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Derived Categories with Applications to Representations of Algebras

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Derived Categories with Applications to Representations of Algebras DERIVED CATEGORIES WITH APPLICATIONS TO REPRESENTATIONSOFALGEBRAS JUN-ICHI MIYACHI Contents 1. Categories and Functors 1 2. Additive Categories and Abelian Categories 5 3. Krull-Schmidt Categories 11 4. Triangulated Categories 14 5. Frobenius Categories 19 6. Homotopy Categories 24 7. Quotient Categories 29 8. Quotient Categories of Triangulated Categories 33 9. Epaisse´ Subcategories 35 10. Derived Categories 40 11. Homotopy Limits 46 12. Derived Functors 52 12.1. Derived Functors 52 12.2. Way-out Functors 55 13. Double Complexes 56 14. Derived Functors of Bi-∂-functors 62 15. Bimodule Complexes 70 16. Tilting Complexes 74 17. Two-sided Tilting Complexes 78 17.1. The Case of Flat k-algebras 78 17.2. The Case of Projective k-algebras 81 17.3. The Case of Finite Dimensional k-algebras 84 18. Cotilting Bimodule Complexes 87 References 90 Index 91 1. Categories andFunctors Definition 1.1 (Category). We define a category C by the following data: 1. A class ObC of elements called objects of C. 2. For a ordered pair (X, Y )ofobjectsasetHomC (X, Y )ofmorphisms is given such that HomC(X,Y )∩HomC (X ,Y )=φ for (X, Y ) =( X ,Y )(anelement f of HomC(X,Y )iscalledamorphism, and denote by f : X → Y ). Date:June2000. This is a seminar note of whichIgavealecture at Chiba University in June 2000. 1 2JUN-ICHI MIYACHI 3. For each triple (X, Y, Z)ofobjectsofC a map θ(X,Y, Z):HomC (X, Y ) × HomC(Y,Z) → HomC(X,Z) (θ is called the composition map) is given. 4. The composition map θ is associative. 5. For each object X of C,thereisamorphism 1X : X → X such that for any g : Y → X, h : X → Z we have 1Xg = g,h1X = h. X ∈ Ob C (often X ∈C)meansthatX is an obje ct of C. Example 1.2. The following often appear in this note. 1. Set is the category consi sti ng of sets as objects and maps as morphisms. 2. Ab is the category consisting of abelian groups as objects and group mor- phisms as morphisms. 3. For a ring A, Mod A is the category consisti ng of ri ght A-modules as obje cts and A-homomorphisms as morphisms. Definition 1.3 (Opposite Category). For a category C,theopposite category Cop of C is defined by 1. Ob Cop =ObC. (for X ∈C,wedenote by Xop ∈Cop the sameobject) 2. For Xop,Yop ∈ Ob Cop, op op HomCop (X ,Y )=HomC (Y, X). op op op (for f ∈ HomC(Y,X), we denote by f ∈ HomCop (X ,Y )) 3. The composition map θop is defined by θop(f op,gop )=θ(g, f)op . Definition 1.4. Let f : X → Y be a morphism inacategory C. 1. f is called a monomorphism if fu = fv implies u = v. 2. f is called an epimorphism if uf = vf implies u = v. 3. f is called a split monomorphism if there is g : Y → X such that gf =1X. 4. f is called a split epimorphism if there is g : Y → X such that fg =1Y . 5. f is called an isomorphism if there is g : Y → X such that gf =1X and fg =1Y . We ofte nwrite for an epimorphism, and for a monomorphism. Definition 1.5 (Functor). For cate gories C and C,acovariantfunctor(resp., con- travariant functor) F : C→C consists of the following data: 1. A map F :ObC→Ob C. 2. For X, Y ∈ Ob C,amap FX,Y :HomC (X, Y ) → HomC (FX,FY) (resp., FX,Y :HomC(X,Y ) → HomC (FY,FX)) such that F(gf)=F(g)F (f)(resp.,F(gf)=F(f)F (g)), F(1X )=1F (X). Here we write simply F(f)instead of FX,Y (f). Example 1.6. In a category C,forX ∈C,wedefine the covariant (resp., con- travariant) functor hX : C→Set (resp., hX : C→Set) X by h (Y )=HomC(X, Y ) (resp., hX (Y )=HomC(Y,X)). DERIVED CATEGORIES AND REPRESENTATIONS OF ALGEBRAS 3 Definition 1.7 (Functorial Morphism). Forcovariant (resp.,contravariant) func- tors F,G : C→C,afunctorial morphism α : F → G consists of thefollowing data: 1. For each X ∈C, αX : FX → GX in C is given. 2. For any morphism f : X → Y (resp., f : Y → X)inC,wehavethefollowing commutative diagram in C FX −−−−αX→ GX F (f) G(f ) FY −−−−αY → GY. In the case that a functorial morphism α is called a functorial isomorphism if αX are isomorphisms for all X ∈C. We denote by Mor(F, G)thecollection of all functorial morphisms from F to G. Lemma 1.8 (Representable Functor). For a covariant (resp., contravariant) func- tor F : C→Set,thefollowing are equivalent for C ∈C. C 1. F is isomorphic to h (resp., hC ). 2. There exists c ∈ F(C) satisfying that for any X ∈Cand x ∈ F(X),thereisauniquef ∈ HomC(C,X) (resp., f ∈ HomC (X, C)) such that x = F(f)(c). Acovariant(resp., contravariant) functor F : C→Set is called representable if there exists C ∈Csuch that F is isomorphic to hC .Similarly, a contravariant functor F : C→Set is called representable if there exists C ∈Csuch that F is isomorphic to hC . Lemma 1.9 (Yoneda’s Lemma). For X ∈Cand a covariant (resp., contravariant) functor F : C→Set,wehave the bijection X θ− : FX → Mor(h ,F) (resp., θ− : FX → Mor(hX ,F)), X where θ− is defined by (θx)Y (f)=F(f)(x) for x ∈ FX, Y ∈C, f ∈ h (Y ) (resp., f ∈ hX (Y )). Corollary1.10. For X, Y ∈C,wehave the bijection − X Y h :HomC (Y, X) → Mor(h ,h ) (resp., h− :HomC(X,Y ) → Mor(hX ,hY )). Definition 1.11. Let F : C→C be a functor. 1. F is called fu l l if FX,Y :HomC (X, Y ) → HomC (FX,FY)aresurjecti ve for all X,Y ∈C. 2. F is called faithful if FX,Y :HomC (X, Y ) → HomC (FX,FY)areinjectivefor all X, Y ∈C. 3. F is called dense if for any Y ∈C,thereisX ∈Csuch that Y is isomorphic to FX. Definition 1.12 (Limit, Colimit). Le t I, C be categori es and X ∈C.Wedenote by XI : I→Cthe constant functor such that XI(i)=X for all i ∈Iand XI(f)=1X for all f ∈ HomI (i, j). For a functor F : I→C,anobjectX of C is called the colimit colim F (resp., the limit lim F)ofF provided that for all Y ∈Cwe have ∼ Mor(F, YI ) = HomC(X,Y ) ∼ (resp., Mor(YI ,F) = HomC(Y,X)). 4JUN-ICHI MIYACHI Definition 1.13 (Filtered Col imit). Asmallcategory I is called afilteredcategory provided that 1. For any i, j ∈I,thereexistsk ∈Iand morphisms i → k, j → k in I. 2. For two morphisms f, g : i → j,thereisamorphism h : j → k such that hf = hg. Foracovari ant (resp., contravariant) functor F : I→Cfrom a filtered category I to a category C,thefiltered colimit− lim→ F (resp., the filtered limit← lim− F )ofF is the colimit colim F (resp., the limit lim F). { } Definition 1.14 (Product, Coproduct). For a collection Xi i∈I of objects in- dexed by a set I, X is called a coproduct i∈I Xi (resp., a product i∈I Xi)of {Xi}i∈I provided that 1. There are a collection of morphisms {qi : Xi → X}i∈I (resp., {pi : X → Xi}i∈I ). 2. For any Y ∈Cand {fi : Xi → Y }i∈I (resp., {pi : Y → Xi}i∈I ), thereexists a unique morphism f : X → Y (resp., f : Y → X)withfi = fqi (resp., f = p f)foral l i. i i { } If a coproduct i∈I Xi is also a product, then it is called a biproduct of Xi i∈I and denoted by i∈I Xi. Definition 1.15 (Bifunctor). Let C1, C2 and D be categori es. The product category C1 ×C2 is the category consisting of pairs (X1,X2)ofobjectsX1 ∈ Ob C1 and X2 ∈ ObC2 as objects, and pairs (f1,f2)ofmorphisms f1 in C1 and f2 in C2 as morphisms. A bifunctor is the functor F : C1 ×C2 →D. For bifunctors F,G : C1 ×C2 →D,abifunctorial morphism α : F → G is a functorial morphismoffunctorsC1 ×C2 →D. Then a bifunctor F : C1 ×C2 →Dconsists of the fol low ing data: 1. For X1 ∈C1, F (X1, −):C2 →Dis a functor. 2. For X2 ∈C2, F (−,X2):C1 →Dis a functor. 3. For a morphism f1 : X1 → Y1 in C1, F (f1, −):F(X1, −) → F(Y1, −)isa functorial morphism. (or equivalently, for a morphism f2 : X2 → Y2 in C1, F(−,f2):F (−,X2) → F (−,X2)isafunctorial morphism.) And a bifunctori al morphism α : F → G consists of the following data: ∈C ×C → D 1. For each (X1,X2) 1 2, α(X1,X2) : F (X1,X2) G(X1,X2)in is given. ∈C − → − 2. For X1 1, α(X1,−) : F(X1, ) G(X1, )isafunctorial mor phism . ∈C − → − 3. For X2 2, α(−,X2) : F( ,X2) G( ,X2)isafunctorial mor phism . In the case that a bifunctorial morphism α is called a bifunctorial isomorphism if ∈C ×C α(X1,X2) are isomorphisms for all (X1,X2) 1 2. Definition 1.16 (Adjoint). Le t F : C→C, G : C →Cbe covariantfunctors. We say that F is a left adjoint of G (or G is a right adjoint of F)(denote by F G)if there is a bifunctorial isomorphism t(−, ?) : HomC (F −, ?) → HomC(−,G?). −1 In this case, let σX = t(X, F X)(1FX)andτY = t(GY, Y ) (1GY )forX ∈C, Y ∈C.(σ : 1C → GF and τ : FG→ 1C are called the adjunction arrows.) Forcontravariant functors F : C→C, G : C →C,apair(F ,G)iscalleda right adjoint pair if there is a bifunctorial isomorphism t (−, ?) : HomC (−,F?) → HomC(?,G−).
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