DOCSLIB.ORG

Diagram Chases: Examples & Exercises

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

DIAGRAM CHASES: EXAMPLES & EXERCISES PETER J. HAINE 1. Examples of Diagram Chases 1.1. Notation. Throughout, we write 푅 for a commutative ring and Mod푅 for the category of 푅-modules. 1.2. Recollection. • Let 푘 be a field. Then Mod푘 is the category Vect푘 of 푘-vector spaces and 푘-linear transformations. • The category Mod퐙 of modules over the ring of integers 퐙 is simply the category Ab of abelian groups. 1.3. Lemma (5-lemma). Consider a commutative diagram 훼1 훼2 훼3 훼4 푀1 푀2 푀3 푀4 푀5 (1.4) 훾1 훾2 훾3 훾4 훾5 ′ ′ ′ ′ ′ 푀1 ′ 푀2 ′ 푀3 ′ 푀4 ′ 푀5 . 훼1 훼2 훼3 훼4 in Mod푅. If the rows are exact, then the following statements hold. (1.3.a) If 훾2 and 훾4 are injective and 훾1 is surjective, then 훾3 is injective. (1.3.b) If 훾2 and 훾4 are surjective and 훾5 is injective, then 훾3 is surjective. Proof. First we prove (1.3.a). So suppose that 훾2 and 훾4 are injective and 훾1 is surjective. Start with 푚3 ∈ 푀3 with the property that 훾3(푚3) = 0. The goal is to show that 푚3 = 0. ′ Then since 훾3(푚3) = 0, we also know that 훼3훾3(푚3) = 0, hence by the commutativity of the diagram (2.5) we have 훾4훼3(푚3) = 0. Since 훾4 is injective, this implies that 훼3(푚3) = 0, hence 푚3 ∈ ker 훼3. Since the top row is exact, ker 훼3 = im 훼2, so there exists an 푚2 ∈ 푀2 such that 훼2(푚2) = 푚3. Then since 훾3(푚3) = 0 we see that 훾3훼2(푚2) = 0. Then by the ′ ′ commutativity of the diagram (1.4) we see that 훼2훾2(푚2) = 0. Therefore 훾2(푚2) ∈ ker 훼2. ′ ′ ′ ′ Since the bottom row is exact, ker(훼2) = im(훼1), so there exists an 푚1 ∈ 푀1 so that ′ ′ ′ 훼1(푚1) = 훾2(푚2). Since 훾1 is surjective, there exists a 푚1 ∈ 푀1 so that 훾1(푚1) = 푚1. Then by the commutativity of the diagram (1.4), we have ′ ′ ′ 훾2훼1(푚1) = 훼1훾1(푚1) = 훼1(푚1) = 훾2(푚2). Thus by the injectivity of 훾2 we have 훼1(푚1) = 푚2. This shows that 훼2훼1(푚1) = 훼2(푚2) = 푚3. Date: January 9, 2018. 1 2 PETER J. HAINE But by the exactness of the top row, we have 훼2훼1(푚1) = 0, so 푚3 = 0, as desired. Hence ker 훾3 = 0, so 훾3 is injective. A summary of this chase is displayed below. 푚1 푚2 푚3 훼3(푚3) ′ 푚1 훾2(푚2) 0 0 . † Now we prove (1.3.b). So suppose that 훾2 and 훾4 are surjective and 훾5 is injective. Start ′ ′ ′ by considering an element 푚3 ∈ 푀3: the goal is to show that 푚3 lies in the image of 훾3. ′ ′ ′ Map 푚3 to 훼3(푚3). Then by the surjectivity of 훾4 there exists an element 푚4 ∈ 푀4 so that ′ ′ 훾4(푚4) = 훼3(푚3). Now consider 훼4(푚4). Notice that by the commutativity of the diagram (1.4), we have ′ ′ ′ 훾5훼4(푚4) = 훼4훾4(푚4) = 훼4훼3(푚3). ′ ′ Then by the exactness of the bottom row, 훼4훼3 = 0, so 훾5훼4(푚4) = 0. Therefore, by the injectivity of 훾5 we have 훼4(푚4) = 0, so 푚4 ∈ ker(훼4). By the exactness of the top row, ker(훼4) = im(훼3), so there exists 푚3 ∈ 푀3 so that 훼3(푚3) = 푚4. ′ Now consider 푚3 − 훾3(푚3). Notice that by the commutativity of the diagram (1.4), ′ ′ ′ ′ ′ ′ ′ 훼3(푚3 − 훾3(푚3)) = 훼3(푚3) − 훼3훾3(푚3) = 훼3(푚3) − 훾4훼3(푚3) ′ ′ ′ ′ ′ ′ = 훼3(푚3) − 훾4(푚4) = 훼3(푚3) − 훼3(푚3) = 0. ′ ′ ′ This shows that 푚3 − 훾3(푚3) is in the kernel of 훼3, which is equal to im 훼2 by the exactness ′ ′ ′ ′ of the lower sequence. Hence there exists an 푚2 so that 훼2(푚2) = 푚3 − 훾3(푚3). Then by the ′ surjectivity of 훾2 there exists an 푚2 ∈ 푀2 so that 훾2(푚2) = 푚2. Now consider 푚3 + 훼2(푚2) and notice that by the commutativity of the diagram (1.4), we have ′ 훾3(푚3 + 훼2(푚2)) = 훾3(푚3) + 훾3훼2(푚2) = 훾3(푚3) + 훼2훾2(푚3) ′ ′ = 훾3(푚3) + (푚3 − 훾3(푚3)) = 푚3. □ This shows that 훾3 is surjective, as desired. 1.5. Remark. The standard application of the 5-lemma in homological algebra is the fol- lowing: in the diagram (1.4) the homomorphisms 훾1, 훾2, 훾4, and 훾5 are isomorphisms, from which we conclude that 훾3 is an isomorphism. Now we prepare to prove the Snake lemma (Proposition 1.14) with a few easy lemmas. First we recall a few definitions. 1.6. Recollection. Let 푅 be a commutative ring and 푓∶ 퐴 → 퐵 a morphism of 푅-modules. Recall that the kernel of 푓 is an 푅-module ker(푓) equipped with a morphism a morphism 푖∶ ker(푓) → 퐴 satisfying the following universal property: the composite 푓푖 is the zero mor- phism and for any morphism 푔∶ 퐾 → 퐴 such that 푓푔 = 0, there exists a unique morphism ̄푔∶ 퐾 → ker(푓) making the triangle 퐾 푔 ̄푔 ker(푓) 퐴 푖 commute. † Actually this follows immediately by duality since Mod푅 is an abelian category and could have been proven in the setting of abelian categories — maybe we’ll talk about these at some point. DIAGRAM CHASES: EXAMPLES & EXERCISES 3 The kernel may explicitly be realized as the submodule of 퐴 defined by ker(푓) ≔ { 푎 ∈ 퐴 | 푓(푎) = 0 } , so that the morphism 푖∶ ker(푓) → 퐴 in the universal property is simply the inclusion ker(푓) ↪ 퐴. 1.7. Remark. This latter description is the description we encounter in a first course in algebra. 1.8. Recollection. Recall that the cokernel of a morphism of 푅-modules 푓∶ 퐴 → 퐵 is an 푅- module coker(푓) equipped with a morphism a morphism 푞∶ 퐵 → coker(푓) satisfying the following universal property: the composite 푞푓 is the zero morphism and for any morphism ℎ∶ 퐵 → 퐶 such that ℎ푓 = 0, there exists a unique morphism ℎ∶̄ coker(푓) → 퐶 making the triangle 퐵 ℎ 푞 coker(푓) 퐴 ℎ̄ commute. The cokernel may explicitly be realized as the quotient of 퐵 by the image of 푓: coker(푓) ≔ 퐵/ im(푓) , so that the morphism 푞∶ 퐵 → coker(푓) in the universal property is simply the quotient map 퐵 ↠ coker(푓). 1.9. Remark. The cokernel is often not introduced in a first course in algebra, which iswhy we introduce it here. The “explicit” description of the cokernel is what is often introduced in commutative algebra courses, but it makes the duality between the kernel and cokernel less obvious than the universal property description. 1.10. Lemma. Let 푅 be a commutative ring and 푓 퐴 퐵 (1.11) 훼 훽 퐴′ 퐵′ 푓′ a commutative square in Mod푅. (1.10.a) The restriction 푓|ker(훼) ∶ ker(훼) → ker(훽) is the unique 푅-module homomorphism making the square 푓| ker(훼) ker(훼) ker(훽) 푖훼 푖훽 퐴 퐵 푓 commute, where 푖훼 and 푖훽 denote the inclusions. Moreover, if 푓 is an injection, then so is 푓|ker(훼). 4 PETER J. HAINE (1.10.b) Dually, there exists a unique 푅-module homomorphism 푓∶̄ coker(훼) → coker(훽) making the square 푓′ 퐴′ 퐵′ 푞훼 푞훽 coker(훼) coker(훽) 푓′̄ commute, where 푞훼 and 푞훽 denote the quotient maps. Moreover, if 푓 is a surjection, then so is 푓′̄ . Proof. Since (1.10.b) is dual to (1.10.a), it suffices to prove (1.10.a). This follows easily from the universal property of the kernel. Consider the composite 훽푓푖훼. By the commutativity of the square (1.11) and the fact that 푖훼 is the inclusion of the kernel of 훼 we have that ′ 훽푓푖훼 = 푓 훼푖훼 = 0 . Thus by the universal property of ker(훽), there exists a unique 푅-module homomorphism ker(훼) → ker(훽) making the square ker(훼) ker(훽) 푖훼 푖훽 퐴 퐵 푓 commute. The fact that the unique morphism ker(훼) → ker(훽) is the restriction 푓|ker(훼) follows immediately from the definition of the restriction. The last thing to check is that if 푓 is an injection, then so is 푓|ker(훼). To see this, notice that if 푓 is an injection, then since 푖훼 is an inclusion, 푓푖훼 is also an injection, so the composite □ 푖훽 ∘ 푓|ker(훼) is an injection, which implies that 푓|ker(훼) is an injection. 1.12. Lemma. Let 푅 be a commutative ring and consider the following diagram of 푅-modules 푓 푔 퐴 퐵 퐶 (1.13) 훼 훽 훾 퐴′ 퐵′ 퐶′ . 푓′ 푔′ (1.12.a) If the top row of (1.13) is exact and 푓′ is an injection, then the sequence 푓| 푔| ker(훼) ker(훼) ker(훽) ker(훽) ker(훾) is exact. (1.12.b) Dually, if the bottom row of (1.13) is exact and 푔 is a surjection, then the sequence 푓′̄ ′̄푔 coker(훼) coker(훽) coker(훾) is exact. Proof. As usual, since (1.12.b) is dual to (1.12.a), it suffices to prove (1.12.a). The only place where we need to show exactness is at ker(훽). First let us show that im(푓|ker(훼)) ⊂ DIAGRAM CHASES: EXAMPLES & EXERCISES 5 ker(푔|ker(훽)). So suppose that 푏 ∈ ker(훽) and 푏 = 푓(푎) for some 푎 ∈ ker(훼). Then since the top row of (1.13) we see that 푔|ker(훽)(푏) = 푔(푏) = 푔푓(푎) = 0 . Hence 푏 ∈ ker(푔|ker(훽)), so we see that im(푓|ker(훼)) ⊂ ker(푔|ker(훽)). Now let us show that ker(푔|ker(훽)) ⊂ im(푓|ker(훼)).
Recommended publications
  • Homological Algebra

    Homological Algebra

    Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of
  • Diagram Chasing in Abelian Categories

    Diagram Chasing in Abelian Categories

    Diagram Chasing in Abelian Categories Daniel Murfet October 5, 2006 In applications of the theory of homological algebra, results such as the Five Lemma are crucial. For abelian groups this result is proved by diagram chasing, a procedure not immediately available in a general abelian category. However, we can still prove the desired results by embedding our abelian category in the category of abelian groups. All of this material is taken from Mitchell’s book on category theory [Mit65]. Contents 1 Introduction 1 1.1 Desired results ...................................... 1 2 Walks in Abelian Categories 3 2.1 Diagram chasing ..................................... 6 1 Introduction For our conventions regarding categories the reader is directed to our Abelian Categories (AC) notes. In particular recall that an embedding is a faithful functor which takes distinct objects to distinct objects. Theorem 1. Any small abelian category A has an exact embedding into the category of abelian groups. Proof. See [Mit65] Chapter 4, Theorem 2.6. Lemma 2. Let A be an abelian category and S ⊆ A a nonempty set of objects. There is a full small abelian subcategory B of A containing S. Proof. See [Mit65] Chapter 4, Lemma 2.7. Combining results II 6.7 and II 7.1 of [Mit65] we have Lemma 3. Let A be an abelian category, T : A −→ Ab an exact embedding. Then T preserves and reflects monomorphisms, epimorphisms, commutative diagrams, limits and colimits of finite diagrams, and exact sequences. 1.1 Desired results In the category of abelian groups, diagram chasing arguments are usually used either to establish a property (such as surjectivity) of a certain morphism, or to construct a new morphism between known objects.
  • Derived Functors and Homological Dimension (Pdf)

    Derived Functors and Homological Dimension (Pdf)

    DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic.
  • UCLA Electronic Theses and Dissertations

    UCLA Electronic Theses and Dissertations

    UCLA UCLA Electronic Theses and Dissertations Title Character formulas for 2-Lie algebras Permalink https://escholarship.org/uc/item/9362r8h5 Author Denomme, Robert Arthur Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Los Angeles Character Formulas for 2-Lie Algebras A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Robert Arthur Denomme 2015 c Copyright by Robert Arthur Denomme 2015 Abstract of the Dissertation Character Formulas for 2-Lie Algebras by Robert Arthur Denomme Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Rapha¨elRouquier, Chair Part I of this thesis lays the foundations of categorical Demazure operators following the work of Anthony Joseph. In Joseph's work, the Demazure character formula is given a categorification by idempotent functors that also satisfy the braid relations. This thesis defines 2-functors on a category of modules over a half 2-Lie algebra and shows that they indeed categorify Joseph's functors. These categorical Demazure operators are shown to also be idempotent and are conjectured to satisfy the braid relations as well as give a further categorification of the Demazure character formula. Part II of this thesis gives a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. The definition of a graded algebra is given whose cate- gory of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite- dimensional modules over an associated degenerate affine Hecke algebra.
  • Lecture 15. De Rham Cohomology

    Lecture 15. De Rham Cohomology

    Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial.
  • On the Cohomology of the Finite Special Linear Groups, I

    On the Cohomology of the Finite Special Linear Groups, I

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 216-238 (1978) On the Cohomology of the Finite Special Linear Groups, I GREGORY W. BELL* Fort Lewis College, Durango, Colorado 81301 Communicated by Graham Higman Received April 1, 1977 Let K be a finite field. We are interested in determining the cohomology groups of degree 0, 1, and 2 of various KSL,+,(K) modules. Our work is directed to the goal of determining the second degree cohomology of S&+,(K) acting on the exterior powers of its standard I + l-dimensional module. However, our solution of this problem will involve the study of many cohomology groups of degree 0 and 1 as well. These groups are of independent interest and they are useful in many other cohomological calculations involving SL,+#) and other Chevalley groups [ 11. Various aspects of this problem or similar problems have been considered by many authors, including [l, 5, 7-12, 14-19, 211. There have been many techni- ques used in studing this problem. Some are ad hoc and rely upon particular information concerning the groups in question. Others are more general, but still place restrictions, such as restrictions on the characteristic of the field, the order of the field, or the order of the Galois group of the field. Our approach is general and shows how all of these problems may be considered in a unified. way. In fact, the same method may be applied quite generally to other Chevalley groups acting on certain of their modules [l].
  • The Left and Right Homotopy Relations We Recall That a Coproduct of Two

    The Left and Right Homotopy Relations We Recall That a Coproduct of Two

    The left and right homotopy relations We recall that a coproduct of two objects A and B in a category C is an object A q B together with two maps in1 : A → A q B and in2 : B → A q B such that, for every pair of maps f : A → C and g : B → C, there exists a unique map f + g : A q B → C 0 such that f = (f + g) ◦ in1 and g = (f + g) ◦ in2. If both A q B and A q B 0 0 0 are coproducts of A and B, then the maps in1 + in2 : A q B → A q B and 0 in1 + in2 : A q B → A q B are isomorphisms and each others inverses. The map ∇ = id + id: A q A → A is called the fold map. Dually, a product of two objects A and B in a category C is an object A × B together with two maps pr1 : A × B → A and pr2 : A × B → B such that, for every pair of maps f : C → A and g : C → B, there exists a unique map (f, g): C → A × B 0 such that f = pr1 ◦(f, g) and g = pr2 ◦(f, g). If both A × B and A × B 0 are products of A and B, then the maps (pr1, pr2): A × B → A × B and 0 0 0 (pr1, pr2): A × B → A × B are isomorphisms and each others inverses. The map ∆ = (id, id): A → A × A is called the diagonal map. Definition Let C be a model category, and let f : A → B and g : A → B be two maps.
  • Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1

    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1

    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0.
  • Math 231Br: Algebraic Topology

    Math 231Br: Algebraic Topology

    algebraic topology Lectures delivered by Michael Hopkins Notes by Eva Belmont and Akhil Mathew Spring 2011, Harvard fLast updated August 14, 2012g Contents Lecture 1 January 24, 2010 x1 Introduction 6 x2 Homotopy groups. 6 Lecture 2 1/26 x1 Introduction 8 x2 Relative homotopy groups 9 x3 Relative homotopy groups as absolute homotopy groups 10 Lecture 3 1/28 x1 Fibrations 12 x2 Long exact sequence 13 x3 Replacing maps 14 Lecture 4 1/31 x1 Motivation 15 x2 Exact couples 16 x3 Important examples 17 x4 Spectral sequences 19 1 Lecture 5 February 2, 2010 x1 Serre spectral sequence, special case 21 Lecture 6 2/4 x1 More structure in the spectral sequence 23 x2 The cohomology ring of ΩSn+1 24 x3 Complex projective space 25 Lecture 7 January 7, 2010 x1 Application I: Long exact sequence in H∗ through a range for a fibration 27 x2 Application II: Hurewicz Theorem 28 Lecture 8 2/9 x1 The relative Hurewicz theorem 30 x2 Moore and Eilenberg-Maclane spaces 31 x3 Postnikov towers 33 Lecture 9 February 11, 2010 x1 Eilenberg-Maclane Spaces 34 Lecture 10 2/14 x1 Local systems 39 x2 Homology in local systems 41 Lecture 11 February 16, 2010 x1 Applications of the Serre spectral sequence 45 Lecture 12 2/18 x1 Serre classes 50 Lecture 13 February 23, 2011 Lecture 14 2/25 Lecture 15 February 28, 2011 Lecture 16 3/2/2011 Lecture 17 March 4, 2011 Lecture 18 3/7 x1 Localization 74 x2 The homotopy category 75 x3 Morphisms between model categories 77 Lecture 19 March 11, 2011 x1 Model Category on Simplicial Sets 79 Lecture 20 3/11 x1 The Yoneda embedding 81 x2 82
  • The Freyd-Mitchell Embedding Theorem States the Existence of a Ring R and an Exact Full Embedding a Ñ R-Mod, R-Mod Being the Category of Left Modules Over R

    The Freyd-Mitchell Embedding Theorem States the Existence of a Ring R and an Exact Full Embedding a Ñ R-Mod, R-Mod Being the Category of Left Modules Over R

    The Freyd-Mitchell Embedding Theorem Arnold Tan Junhan Michaelmas 2018 Mini Projects: Homological Algebra arXiv:1901.08591v1 [math.CT] 23 Jan 2019 University of Oxford MFoCS Homological Algebra Contents 1 Abstract 1 2 Basics on abelian categories 1 3 Additives and representables 6 4 A special case of Freyd-Mitchell 10 5 Functor categories 12 6 Injective Envelopes 14 7 The Embedding Theorem 18 1 Abstract Given a small abelian category A, the Freyd-Mitchell embedding theorem states the existence of a ring R and an exact full embedding A Ñ R-Mod, R-Mod being the category of left modules over R. This theorem is useful as it allows one to prove general results about abelian categories within the context of R-modules. The goal of this report is to flesh out the proof of the embedding theorem. We shall follow closely the material and approach presented in Freyd (1964). This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects. This approach is summarised as follows: • the functor category rA, Abs is abelian and has injective envelopes. • in fact, the same holds for the full subcategory LpAq of left-exact functors. • LpAqop has some nice properties: it is cocomplete and has a projective generator. • such a category embeds into R-Mod for some ring R. • in turn, A embeds into such a category. 2 Basics on abelian categories Fix some category C. Let us say that a monic A Ñ B is contained in another monic A1 Ñ B if there is a map A Ñ A1 making the diagram A B commute.
  • Introduction to Homology

    Introduction to Homology

    Introduction to Homology Matthew Lerner-Brecher and Koh Yamakawa March 28, 2019 Contents 1 Homology 1 1.1 Simplices: a Review . .2 1.2 ∆ Simplices: not a Review . .2 1.3 Boundary Operator . .3 1.4 Simplicial Homology: DEF not a Review . .4 1.5 Singular Homology . .5 2 Higher Homotopy Groups and Hurweicz Theorem 5 3 Exact Sequences 5 3.1 Key Definitions . .5 3.2 Recreating Groups From Exact Sequences . .6 4 Long Exact Homology Sequences 7 4.1 Exact Sequences of Chain Complexes . .7 4.2 Relative Homology Groups . .8 4.3 The Excision Theorems . .8 4.4 Mayer-Vietoris Sequence . .9 4.5 Application . .9 1 Homology What is Homology? To put it simply, we use Homology to count the number of n dimensional holes in a topological space! In general, our approach will be to add a structure on a space or object ( and thus a topology ) and figure out what subsets of the space are cycles, then sort through those subsets that are holes. Of course, as many properties we care about in topology, this property is invariant under homotopy equivalence. This is the slightly weaker than homeomorphism which we before said gave us the same fundamental group. 1 Figure 1: Hatcher p.100 Just for reference to you, I will simply define the nth Homology of a topological space X. Hn(X) = ker @n=Im@n−1 which, as we have said before, is the group of n-holes. 1.1 Simplices: a Review k+1 Just for your sake, we review what standard K simplices are, as embedded inside ( or living in ) R ( n ) k X X ∆ = [v0; : : : ; vk] = xivi such that xk = 1 i=0 For example, the 0 simplex is a point, the 1 simplex is a line, the 2 simplex is a triangle, the 3 simplex is a tetrahedron.
  • Classifying Categories the Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories

    Classifying Categories the Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories

    U.U.D.M. Project Report 2018:5 Classifying Categories The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories Daniel Ahlsén Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2018 Department of Mathematics Uppsala University Classifying Categories The Jordan-Holder¨ and Krull-Schmidt-Remak theorems for abelian categories Daniel Ahlsen´ Uppsala University June 2018 Abstract The Jordan-Holder¨ and Krull-Schmidt-Remak theorems classify finite groups, either as direct sums of indecomposables or by composition series. This thesis defines abelian categories and extends the aforementioned theorems to this context. 1 Contents 1 Introduction3 2 Preliminaries5 2.1 Basic Category Theory . .5 2.2 Subobjects and Quotients . .9 3 Abelian Categories 13 3.1 Additive Categories . 13 3.2 Abelian Categories . 20 4 Structure Theory of Abelian Categories 32 4.1 Exact Sequences . 32 4.2 The Subobject Lattice . 41 5 Classification Theorems 54 5.1 The Jordan-Holder¨ 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 of their research into algebraic topology. One of their aims was to give an axiomatic account of relationships between collections of mathematical structures. This led to the definition of categories, functors and natural transformations, the concepts that unify all category theory, Categories soon found use in module theory, group theory and many other disciplines. Nowadays, categories are used in most of mathematics, and has even been proposed as an alternative to axiomatic set theory as a foundation of mathematics.[Law66] Due to their general nature, little can be said of an arbitrary category.