Chapter 5. Localization. 5.1 Localization of categories

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1 Chaper 5 Localizaion Consider a caegory C and a amily o morphisms in C. The aim o localizaion is o ind a new caegory C and a uncor Q : C C which sends he morphisms belonging o o isomorphisms in C, (Q, C ) being universal or such a propery. In his chaper, we shall consruc he localizaion o a caegory when saisies suiable condiions and he localizaion o uncors. The reader shall be aware ha in general, he localizaion o a U-caegory C is no more a U-caegory (see Remark ). Localizaion o caegories appears in paricular in he consrucion o derived caegories. A classical reerence is [6]. 5.1 Localizaion o caegories Le C be a caegory and le be a amily o morphisms in C. Deiniion A localizaon o C by is he daa o a caegory C and a uncor Q : C C saisying: (a) or all s, Q(s) is an isomorphism, (b) or any uncor F : C A such ha F(s) is an isomorphism or all s, here exiss a uncor F : C A and an isomorphism F F Q, C F A Q F C 89

2 90 CHAPTER 5. LOCALIZATION (c) i G 1 and G 2 are wo objecs o Fc(C, A), hen he naural map (5.1) Hom Fc(C,A) (G 1, G 2 ) Hom Fc(C,A) (G 1 Q, G 2 Q) is bijecive. Noe ha (c) means ha he uncor Q : Fc(C, A) Fc(C, A) is ully aihul. This implies ha F in (b) is unique up o unique isomorphism. (i) I C exiss, i is unique up o equivalence o ca- Proposiion egories. (ii) I C exiss, hen, denoing by op he image o in C op by he uncor op, (C op ) op exiss and here is an equivalence o caegories: (C ) op (C op ) op. Proo. (i) is obvious. (ii) Assume C exiss. e (C op ) op := (C ) op and deine Q op : C op (C op ) op by Q op = op Q op. Then properies (a), (b) and (c) o Deiniion are clearly saisied. Deiniion One says ha is a righ muliplicaive sysem i i saisies he axioms 1-4 below. 1 For all X C, id X. 2 For all, g, i g exiss hen g. 3 Given wo morphisms, : X and s : X X wih s, here exis : and g : X wih and g s =. This can be visualized by he diagram: X X g s s X X 4 Le, g : X be wo parallel morphisms. I here exiss s : W X such ha s = g s hen here exiss : Z such ha = g. This can be visualized by he diagram: W s X g Z

3 5.1. LOCALIZATION OF CATEGORIE 91 Noice ha hese axioms are quie naural i one wans o inver he elemens o. In oher words, i he elemen o would be inverible, hen hese axioms would clearly be saisied. Remark Axioms 1-2 assers ha is he amily o morphisms o a subcaegory o C wih Ob( ) = Ob(C). Remark One deines he noion o a le muliplicaive sysem by reversing he arrows. This means ha he condiion 3 is replaced by: given wo morphisms, : X and :, wih, here exis s : X X and g : X wih s and g = s. This can be visualized by he diagram: X X g s X and 4 is replaced by: i here exiss : Z such ha = g hen here exiss s : W X such ha s = g s. This is visualized by he diagram W s X g In he lieraure, one oen calls a muliplicaive sysem a sysem which is boh righ and le muliplicaive. Many muliplicaive sysems ha we shall encouner saisy a useul propery ha we inroduce now. Deiniion Assume ha saisies he axioms 1-2 and le X C. One deines he caegories X and X as ollows. Z Ob( X ) = {s : X X ; s } Hom X((s : X X ), (s : X X )) = {h : X X ; h s = s } Ob( X ) = {s : X X; s } Hom X ((s : X X), (s : X X)) = {h : X X ; s h = s}. Proposiion Assume ha is a righ (resp. le) muliplicaive sysem. Then he caegory X (resp. op X ) is ilran. Proo. By reversing he arrows, boh resuls are equivalen. We rea he case o X.

4 92 CHAPTER 5. LOCALIZATION (a) Le s : X X and s : X X belong o. By 3, here exiss : X X and : X X such ha s = s, and. Hence, s by 2 and (X X ) belongs o X. (b) Le s : X X and s : X X belong o, and consider wo morphisms, g : X X, wih s = g s = s. By 4 here exiss : X W, such ha = g. Hence s : X W belongs o X. One deines he uncors: α X : X C (s : X X ) X, β X : op X C (s : X X) X. We shall concenrae on righ muliplicaive sysem. Deiniion Le be a righ muliplicaive sysem, and le X, Ob(C). We se Hom C r (X, ) = lim ( ) Hom C (X, ). Lemma Assume ha is a righ muliplicaive sysem. Le C and le s : X X. Then s induces an isomorphism Hom C r (X, ) s Hom C r (X, ). Proo. (i) The map s is surjecive. This ollows rom 3, as visualized by he diagram in which s,, : X s X (ii) The map s is injecive. This ollows rom 4, as visualized by he diagram in which s,, : X s X g

5 5.1. LOCALIZATION OF CATEGORIE 93 Using Lemma 5.1.9, we deine he composiion (5.2) as Hom C r (X, ) Hom C r (, Z) Hom C r (X, Z) lim Hom C (X, ) lim Hom C (, Z ) Z Z ( lim Hom C (X, ) lim Hom C (, Z ) ) Z Z ( lim Hom C (X, ) lim Hom C (, Z ) ) Z Z lim lim Hom C (X, Z ) Z Z lim Hom C (X, Z ) Z Z Lemma The composiion (5.2) is associaive. The veriicaion is le o he reader. Hence we ge a caegory C r whose objecs are hose o C and morphisms are given by Deiniion Le us denoe by Q : C C r he naural uncor associaed wih Hom C (X, ) lim ( ) Hom C (X, ). I here is no risk o conusion, we denoe his uncor simply by Q. Lemma I s : X belongs o, hen Q(s) is inverible. Proo. For any Z C r, he map Hom C Z) Hom (, r C r (X, Z) is bijecive by Lemma A morphism : X in C r is hus given by an equivalence class o riples (,, ) wih :, and : X, ha is: X he equivalence relaion being deined as ollows: (,, ) (,, ) i here exiss (,, ) (,, ) and a commuaive diagram: (5.3) X

6 94 CHAPTER 5. LOCALIZATION Noe ha he morphism (,, ) in C r is Q() 1 Q( ), ha is, (5.4) = Q() 1 Q( ). For wo parallel arrows, g : X in C we have he equivalence (5.5)Q() = Q(g) C r here exis s :, s wih s = s g. The composiion o wo morphisms (,, ) : X and (Z, s, g ) : Z is deined by he diagram below in which, s, s : h W s X g Z s Z Theorem Assume ha is a righ muliplicaive sysem. (i) The caegory C r and he uncor Q deine a localizaion o C by. (ii) For a morphism : X, Q() is an isomorphism in C r i and only i here exis g : Z and h : Z W such ha g and h g. Noaion From now on, we shall wrie C insead o C r. This is jusiied by Theorem Remark (i) In he above consrucion, we have used he propery o o being a righ muliplicaive sysem. I is a le muliplicaive sysem, one ses Hom C l (X, ) = lim Hom C (X, ). (X X) X By Proposiion (i), he wo consrucions give equivalen caegories. (ii) I is boh a righ and le muliplicaive sysem, Hom C (X, ) lim (X X) X,( ) Hom C (X, ). Remark In general, C is no more a U-caegory. However, i one assumes ha or any X C he caegory X is small (or more generally, coinally small, which means ha here exiss a small caegory coinal o i), hen C is a U-caegory, and here is a similar resul wih he X s.

7 5.2. LOCALIZATION OF UBCATEGORIE Localizaion o subcaegories Proposiion Le C be a caegory, I a ull subcaegory, a righ muliplicaive sysem in C, T he amily o morphisms in I which belong o. (i) Assume ha T is a righ muliplicaive sysem in I. Then I T well-deined. C is (ii) Assume ha or every : X,, I, here exiss g : X W, W I, wih g. Then T is a righ mulilplicaive sysem and I T C is ully aihul. Proo. (i) is obvious. (ii) I is le o he reader o check ha T is a righ mulpiplicaive sysem. For X I, T X is he ull subcaegory o X whose objecs are he morphisms s : X wih I. By Proposiion and he hypohesis, he uncor T X X is coinal, and he resul ollows rom Deiniion Corollary Le C be a caegory, I a ull subcaegory, a righ muliplicaive sysem in C, T he amily o morphisms in I which belong o. Assume ha or any X C here exiss s : X W wih W I and s. Then T is a righ mulpiplicaive sysem and I T is equivalen o C. Proo. The naural uncor I T C is ully aihul by Proposiion and is essenially surjecive by he assumpion. 5.3 Localizaion o uncors Le C be a caegory, a righ muliplicaive sysem in C and F : C A a uncor. In general, F does no send morphisms in o isomorphisms in A. In oher words, F does no acorize hrough C. I is however possible in some cases o deine a localizaion o F as ollows. Deiniion A righ localizaion o F (i i exiss) is a uncor F : C A and a morphism o uncors τ : F F Q such ha or any uncor G : C A he map (5.6) Hom Fc(C,A) (F, G) Hom Fc(C,A) (F, G Q) is bijecive. (This map is obained as he composiion Hom Fc(C,A) (F, G) Hom Fc(C,A) (F Q, G Q) τ Hom Fc(C,A) (F, G Q).) We shall say ha F is righ localizable i i admis a righ localizaion.

8 96 CHAPTER 5. LOCALIZATION One deines similarly he le localizaion. ince we mainly consider righ localizaion, we shall someimes omi he word righ as ar as here is no risk o conusion. I (τ, F ) exiss, i is unique up o unique isomorphisms. Indeed, F is a represenaive o he uncor G Hom Fc(C,A) (F, G Q). (This las uncor is deined on he caegory Fc(C, A) wih values in e.) Proposiion Le C be a caegory, I a ull subcaegory, a righ muliplicaive sysem in C, T he amily o morphisms in I which belong o. Le F : C A be a uncor. Assume ha (i) or any X C here exiss s : X W wih W I and s, (ii) or any T, F() is an isomorphism. Then F is righ localizable. Proo. We shall apply Corollary Denoe by ι : I C he naural uncor. By he hypohesis, he localizaion F T o F ι exiss. Consider he diagram: Denoe by ι 1 Q Q C C ι Q T Q I I T F T F ι A ι a quasi-inverse o ι Q and se F := F T ι 1 Q. Le us show ha F is he localizaion o F. Le G : C A be a uncor. We have he chain o morphisms: Hom Fc(C,A) (F, G Q ) F λ Hom Fc(I,A) (F ι, G Q ι) Hom Fc(I,A) (F T Q T, G ι Q Q T ) Hom Fc(IT,A) (F T, G ι Q ) Hom Fc(C,A) (F T ι 1 Q, G) Hom Fc(C,A) (F, G). We shall no prove here ha λ is an isomorphism. The irs isomomorphism above (aer λ) ollows rom he ac ha Q T is a localizaion uncor (see Deiniion (c)). The oher isomorphisms are obvious.