U.U.D.M. Project Report 2018:5 Classiying Categories The Jordan-Hölder and Krull-Schmidt-Remak Theorems or belian Categories Daniel hlsén Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2018 Department o Mathematics Uppsala University
Classiying Categories The Jordan-Hölder and Krull-Schmidt-Remak theorems or abelian categories Daniel hlsén Uppsala University June 2018 bstract The Jordan-Hölder and Krull-Schmidt-Remak theorems classiy inite groups, either as direct sums o indecomposables or by composition series. This thesis deines abelian categories and extends the aorementioned theorems to this context. 1
Contents 1 Introduction 3 2 Preliminaries 5 2.1 Basic Category Theory............................ 5 2.2 Subobjects and Quotients........................... 9 3 belian Categories 13 3.1 dditive Categories.............................. 13 3.2 belian Categories.............................. 20 4 Structure Theory o belian Categories 32 4.1 Exact Sequences................................ 32 4.2 The Subobject Lattice............................. 41 5 Classiication Theorems 54 5.1 The Jordan-Hölder Theorem......................... 54 5.2 The Krull-Schmidt-Remak Theorem.................... 60 2
1 Introduction Category theory was developed by Eilenberg and Mac Lane in the 1942-1945, as a part o their research into algebraic topology. One o their aims was to give an axiomatic account o relationships between collections o mathematical structures. This led to the deinition o categories, unctors and natural transormations, the concepts that uniy all category theory, Categories soon ound use in module theory, group theory and many other disciplines. Nowadays, categories are used in most o mathematics, and has even been proposed as an alternative to axiomatic set theory as a oundation o mathematics.[law66] Due to their general nature, little can be said o an arbitrary category. Instead, mathematical theory must ocus on a speciic type o category, the choice o which is largely dependent on ones interests. In this work, the categories o choice are abelian categories. These categories were independently developed by Buchsbaum[Buc55] and Grothendieck[Gro57]. Grothendieck s work was especially groundbreaking, as he uniied the cohomology theories or groups and or sheaves, which had similar properies but lacked a ormal connection. This showed that abelian categories was the basis o general ramework or cohomology theories, a powerul incentive or research. belian categories are highly structured, possessing both a matrix calculus and various generalizations o the isomorphism theorems. This gives rise to a reined structure theory, which is the topic o this thesis. O special interest here is the structure o subobjects to an object in an abelian category, since this structure contains a lot o inormation about the objects themselves. The ultimate aim o a structure theory is to provide theorems that classiy some collection o objects up to isomorphism. Here, two results pertaining to such theorems are presented. The irst is the Jordan-Hölder theorem, which classiies objects by maximal chains o subobjects. The second is the Krull-Schmidt-Remak theorem, which gives a classiication o objects by linearly independent components. These theorems do not provide a universal classiication theorem or all abelian categories. The problem with the Jordan-Hölder theorem is that not all objects in an abelian category has a maximal chain o subobjects, while the problem or the Krull- Schmidt-Remak theorem is that is requires that the endomorphisms o certain objects are o a particular orm, which is not true or alla objects in an abelian category. 3
Fortunately, one can show that every object that can be classiied using Jordan-Hölder can also be classiied using Krull-Schmidt-Remak. The extent to which Krull-Schmidt- Remak can be extended is not discussed urther. The thesis is divided into our chapters, each divided into two sections. The irst chapter covers the basics o category theory and deines subobjects and quotients in general categories. The aim is to set up the the coming chapters, and ix terminology etc. The second chapter deines additive categories, and gives an account o the matrix calculus it contains. Then, abelian categories are deined and some undamental properties are proven, so as to set up the third chapter, which urther develops the theory. In the third chapter, the ocus is on developing the theory o exact sequences, an important tool in the study o abelian categories, and to urther deepen our understanding the subobject structure o abelian categories. In the ourth and inal chapter, the theory is used to prove the Jordan-Hölder and Krull-Schmidt-Remak theorems. 4
2 Preliminaries Categories is a general ramework or studying mathematical structures and how they relate to one another. 2.1 Basic Category Theory This section covers the basics o category theory, in order to ix terminology and notation. Proos and detailed examples are omitted. The interested reader should consider the introductory chapter in Leinster s book Basic Category Theory[Lei14]. Deinition 2.1. category C consists o a class o objects and a class o morphisms Hom C (,B) or every object and B in C, subject to the ollowing constraints. (i) For each in Hom C (,B) and g in Hom C (B,C) there is a morphism g in Hom C (,C), called the composition o and g. (ii) For all in Hom C (,B), g in Hom C (B,C) and h in Hom C (C,D), we have h (g ) = (h g). In other words, the composition is associative. (iii) For all objects in C, there is an morphism id in Hom C (,), called the identity on, such that id = and id g = g or all morphisms and g. The composition g is written as g most o the time, and one usually writes Hom instead o Hom C. morphism in Hom(,) is called an endomorphism, and the collection o endomorphisms on is denoted End(). Composition turns End() into a monoid, with the identity as unit object. I is a morphism in End(), the morphism n is the endomorphism on deined by By convention, 0 = id.. } {{ } n times 5
Deinition 2.2. Let be a morphism in Hom(,B) in some category. Then is called the domain o, while B is called the codomain o, written : B. Example 2.3. (i) Set is the category o all sets and unctions under composition. (ii) Vect K is the category o all vector spaces and linear transormations (over a ield K) under composition. (iii) R-Mod is the category o all R-modules and module morphisms (over a ring R) under composition. (iv) Grp is the category o all groups and group morphisms under composition. The deinition o category does not assume that the collections o objects and morphisms are sets. In some circumstances this can be problematic. For details, consider [ML98] or [Lei14]. From old categories, new ones arise. Deinition 2.4. Let C and D be categories. The product category C D o C and D is the category such that (i) the objects o C D are the pairs (,B) with rom C and B rom D. (ii) the morphisms rom (,B) to (,B ) are pairs o morphisms (,g) rom C and D, with : and g : B B. (iii) the identity morphisms id (,B) are (id,id B ). (iv) the composition o morphisms (,g) and (,g ) is (,g g). Deinition 2.5. Let C be a category. The opposite category C op o C is the category such that (i) the objects o C op are the objects in C. (ii) or every morphism : B, there is a morphism op : B. (iii) the identity morphisms id are id. (iv) the composition o morphisms op : B and g op : B C in C op is the morphism ( g) op : C. The opposite category is also called the dual category. For any property o morphisms and objects in a category, there is a corresponding dual property in the dual category where the morphisms are reversed. So, i a property holds in a category, then the dual property holds in the dual category. Since any category is the dual category o its dual category, this means that i a property holds or all categories, the dual property holds or all categories. In particular, or every theorem, there is a dual theorem that holds in the dual categories (see [ML98, p.33-35] or a more complete discussion o this). 6
Some caution is needed. I a property holds or a category, the dual property holds or the dual category. This does not mean that dual property hold or the original category! Since there is no guarantee that categories and dual categories have similar properties, this limits the scope o the duality principle. In an arbitrary category, morphisms are not unctions. Thus, there is no concept o surjective, injective or bijective morphisms. Instead, one uses a dierent terminology. Deinition 2.6. morphism is (i) an epimorphism i g = h implies g = h, or all morphisms g,h. Then is called epic. (ii) a monomorphism i g = h implies g = h, or all morphisms g,h. Then is called monic. (iii) an isomorphism i there is a morphism g such that g = id B and g = id. Then, and B are isomorphic, denoted B. n isomorphism rom an object to itsel is called an automorphism. Note that epimorphisms and monomorphisms are dual concepts: a monomorphism in a category is an epimorphism in the dual category and vice versa. Isomorphisms are dual to themselves: isomorphisms are isomorphisms in the dual category as well. Remark. Not all epimorphisms are surjective, nor are all monomorphisms injective. lso, bijective morphisms and isomorphisms do not coincide in general. See [IB68, p.3-7] and [Lei14, p.12] or details. The ollowing acts will be used liberally throughout the thesis. Proposition 2.7. Let : B and g : B C be morphisms in a category. (i) I and g are epic, so is g. (ii) I g is epic, so is g. (iii) I and g are monic, so is g. (iv) I g is monic, so is. (v) I and g are isomorphisms, so i g. (vi) ll isomorphisms are epic and monic. Remark. The converse o (vi) is not true: there are morphisms that are not isomorphisms, yet still epic and monic.[lei14, p.12] Maps between categories that preserve composition are called unctors. Deinition 2.8. Let C and D be categories. covariant unctor F : C D is an assignment o objects in C to D and morphisms in C to morphisms in D such that 7
(i) or all and B in C and : B, we have F( ) : F() F(B). (ii) or all in C, we have F(id ) = id F(). (iii) or all : B and g : B C, we have F(g ) = F(g) F( ). contravariant unctor rom C to D is a covariant unctor rom C op to D. Deinition 2.9. biunctor on a category C is a unctor rom C C to C. Functors are assignments between categories, and can be composed pointwise on objects and morphisms. This composition has an identity and is associative. Hence, the collection o categories and unctors behaves like a category. Functors are maps between categories - natural transormations are maps between unctors. Deinition 2.10. Let C and D be categories and F and G unctors rom C to D. natural transormation rom F to G assigns a morphism η : F() G() in D or all in C, such a that i is a morphism in C between and B, the diagram F() η G() F( ) G( ) F(B) η B G(B) commutes, i.e η B F( ) = G( )η. I η is an isomorphism or every, then η is a natural isomorphism, denoted F G. Just as with unctors, natural transormations can be composed pointwise. Once again, the composition o two natural transormations is a natural transormation and the composition is associative. Two objects in a category are isomorphic to each other i there are invertible morphisms between them. There is a natural analogue to this condition or categories. Deinition 2.11. Two categories C and D are isomorphic i there are unctors F : C D and G : D C such that FG = id D and GF = id C. Two categories are isomorphic i and only i they are isomorphic as objects in the category o categories. However, isomorphic categories rarely occur in practice. Instead, a weaker notion is used. Deinition 2.12. Two categories C and D are equivalent i there are unctors F : C D and G : D C such that FG id D and GF id C. 8
Deinition 2.13. unctor F rom a category C to a category D is aithul i the induced map rom Hom(,B) to Hom(F(),F(B)) deined by mapping to F( ) is injective. ull i the induced map rom Hom(,B) to Hom(F(),F(B)) deined by mapping to F( ) is surjective. dense i all objects in D is isomorphic to F() or some in C. Proposition 2.14. unctor is an equivalence i and only i it is aithul, ull and dense. 2.2 Subobjects and Quotients This thesis is concerned with classiies objects in a category using subobjects. But how can one speak o subobjects without sets? The idea to deine a subobject o an object as an equivalence class o morphisms. Let Mono() denote the class o monomorphisms with codomain. Mono(), the domain o i is denoted i. I i lies in Deinition 2.15. Let be an object in a category, and suppose that i and j are morphisms in Mono(). Then i contains j via a morphism, denoted j i, i there is a morphism such that the diagram i i j j commutes, i.e i i = j. I i and j contain each other, they are equivalent, denoted i j. Otherwise, the containment is proper, denoted j < i. Proposition 2.16. Suppose that i and j are monomorphisms in Mono() and that i contains j via. Then is a monomorphism, and i satisy i = j, then =. Proo. I i = j = i, cancel i on both sides and obtain =. That is monic ollows rom (iv) in Proposition 2.7. Proposition 2.17. Let be an object in a category and suppose that i and j are morphisms in Mono(). Then i and j are equivalent i and only there is an isomorphism such that i contains j via. 9
Proo. I there is such an isomorphism, then i = j and i = j 1, so i and j are equivalent. I i and j are equivalent, there are morphisms 1 and 2 such that i = j 2 and j = i 1. Thus, i = i 1 2 id i = 1 2 j = j 2 1 id i = 2 1, so 1 and 2 are isomorphisms. preorder is a transitive and relexive relation on a class o objects. partial order is preorder which is antisymmetric. ny preorder induces an equivalence relation on its underlying set via x y i and only i x y and y x. The set o equivalence classes o is partially ordered by comparing representatives using. Proposition 2.18. The relation is a preorder on Mono() or every object Proo. To show that any object is contained in itsel, take to be the identity. For transitivity, suppose that i, j and k in Mono() satisy i j and j k. By assumption, there are morphisms and g be such that k = j and j = ig. Then g satisies ig = j = k, and so i is contained in k. By deinition, is the equivalence relation induced by the preorder. Deinition 2.19. subobject o an object is an equivalence class o Mono() under the relation. The class o subobjects o an object is denoted S. By previous remarks, S is partially ordered by. The dual to a subobject is a quotient. s with subobjects, quotients are deined as equivalence classes o morphisms. I is an object, let Epi() denote the class o epimorphisms out o. Deinition 2.20. Let be an object in a category, and suppose that p and q are epimorphisms in Epi(). Then p contains q, denoted q p, i there is a morphism such that the diagram q p commutes, i.e i p = q. I p and q contain each other, they are equivalent, denoted p q, otherwise it is proper, denoted q < p p q 10
Quotients are entirely dual to subobjects, so the ollowing proos are omitted. Proposition 2.21. Suppose that p and q are epimorphisms in Epi() and that i contains j via. Then is an epimorphism, and i satisy p = q, then =. Proposition 2.22. Let be an object in a category and suppose that p and q are morphisms in Epi(). Then p and q are equivalent i and only there is an isomorphism such that p contains q via. Proposition 2.23. Let be an object in a category. Then is a preorder on Epi(). Deinition 2.24. quotient object o an object is an equivalence class o Epi() under the relation. The class o quotients o is denoted Q. Remark. Every object has at least one subobject and quotient, represented by the identity morphism. This subobject is identiied with itsel, so that one may speak o as a subobject and quotient o itsel. s a subobject, it contains every subobject and as a quotient object it is contained in every quotient object. In other words, is a lowest upper bound in S and a greatest lower bound in Q. Example 2.25. Every subgroup o the abelian group Z is o the orm nz = { 2n, n,0,n,2n, } or some unique natural number n. The inclusions j n : nz Z are deined by j n (x) = x. Each subobject o Z is an equivalence class o monomorphisms into Z. Each class contains precisely one o the morphisms j n. Moreover, j m contains j n i and only i there is a morphism : nz mz so that j m = j n, i.e m (x) = nx or all integers x in Z. This happens only when m divides n, in which case is deined via (x) = (n/m)x. Hence, j n j m m n and S Z is isomorphic as a partial order to Z under the reversed divisibility order. What about quotients? Let p : Z G be a surjective group homomorphism. Then G = im(p) Z/ ker(p). The kernel o p is a subgroup o Z, and so there is a natural number n such that ker(p) nz. Consequently, G is isomorphic to C n = Z/nZ, the cyclic group on n elements. Thus every quotient o Z is represented by a unique epimorphism p n : Z C n, deined by p n (x) = x + nz Suppose that the quotient object p n : Z C n contains another quotient p m : Z C m. Then there is an epimorphism : C n C m, so that p n = p m, i.e (p n (x)) = p m (x) (x + nz) = x + mz. 11
This holds i and only i m divides n, and so p m p n m n and Q Z is isomorphic as a partial order to Z under the divisibility order. Notice that S Z and Q Z are order isomorphic up to reversal o the order. This is not coincidental: it is a special case o Proposition 3.27. 12
3 belian Categories belian categories are additive categories with additional structure. 3.1 dditive Categories dditive categories can be seen as the most general type o category that retains a kind o matrix calculus. Deinition 3.1. n object in a category C is (i) initial i or every object B in C there is exactly one morphism rom to B. (ii) terminal i or every object B in C there is exactly one morphism rom B to. (iii) null i it is both initial and terminal. Proposition 3.2. Initial, terminal and null objects are unique up to a unique isomorpism. Proo. By deinition, the only endomorphism on an initial object is the identity morphism. Let I and J be initial objects. Then, there are unique morphisms : I J and g : J I, and g = id J and g = id I, so I and J are isomorphic. The proos or terminal and null objects are dual. Example 3.3. (i) In Set, the empty set is an initial object and singleton set is a terminal object. There is no null object. (ii) In Grp, the trivial group is a null object. Similarly, the zero module is a null object in R-Mod. Deinition 3.4. Let 0 be a null object in a category and and B be objects in the same category. The null morphism between and B is the unique morphism given by the composition o the morphisms 0 and 0 B. Example 3.5. In Grp, the null morphism between two groups G and H is the morphism rom G to H deined by mapping every element in G to 1 H. 13
Deinition 3.6. category is preadditive i it has a null object and every set o morphisms between two objects orm an abelian group, such that composition is biadditive. That is, (g + h) = g + h and ( + g)h = h + gh or all morphisms. In a preadditive category, the set o endomorphisms on an object is a ring, with morphism addition as addition and composition as the ring multiplication. The endomorphism ring is a Z-bimodule, via n times (i n is negative.) n = n = + + n times (i n is positive.) 0 (i n is 0.) Proposition 3.7. In preadditive categories, the ollowing are equivalent or an object : (i) is initial. (ii) is terminal. (iii) id is the additive identity in the endomorphism ring. (iv) The endomorphism ring is trivial. Proo. See [ML98, p.194]. When there is a null object in a category, the null morphism 0 : B and the additive identity in Hom(,B) coincide, since 0 is the composition o the additive identity in Hom(,0) and Hom(0,). Deinition 3.8. direct sum o objects 1,..., n in a preadditive category is an object S along with morphisms k i k S p l l such that and n i k p k = id S. k=1 id p l i k = δ lk = k i l = k 0 otherwise. The morphisms i k are called injection morphisms, while the morphisms p l are called projection morphisms. The objects 1,..., n are called direct summands o S, and the 14
collection o objects 1,..., n, S along with the morphisms is called a direct sum system. direct sum is trivial i every summand is isomorphic to either or the zero object, and nontrivial otherwise. Every direct summand is a subobject o the direct sum, and a proper one i and only i the direct sum is nontrivial. Not all subobjects are direct summands. For example, Z cannot be written as a non-trivial direct sum, but has a lot o subobjects. Direct sums are sel-dual, since every direct sum system gives rise to a direct sum system in the dual category, by switching projection and injection morphisms. Proposition 3.9. ny direct sum in a preadditive category is unique up to isomorphism. Proo. Let S and S be direct sums o 1,..., n, and p k, i k, p k and i k the corresponding injection and projection morphisms. Deine rom S to S by and g rom S to S by Then and g = g = n i j p j j=1 n i j p j j=1 k=1 = g = n i k p k k=1 n i k p k. k=1 n n n n i k p k = i j p ji k p k = i k p k = id S n i k p k = k=1 Hence, S and S are isomorphic. j=1 k=1 n j=1 k=1 n i j p j i k p k = k=1 n i k p k = id S. The above proposition allows us talk about the direct sum o 1,..., n, denoted j. Note that the isomorphism between two direct sums is not unique. k=1 Deinition 3.10. category is additive i it is preadditive and every set o objects has a direct sum. Example 3.11. I R is a ring, the category R-Mod is additive. The null object is the zero module, and direct sum is cartesian product. Direct sums extends to morphisms. 15
Proposition 3.12. Let 1,..., n and 1,..., n be objects in an additive category with corresponding projection and injection morphisms p k, i k, p k and i k. Suppose j is a morphism rom j to j, or every j = 1,...,n. Then there is a unique morphism, denoted j, rom j to j, such that the diagram j j j p k k k k p k commutes or every k. Proo. Let Then p k j = n i j jp j. j=1 ) n ( j = p k i j jp j = j=1 n p k i j jp j = k p k or all k. To prove uniqueness, suppose that p j g = jp j = p j g or all j. Then or all j. Summing over j gives n n n i j p j g = i j p j j=1 j=1 j=1 i j p j i j p j g = i j p j n g = j=1 j=1 i j p j id S g = id S g =. Remark. There is an dual deinition o j, where one replaces the projection morphisms with the injection morphisms in the opposite direction, resulting in the diagrams j i k j i k j k k k and equations ( j ) ik = k i k. 16
The process o constructing morphisms between direct sums rom morphisms between the summands can be inverted. Deinition 3.13. Suppose 1,..., n and 1,..., m are objects in an additive category, and that is a morphism rom j to k. The component jk o is the morphism rom j to k deined by jk = p k i j. The matrix o is the matrix 11 1m [ ] =..... n1 nm. Example 3.14. Let be the direct sum o objects 1,..., n in an additive category. Then id 1 0 0 p 1 id i 1 p n id i 1 [id ] =..... = 0 id 2 0... p 1 id i n p n id i n. 0.. 0 0 0 id n Similarly, the matrices o the injection and projection morphisms are [i j ] = [ 0 0 id j 0 0 ] and [p k ] = [ 0 0 id k 0 0 ] T. Matrices o morphisms can be seen as elements o the abelian group n j=1 k=1 m Hom( j, k ) with addition deined componentwise. The identity o this group given by the matrix whose entries are all zero morphism. Proposition 3.15. Let 1,..., n and 1,..., m be objects in an additive category. Then as abelian groups. Hom ( ) n j, k j=1 k=1 m Hom( j, k ). 17
Proo. Deine ϕ : Hom ( ) n j, k j=1 k=1 by mapping the morphism : j k to its matrix 11 1m ϕ( ) = [ ] =..... n1 nm. m Hom( j, k ). I = 0, then jk = 0 or all j and k, and hence ϕ preserves the zero matrix. Moreover, i and g are morphisms rom j to k, then ( + g) jk = p k ( + g)i j = p k i j + p k gi j = jk + g jk, so ϕ is a group morphism. Next, deine the map by ψ : n m j=1 k=1 Hom( j, k ) Hom( ) j, k g 11 g 1m n m ψ..... = i j g jkp k. g n1 g nm, j=1 k=1 Let be a morphism rom j to k. Then 11 1m p 1 i 1 p m i 1 ψ(ϕ( )) = ψ..... = ψ..... n1 nm, p 1 i = n p m i n, n m m n = i j p j i kp k = i j p j i k p k =. j=1 k=1 Similarly, one can show that ψ(ϕ([ jk ])) = [ jk ] or all matrices, which establishes that ϕ is an isomorphism. The above proposition shows that every morphism in an additive can be viewed as a matrix. It turns out that composition can be transered as well. Proposition 3.16. Let 1,..., n, 1,..., m, and 1,..., p be objects in an additive category, with morphisms : j j=1 k=1 k and g : k l. 18
Then, the matrix o the composition is given by the matrix where or all i and j. g : j l h 11 h 1p [g ] =..... h n1 h np h ij = m g ki jk Proo. By deinition, ik = p k i i and g kj = p j gi k. Thus, (g ) ij = p j g i i = ( m m m p j g) i k p k ( i i) = p j gi k p k i i = g kj ik. k=1 k=1 k=1 k=1 Not only is there a matrix calculus in additive categories, the direct sum is also unctorial. Proposition 3.17. Let j, k and l be three n-tuples o objects in an additive category, and j : j j and j : j j two n-tuples o morphisms. Then ( ) ( ) ( j j = j ) j and j=1 id j = id j. Proo. Straightorward calculation gives ( ) ( ) n n j j = i j j p j i k kp k = and = n j=1 k=1 i j ( j j)p j = id j = n ( j j). n i j p j = id j. j=1 n j=1 k=1 i j j p j i k kp k = 19
The above result shows that in an additive category C, are unctors or every n, deined by and n : C n C n ( 1,..., n ) = n ( 1,..., n ) = n i=1 i n i. Since the composition o two unctors is a unctor, iteration yields a myriad o unctors that purports to be the direct sum o n. Even i one restricts onesel to iteration o the biunctor 2, the number o dierent direct sum unctors o n variables is (2n)! (n + 1)!n!, each corresponding to a unique bracketing o n variables. Can any sense be made o this? The answer is yes - one can show that the direct sum is a monoidal product on C. Such categories are subject to a coherence theorem, which essentially states that it does not matter how one places the brackets in a direct sum. detailed treatment o these issues is beyond the scope o this thesis, and the reader is reerred to [ML98, p.161-170]. From now on, all direct sums o the same objects and morphisms are treated as equal, and it is assumed that no problems can arise due to bracketing o direct summands. i=1 3.2 belian Categories Every morphism between two modules can be described uniquely by its kernel and image. It is desirable to ind a similar decomposition or additive categories. For this to work, the concept o kernel and image must be redeined using morphisms. It turns out that, even with a proper account o these concepts, a morphism in additive category does not necessarily have a kernel or an image. dditive categories that do are called abelian categories. Deinition 3.18. Let C be a category with a null object and null morphism 0. kernel o an morphism : B is a morphism k : K such that k = 0, and or every 20
h : C such that hk = 0, there is a unique h : C K such that h = kh. K k=0 h k h B. C h=0 Example 3.19. Suppose that : B is a morphism o abelian group, and let k : K be the inclusion o the preimage o identity. Clearly, k = 0. I k : K satisies k = 0, then the image o k is contained (as a set) in the image o k. Since inclusions are injective, each g in the image o k has a unique preimage k 1 (g), such that k(k 1 (g)) = g. Deine h : K K by Then i.e kh = k. Thus k is the kernel o. h(x) = k 1 (k (x)). (kh)(x) = k(k 1 (k (x))) = k (x), Deinition 3.20. Let C be a category with a null object and null morphism 0. cokernel o an morphism : B is an object C and morphism c : C C such that c = 0, and or every h : B D such that ch = 0, there is a unique h : C D such that h = h c. c =0 C B c h h =0 h D Example 3.21. Let : V W be a linear transormation, and let c : V W / im( ) be deined by c(v) = v + im( ). Then i.e the composition c is 0. c (v) = c( (v)) = (v) + im( ) = im( ) = 0, Moreover, i c rom B to C satisies c = 0, deine h : W / im( ) C by h(v + im( )) = c (v). 21
To show that this is well deined, suppose that v and w lie in the same equivalence class o the quotient W / im( ). Then there is some u in V such that (u) = v w, and hence c (v) c (w) = c (v w) = c ( (u)) = 0 since c = 0 by assumption. Clearly c = hc, and so c is the cokernel o. Remark. Kernels and cokernels are duals: the kernel o a morphism is the cokernel o the dual morphism and vice versa. Proposition 3.22. Kernels are monic and cokernels are epic. Proo. Let k be a kernel o a morphism and suppose that g = kg 1 = kg 2. By deinition g = 0, and the diagram. K k=0 g 2 g 1 k g B C g=0 commutes. The uniqueness condition guarantees that g 1 = g 2 and so k is monic. The proo that cokernels are epic is dual. Proposition 3.23. Kernels and cokernels are unique up to a unique isomorphism. Proo. Let k and k be kernels o. Then k and k are both 0, and so there are morphisms h and h such that kh = k and k h = k. Consequently, k = kh = k h h and k = k h = khh and since k is monic one can cancel on both sides and ind that h and h are isomorphisms. The proo or cokernels is dual. Since kernels and cokernels are unique, one speaks o the kernel and cokernel o a morphism : B, denoted ker( ) and cok( ) respectively. One way o thinking about the kernel o is as the largest subobject o the domain that is mapped to zero by. Dually, one can think o the cokernel as the smallest quotient that maps to zero. Deinition 3.24. n additive category is abelian i (i) every morphism in the category has a kernel and cokernel. (ii) every monomorphism is a kernel and every epimorphism is a cokernel. 22
Since direct sums are sel-dual, and kernels and monomorphisms are dual to cokernels and epimorphisms, respectively, the dual o an abelian category is also abelian. Hence, every theorem or general abelian categories has a dual theorem, obtained by reversing the morphisms and substituting monic or epic and kernel or cokernel, and vice versa. Proposition 3.25. Let be a morphism in an abelian category. Then (i) ker( ) = 0 i and only i is monic. (ii) i g is a monomorphism, then ker(g ) = ker( ). (iii) cok( ) = 0 i and only i is epic. (iv) i is epic and g is a morphism, then cok(g ) = cok(g). Proo. (i) Suppose satisy ker( ) = 0 and that two morphisms g and h satisy g = h. Let l = g h. Then l = (g h) = g h = 0. Thus, there is a morphism h such that g h = h 0 = 0. Thus g = h, so is monic. Conversely, suppose that is monic. Let k satisy k = 0 = 0. Since is monic, cancellation yields k = 0, so ker( ) = 0. (ii) Suppose that g is monic. Then g k = 0 g k = g0 k = 0. or all morphisms k, so ker(g ) = ker( ). The proos or (iii) and (iv) are dual. In abelian categories, kernel and cokernels induces maps between the class o subobjects and the class o quotients o an object. Proposition 3.26. Let be an object in an abelian category. (i) I i and j in Mono() are equivalent, so are cok(i) and cok(j). (ii) I p and q in Epi() are equivalent, so are ker(p) and ker(q). Proo. I i and j are equivalent there is an isomorphism such that i = j, and hence cok(i) = cok(j ) = cok(j) by Proposition 2.7. The second point is done similarly. Deine ker : Q S and cok : S Q, by mapping p and i to ker(p) and cok(i) respectively. 23
Proposition 3.27. The unctions ker and cok are mutually inverse and order reversing. Proo. Let i and j be subobjects o, such that i j via a morphism. Let c 1 be the cokernel o i and c 2 be the cokernel o j. Then c 2 i = c 2 j = 0, so there is a morphism g such that gc 2 = c 1. In other words, cok(j) cok(i). That ker is order reversing is proved similarly. Let i be a monomorphism in Mono(). Then it is the kernel o some map. Let c be the cokernel o i and k the kernel o c. By assumption i = 0, and hence there is a map g such that gc =. lso, ci = 0, so there is a map h 1 such that kh 1 = i. Finally, so there is a map h 2 such that ih 2 = k. k = gck = 0, i c C h 2 h 1 g K Thus i and k are equivalent and represent the same subobject, and thus k B i = k = ker(c) = ker(cok(i)) as subobjects. The other direction is proved is similarly. The above proposition generalize Example 2.25 - the subobject and quotient structure o an objects are mirror images o each other. Proposition 3.28. morphism in an abelian category is an isomorphism i and only i it is monic and epic. Proo. Let be a morphism that is both monic and epic. Since is monic, the kernel o is zero. Hence, a cokernel o ker( ) is the identity. However, Proposition 3.27 assures us that is a cokernel o ker( ). Hence, there is a morphism g such that g is the identity. The exact same reasoning gives that the kernel o the cokernel o is the identity, and that there exists a morphism h such that h is the identity. So is both right and let invertible, and hence an isomorphism. The other direction is (vi) in Proposition 2.7. 24
Cokernels can be extended to subobjects. Proposition 3.29. Let i, i, j and j be monomorphisms, such that i i and j j via isomorphisms ϕ i and ϕ j, respectively. Suppose that the subobject represented by i and i is contained in the subobject represented by j and j via monomorphisms and. Then the codomains o cok( ) and cok( ) are isomorphic. Proo. Consider the diagram i ϕ i i i i j j j ϕ j j. The assumptions that i i and j j imply that ϕ i and ϕ j are isomorphisms and the upper and lower triangle commute. The assumption that i is contained in j means that and are monomorphisms and that the let and right triangle commute. Hence, and i = i ϕ i = j ϕ i i = j = j ϕ j. Equating these expressions and cancelling j yields ϕ i = ϕ j. Let c and c be the cokernels o and respectively. Then and c ϕ j = c ϕ i = 0ϕ i = 0 cϕj 1 = c ϕi 1 = 0ϕi 1 = 0. Hence there are morphisms h and h so that the diagram i j c C ϕ i ϕ j h i j c C h commutes. Thus, h c ϕ j = c h hc = c hcϕj 1 = c hh c = c h h = id C hh = id C. 25
Deinition 3.30. Let i and j be two subobjects with domains i and j, such that i is contained in j via a morphism. The quotient o j by i, denoted j / i, is the codomain o the cokernel o. In abelian categories, all morphisms can be decomposed into monomorphisms and epimorphisms. Proposition 3.31. Let be a morphism in an abelian category. Then = me or an epimorphism e and monomorphism m. Moreover, m is the kernel o the cokernel o and e is the cokernel o the kernel o. K k e D m B c C Proo. Let be a morphism in an abelian category. Let c be the cokernel o and let m to be the kernel o c. By deinition, c = 0, and since m is the kernel o c there is a morphism e such that = me. By Proposition 3.22, e is epic and m is monic. Moreover, since m is monic. e = cok(ker(e)) = cok(ker(me)) = cok(ker( )) The canonical decomposition transers to morphisms. Proposition 3.32. Consider the commutative square B g h B in an abelian category, and let = me and = m e be a canonical decomposition. Then there is a unique ϕ such that diagram commutes. g e D m ϕ B e D m B h 26
Proo. Let,, g and h be given as above. By Proposition 3.31, there are decompositions = me and = m e. Let u be the kernel o. By deinition h u = 0, and thus m e gu = 0. Since m is monic, e gu = 0, and since e is the cokernel o u, there is a unique morphism ϕ such that e g = ϕe. K u e g D ϕ m B e D m B. h Moreover, and since e is epic, m ϕ = hm. m ϕe = m e g = hme, Proposition 3.33. The canonical decomposition o a morphism in an abelian category is unique up to a unique isomorphism. Proo. pply Proposition 3.32 to the square to ind the isomorphism ϕ. id e D m ϕ B e D m B Deinition 3.34. Let be a morphism in an abelian category, and = me its canonical decomposition. The image o, denoted im( ) is the monomorphism m. The coimage o, denoted coim( ), is the epimorphism e. Since the coimage and image o a morphism are unique up to a unique isomorphism, they deine a quotient and a subobject o and B respectively. Deinition 3.35. span into an object in a category is a pair o morphisms with common codomain. cospan rom an object B is a pair o morphisms with common domain B. id B 27
Deinition 3.36. Let and g be a span into. pullback o and g is a cospan and g such that g = g, with the property that i and g is a cospan such that g = g, then there is a unique morphism h such that = h and g = g h. g D h D Deinition 3.37. Let h and k be a cospan rom. pushout o h and k is a span h and k such that k h = h k, with the property that i h and k is a span that satisy hk = kh, then there is a unique morphism p such that ph = h and pk = k. h B k D g k B C p g D k C h h Pullbacks and pushout are dual to each other, as are span and cospans. Proposition 3.38. Pullbacks and pushouts are unique up to a unique isomorphism. Proo. Suppose that there are two pullbacks o the same span. Then there are unique maps h and h such that the diagram P commutes. Hence = hh and g = ghh, and the diagram P h h h h g g P P g g B C B g C g 28
commutes. I one replaces h h by id P, the diagram still commutes and thus the uniqueness criterion implies that h h = id P. Similarly, hh = id P, and so h is an isomorphism. Pullbacks and pushouts can be composed. Proposition 3.39. Suppose that the diagram C g E u B u D g F. u commute. Then (i) i the two inner squares are pullback/pushouts, then so is the outer square. (ii) i the inner let-hand square is a pushout, the outer square is a pushout i and only i the inner right-hand square is a pushout. (iii) i the inner right-hand square is a pullback, the outer square is a pullback i and only i the inner let-hand square is a pullback. Proo. (i) Suppose that the two inner square are pushouts. Suppose that h and h satisies h u = hg. Since the let-hand square is a pushout, there is a unique morphism ϕ so that ϕ = h and ϕu = hg. Since the right-hand square is a pushout as well, there is a unique ψ so that ψg = ϕ and ψu = h. But then ψg = ϕ = h. Since ψ is unique, this mean that the outer square is a pushout. The proo or pullbacks is dual. (ii) Suppose that the inner let-hand and the outer squares are pushouts. Let h and h be such that h u = hg. Then hg = h u = h u and since the outer square is a pushout, there is a unique ϕ so that ϕg = h and ϕu = h. Then ϕg u = h u = hg and ϕu g = hg = ϕg u = h u. 29
So there is a pushout C hg u B u D h ϕg ϕg P and uniqueness implies that ϕg = h, so the right-hand square is a pushout. The other implication is proved in (i). (iii) Dual to (ii). Proposition 3.40. In an abelian category, every span have a pullback and every cospan have a pushout. Proo. Let and g be a span with domains B and C and common codomain. Consider the direct sum system B i p B C and let h = p gq. Let k be the kernel o h. Since k is the kernel o h, 0 = hk = ( p gq)k = pk gqk, i.e pk = gqk. Moreover, i g and are such that g = g, let h = ig j. Then hh = ( p gq)(ig j ) = g g = 0, and since k is the kernel o h there is a unique map h such that kh = ig j. q j C g K h K k p B i B C h j C q g This yields pkh = g and qkh = g, which shows that pk and qk is the pullback o and g. 30
For the pushout, suppose that and g has common domain and codomains B and C respectively. Let h = i jg and c be the cokernel o h. One can show, using a similar argument as above, that ci and cj is the pushout o and g. 31
4 Structure Theory o belian Categories The topic o chapter is the structure theory o abelian categories, a preparation or the proos o the Jordan-Hölder and Krull-Schmidt-Remak theorems. 4.1 Exact Sequences Exact sequences are the bread and butter o abelian categories. Deinition 4.1. sequence o morphisms 2 d 2 1 d 1 0 d 0 1 d 1 in an abelian category is exact at n i im(d n ) = ker(d n 1 ). sequence is exact i it is exact at every object in the sequence. Many properties o morphisms are characterized via exact sequences. Proposition 4.2. Let 0 B g C 0 be a sequence o morphisms. Then (i) 0 B is exact i and only i is monic. (ii) 0 B C is exact i and only i is the kernel o g. (iii) B 0 is exact i and only i is epic. (iv) B C 0 is exact i and only i g is the cokernel o. (v) 0 B 0 is exact i and only i is an isomorphism. (vi) 0 B C 0 is exact i and only i is the kernel o g and g is the cokernel o. 32
Observe that i a sequence o morphisms is exact, the dual o that sequence is also exact. Deinition 4.3. n exact sequence o the orm 0 0 B g C 0 0 is called a short exact sequence. The simplest examples o short exact sequences are o the orm 0 i B p B 0 where p and i are the projection and injection maps. They are characterized thus. Proposition 4.4 (Splitting lemma). For all exact sequences 0 B g C 0 the ollowing statements are equivalent. (i) The middle object B is a direct sum o and C, such that is an injection morphism and g a projection morphism. (ii) There is a morphism l (called a let split) rom B to such that l = id. (iii) There is a morphism r (called a right split) rom C to B such that gr = id C Proo. (i) In a direct sum system, injection morphisms and projection morphism is are right/let splits respectively. (ii) Suppose that an exact sequence 0 B g C 0 has a right split r such that gr = id C. Let p = id B rg. By deinition, gp = g grg = g g = 0. Since is the kernel o g, there is a morphism l, such that l = p, i.e l = id B rg. Thus l +rg = id B. Since gr = id C by assumption, it suices to prove that l = id and lr = 0 to show that B is the direct sum o and C. Yet and since is monic, l = id. lso, l = (id B rg) = rg = 0 =, lr = (id B rg)r = r rgr = r r = 0, and since is monic lr = 0. This shows that l, r, and g orm a direct sum system or C. 33
(iii) Dual to (ii). Remark. The splitting lemma implies that i g is such that h = g is an automorphism, then is a direct summand o. For i c is the cokernel o, the sequence 0 c B 0 is exact, and h 1 g, is a let split o, i.e B. Pullbacks and pushouts in abelian categories can be described in terms o exact sequences. Proposition 4.5. Consider the diagram C g B g D. and the direct sum system B i p B C q j C. Let t = j + ig and s = p gq. Then (i) the square commutes i and only i st = 0. (ii) the square is a pullback i and only i 0 t B C s D is exact, i.e t is the kernel o s. (iii) the square is a pushout i and only i t B C s D 0 is exact, i.e s is the cokernel o t. Proo. Note that pt = g and qt =, and si = and sj = g. 34
(i) The square is commutative i and only i 0 = g g = sipt + sjqt = s(ip + jq)t = st. (ii) ssume that the square is a pullback. Suppose that k is such that sk = 0. Then 0 = sk = s(ip + jq)k = sipk + sjqk = pk gqk, i.e pk = gqk. Since the square is a pullback, there is a unique morphism h such that h = qk and g h = pk. Hence qth = qk and pth = pk, and so th = (jq + ip)th = jqth + ipth = jqk + ipk = (jq + ip)k = k. This shows that t is a kernel o s, so the sequence is exact. Conversely, suppose that t is the kernel o s and that there are morphism and g such that g = g. Deine r = ig + j. Then pr = g, qr =, and sr = ( p gq)(ig + j ) = pig gqj = g g = 0. Since t is the kernel o s, there is a unique m : U such that tm = r, so ptm = pr and qtm = qr, i.e g m = g and m =. This shows that the square is a pullback. (iii) Dual to the proo above. Proposition 4.6. Consider the pullback P g B C. g in an abelian category. I is monic, so is, and i is epic, so is. Proo. Suppose that is monic. Let h and h be morphisms rom P to P, such that h = h. Then g h = g h, and since the diagram commutes g h = g h. Since is monic, g h = g h. Since the diagram is a pullback, the uniqueness property guarantees that h = h. Suppose that is epic and consider the direct sum system B i p B C q j C. 35
In the proo o Proposition 3.40, it was shown that i k is the kernel o h = p gq, then = qk and g = pk. Suppose that uh = 0 or some morphism u. Then 0 = uh = uhi = u( p gq)i = u pi = u and since is epic, u = 0. Thus, h is epic as well. Thus, the sequence 0 P k B C h 0 is exact, i.e h is the cokernel o k. Suppose that u = 0 or some morphism u. Then 0 = u = uqk and hence there is morphism u such that uq = u h. Thus Since is epic, u is 0, and 0 = uqi = u hi = u ( p gq)i = u pi = u. uq = u h = 0. Since q is epic u = 0, which shows that is epic. The nine lemma is a generalization o the isomorphism theorems. The ollowing proo is due to Popescu [Pop73] and [Fre64]. Proposition 4.7. Consider the commutative diagram g C h 0 B g D h E such that the bottom row is exact. The square is a pullback i and only i the sequence is exact, i.e g is the kernel o h. 0 g C h E Proo. Suppose that the square is a pullback. Since the diagram is commutative and the bottom row is exact, h g = hg = 0. Let s be a morphism such that h s = 0. Since the bottom row is exact, g is the kernel o h. Since h s = 0 by assumption, there is a unique morphism t so that s = g t. 36
Moreover, since the square is a pullback, there is a unique r so that gr = s. Thus, g is the kernel o h. Conversely, suppose that g is the kernel o h. Let s and t be morphisms such that s = g t. Since the diagram is commutative, h s = hg t = 0, and since g is the kernel o h, there is a unique morphism r so that s = gr. U s t r g C 0 B g h D h E The diagram is commutative, so g r = gr = s = g t and since g is monic, cancellation yields t = r. This shows that the square is a pullback. Proposition 4.8. Consider the commutative diagram g C p E 0 0 B g D p s F 0 such that the right square is a pullback and the rows are exact. Then s is monic. I is epic then s is an isomorphism. Proo. Let r be such that sr = 0. Let u and v be the pullback o p and r. Since the right square is a pullback and the bottom row is exact, Proposition 4.7 implies that g is the kernel o p. Moreover, p v = spv = sru = 0. 37
Hence, there is a map t so that gt = v. U u K t g C v p E r 0 0 B g D p s F 0 Thus, ru = pv = pgt = 0 since p is the cokernel o g. Moreover, the morphism p is epic, and thus u is epic, so r = 0, which show that s is monic. I is epic, the composition p = sp is epic, and so s is epic. Since s is always monic, s is an isomorphism. Proposition 4.9. Consider the commutative diagram 0 0 0 0 h D k G 0 B h E k H g g 0 C h F 0 with exact columns and exact middle row. Then the upper row is exact i and only i the bottom row is exact (i.e h is monic). Proo. Suppose that the upper row is exact. The right column is exact, the diagram commutes and is monic, so h = ker(k) = ker( k) = ker( k ). 38
Hence, Proposition 4.7 implies that the square h D B h E is a pullback. Thus, the diagram 0 B g C 0 h 0 D E h g F h has exact rows, with the right square being a pullback diagram. Thus h is monic and the bottom row is exact. Conversely, suppose that the bottom row is exact, i.e h is monic. Then, so the sequence = ker(g) = ker(h g) = ker(g h ), 0 B g h F is exact. Thus, the top right square is a pullback. Let r be a morphism such that kr = 0. Then k r = kr = 0, and since the middle row is exact, h is the kernel o k, and and there is a unique morphism t such that h t = t. Since the top right square is a pullback, there is a unique morphism s so that hs = r, which show that h is the kernel o k and the top row is exact. 39
Proposition 4.10 (Nine lemma). Consider the commutative diagram 0 0 0 0 D 0 B E 0 C F G H I 0 0 0 0 0 0 with exact columns and exact middle row. Then the top row is exact i and only i the bottom row is exact. Proo. Direct application o Proposition 4.9 and its dual yields the conclusion. The strength o the nine lemma is evident in ease o which it proves the second isomorphism theorem. Proposition 4.11 (Second isomorphism theorem). Suppose that i and j represent subobjects o, so that i contains j via a morphism : j i. Then exists there a commutative diagram 0 i i c / i 0 u 0 i / j k u / j p u (/ j )/( i / j ) 0 such that the rows are exact, u is an isomorphism, and u and u are the cokernels o and j respectively. Moreover, the morphisms k and p are unique with this property. Proo. Let u = cok( ), u = cok(j) and c = cok(i). By assumption, j = i. Hence u i = uj = 0, and since u is the cokernel o, there is a unique morphism k so that u i = ku. Let p be the cokernel o k. Then u i = ku implies that pu i = pku = 0, 40
and hence there is a unique morphism u that satisies u p = pu. Thus, the diagram 0 0 0 0 j i u i / j 0 id 0 j j i k u / j 0 0 0 / i c u (/ j )/( i / j ) p 0 0 0 is commutative. By assumption, the columns and the middle and upper rows are exact. Hence the lowest is exact as well, and u is an isomorphism. 4.2 The Subobject Lattice Subobjects and quotients were deined or general categories in Section 2.2. It is time to return to topic in the case o abelian categories. But irst, some order theory is required. Until urther notice, reers to an arbitrary partial order on some underlying set. Deinition 4.12. Let x and y be objects in a partial order. greatest lower bound o x and y is an element z such that z x and z y, with the property that i w satisy w x and w y, then w z. Dually, a lowest upper bound o x and y is an element z so that x z and y z, with the property that i w satisy x w and y w, then z w. Deinition 4.13. lattice is a partial order in which every pair o elements have a greatest lower bound and lowest upper bound. The greatest lower bound and lowest upper bound o two elements x and y are unique i they exist, and are denoted x y and x y, respectively. The symbols and are known as meet and join, respectively. Example 4.14. The power set o a set is a lattice, where the meet is intersection and the join is union. 41
Lattices have been studied extensively by Birkho in [Bir67]. In the introduction, proos can be ound o the ollowing three propositions. Proposition 4.15. In any lattice, the meet and join satisies (i) x x = x and x x = x. (ii) x y = y x and x y = y x. (iii) x (y z) = (x y) z and x (y z) = (x y) z. (iv) x (x y) = x (x y) = x or all lattice elements x,y and z. Moreover, x y is equivalent to each o the conditions x y = x and x y = y. Proposition 4.16. For all elements x,y and z in a lattice, i y z, then x y x z and x y x z. Proposition 4.17 (Modular inequality). For all elements x,y and z in a lattice, i x z, then x (y z) (x y) z. Remark. In a lattice, the meet and the join can be seen as binary operations. Indeed, lattices can be characterized as a set with two binary operations that satisy (i)-(iv) o Proposition 4.15.[Bir67, p.10] Deinition 4.18. lattice is modular i x z implies that or all lattice elements x, y and z. x (y z) = (x y) z. The term modular lattice comes rom module theory: the set o submodules o a module is a modular lattice. Deinition 4.19. n element in a lattice is called a top i x or every element x in the lattice. n element is a bot i x or every x in the lattice. lattice with a top and bot is called bounded. Example 4.20. The lattice o subsets o a set S has both a top and bot, given by = S and =. The top and bot are unique i they exist. Every element x in such lattices satisy x = x, x =, x = x and x =. Deinition 4.21. complement o an element x in a bounded lattice is an element c such that x c = and x c =. 42