(α) = 0 for α ∈ Hom(A,B′). Then ϕ(α(a)) = 0 for all a ∈ A. However, since ϕ injective, we therefore have α(a) = 0 for all a ∈ A. As such, α is the zero homomorphism, and ϕ∗ is injective. 62 Then for ker(ψ∗) = im(ϕ∗), we utilize double containment. For im(ϕ∗) ⊂ ker(ψ∗) we consider ψ∗(ϕ∗(α)), and have ψ∗(ϕ∗(α)) = ψϕα. Then since im(ϕ) = ker(ψ), we rewrite as 0α = 0. Thus, im(ϕ∗) ⊂ ker(ψ∗). For ker(ψ∗) ⊂ im(ϕ∗), let β ∈ Hom(A,B) be in ker(ψ∗). Thus ψβ(a) = 0 for all a ∈ A. As such, im(β) ⊂ ker(ψ). Since ker(ψ) = im(ϕ), we then have im(β) ⊂ im(ϕ). Then since ϕ injective, it is an isomorphism between B′, im(ϕ). Thus, ϕ−1 (considered as a homomorphism im(ϕ)→ B′) is an isomorphism as well. Since im(β) ⊂ im(ϕ) we have that ϕ−1β is a homomorphism α : A → B′ such that ϕα = ϕϕ−1β = β. Thus, β = ϕα for some α : A → B′, and β ∈ im(ϕ∗). As such, we have ker(ψ∗) ⊂ im(ϕ∗), completing our double containment. As such, the sequence Hom(A,B′) ϕ∗ ↣Hom(A,B) ψ∗−→ Hom(A,B′′) is left exact. The proof that Hom(−, B) is left exact is more involved, and requires a lemma, which we present here. Lemma 3.5.2. Let ϕ : G→ H be a homomorphism between abelian groups G,H. Then for quotient groups G/K,H/K ′, if ϕ(K) ⊂ K ′, ϕ induces a homomorphism ϕ : G/K → H/K ′ defined by ϕ(g +K) = ϕ(g) +K ′. Proof. We must show that ϕ is well-defined, and a homomorphism. To show ϕ is well- defined, we suppose g1 +K = g2 +K and seek to show that ϕ(g1 +K) = ϕ(g2 +K). Since g1 +K = g2 +K, we have g1 − g2 ∈ K. Then ϕ(g1 − g2) ∈ K ′ by ϕ(K) = K ′, which gives ϕ(g1)− ϕ(g2) ∈ K ′ as well. It is a homomorphism, as ϕ((g1 + g2) +K) = ϕ(g1 + g2) +K ′ = (ϕ(g1) + ϕ(g2)) +K ′ = (ϕ(g1) +K ′) + (ϕ(g2) +K ′) 63 which is equal to ϕ(g1 + K) + ϕ(g2 + K) as desired. Thus, ϕ(g1) + K ′ = ϕ(g2) + K ′ and therefore ϕ(g1 +K) = ϕ(g2 +K) as desired. Theorem 3.5.3 ([HW89]). Hom(−, B) is a left exact functor. Proof. Consider the right exact sequence A′ ϕ−→ A ψ ↠A′′ We seek to show that its image under Hom(−, B) Hom(A′′, B) ψ∗ ↣Hom(A,B) ϕ∗−→ Hom(A′, B) is left exact. Note that since Hom(−, B) is contravariant, and thus a functor Abop → Ab, we must begin with a right exact sequence in order to get a left exact sequence as the image. This is because we desire our sequence to be left exact in Abop, the domain of Hom(−, B). We must show that ψ∗ is injective, and that im(ψ∗) = ker(ϕ∗). For injectivity, let ψ∗(α) = 0 for α ∈ Hom(A′′, B). We then have αψ(a) = 0 for all a ∈ A. Since ψ is surjective, we therefore have α(a′′) = 0 for all a′′ ∈ A′′, and therefore α is the zero homomorphism, giving that ψ∗ is injective. For im(ψ∗) = ker(ϕ∗) we again utilize double containment. For im(ψ∗) ⊂ ker(ϕ∗) we let ψ∗(α) ∈ im(ψ∗). Then ϕ∗(ψ∗(α)) = ψ∗(α)ϕ = α(ψϕ) = α0 = 0 by exactness. Thus, im(ψ∗) ⊂ ker(ϕ∗). The process of showing ker(ϕ∗) ⊂ im(ψ∗) is considerably more difficult. Let β ∈ Hom(A,B) be in ker(ϕ∗). Then βϕ(a′) = 0 for all a′ ∈ A′. This gives that im(ϕ) ⊂ ker(β). Since im(ϕ) = ker(ψ) by exactness, we then have ker(ψ) ⊂ ker(β). Since β : A → B is a homomorphism, by the fundamental homomorphism theorem, we therefore have an induced injective homomorphism β′ : A/ ker(β) → B with the property β = β′πβ where πβ : A→ A/ ker(β) is the canonical projection. 64 Then ψ similarly induces a homomorphism ψ′ : A/ ker(ψ) → A′′ satisfying ψ = ψ′πψ with πψ the canonical projection πψ : A→ A/ ker(ψ). Since ψ is surjective by exactness, so is ψ′. Given that ψ′ injective, it is bijective, therefore an isomorphism, and we may take (ψ′)−1 : A′′ → A/ ker(ψ). Since ker(ψ) ⊂ ker(β), by Lemma 3.5.2 the identity on A induces a homomorphism πψ→β : A/ ker(ψ)→ A/ ker(β) and we may form the composition β′πψ→β(ψ ′)−1 : A′′ → B which is further explained by the diagram A A′′ A/ ker(ψ) B A/ ker(β) ψ πβ πψ β (ψ′)−1ψ′ πψ→β β′ We have β′πψ→β(ψ ′)−1 ∈ Hom(A′′, B), so we may apply ψ∗ to receive ψ∗(β′πψ→β(ψ ′)−1) = β′πψ→β(ψ ′)−1ψ = β′πψ→β(ψ ′)−1ψ′πψ = β′πψ→βπψ Then since πψ→β : A/ ker(ψ) → A/ ker(β), and πψ : A → A/ ker(ψ), we have πψ→βπψ : A → A/ ker(β), which is precisely πβ. Thus, β′πψ→βπψ = β′πβ = β. Therefore, for any β ∈ ker(ϕ∗), there exists β′πψ→β(ψ ′)−1 ∈ Hom(A′′, B) s.t. ψ∗(β′πψ→β(ψ ′)−1) = β and as such ker(ϕ∗) ⊂ im(ψ∗), which gives ker(ϕ∗) = im(ψ∗). Thus, the sequence A′ ϕ−→ A ψ ↠A′′ 65 being right exact, ensures that the induced sequence Hom(A′′, B) ψ∗ ↣Hom(A,B) ϕ∗−→ Hom(A′, B) is left exact, as desired. We now examine Hom’s right exactness, or lack thereof, with the following examples. Example 3.5.4 (Hom(A,−) fails to be right exact.). Consider the right exact sequence in Ab Z µ−→ Z ε ↠Z4 → 0 where µ is multiplication by 4, and ε is the homomorphism Z→ Z4 defined as x 7→ x mod 4. We then let A = Z2, and find the induced sequence Hom(Z2,Z) µ∗−→ Hom(Z2,Z) ε∗−→ Hom(Z2,Z4)→ 0 Since Z contains no elements of finite order, Hom(Z2,Z) ∼= 0. Additionally, we have previ- ously proven that Hom(Zm,Zn) ∼= Zgcd(m,n), so Hom(Z2,Z4) ∼= Z2. We then rewrite this as 0 µ∗−→ 0 ε∗−→ Z2 → 0 From a group theory perspective, this sequence is exact if and only if we have