Triangulated categories and localization

Transcription

1 Trianulated cateories and localization Karin Marie Jacobsen Master o Science in Physics and Mathematics Submission date: Januar 212 Supervisor: Aslak Bakke Buan, MATH Norweian University o Science and Technoloy Department o Mathematical Sciences

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3 Problem Description The student should ive a description o the ollowin topics: abstract trianulated cateories localisation with respect to arbitrary sets o morphisms localisation as used in the construction o derived cateories o module cateories Moreover, the student should ive detailed examples o derived cateories or inite dimensional alebras and other concrete trianulated cateories. i

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5 Abstract We study Gabriel-Zisman localization, localization by a multiplicative system and by a null system. We deine the trianulated cateory and the derived cateory. Finally we describe a scheme or localization rom a trianulated cateory to a module cateory. iii

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7 Preace This thesis was written or the course TMA49 - Mathematics, Master Thesis. I irst learned about cateory theory durin my exchane year at the University o Helsinki, and ell in love with the abstract race and versatility o the material. My specialization project durin the sprin o 211 allowed me to study it in the context o abelian cateories, and in this thesis I have studied subjects connected to the concept o localization, i.e. the creation o inverses in cateories. I was particularly happy to be able to include some current research by my advisor in the inal chapter. I would like to thank my advisor, Aslak Bakke Buan. He has been reassurin, helpul and supportive, whilst also bein a master hunter o typos and ambiuities in the manuscript Second, I must thank my parents, who have always been encourain and always there or me, and my almost odmother proessor Gunilla Boreors who is a constant source o intellectual inspiration. Finally I want to thank two people who over the past year have listened patiently to me panic, complain and enthuse about cateory theory: My roommate Anna and my beloved Petter. Karin Marie Jacobsen Trondheim, January 212 v

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9 CONTENTS vii Contents 1 Introduction The Ore condition Localization o a eneral cateory A simple proo o existence Skeletally small cateories Multiplicative systems and roos Localization o a trianulated cateory Trianulated cateories Localization by a null system Derived cateories Trianulation o K(C) Derived cateories Localization o subcateories Projective resolutions Example: Derived module cateory o a hereditary alebra Localization to a module cateory 47 6 Summary 51 A Diaram lemmas or abelian cateories 52 A.1 The ive lemma A.2 The snake lemma A.3 Technical lemma or theorem

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11 1 1 Introduction This thesis will consider a concept in cateory theory called localization. We will take a cateory C and a class o morphisms S in the cateory, and construct a new cateory C S in which the morphisms o S have become isomorphisms. Moreover, we equip C S with a suitable universal property, so that we are able to call it unique. Ater an introductory study o the Ore condition in rin theory we will consider the very eneral Gabriel-Zisman localization. We will deine a multiplicative system, a trianulated cateory and a null system. All this will set the stae or deinin the derived cateory. The construction o the derived cateory moves us rom an abelian cateory to a trianulated cateory. As the inal part o the thesis, we will study a recent article by Aslak Bakke Buan and Robert J. Marsh [3], in which a trianulated cateory is turned into an abelian cateory throuh the process o localization. In some ways localization is an old idea. The concept is used in a dierent uise in set theory when constructin the inteers Z; take the set o natural numbers N, and consider the set o unctions n, where n (m) = m + n. When all these unctions are turned into isomorphisms, that is when N is localized with respect to the set { n n N}, we can identiy n = n () and n = n 1 (), and thus we have deined Z. Similarly, the construction o the rational numbers rom the inteers can also be described as a localization. We will study the latter in an example in the ollowin subsection. 1.1 The Ore condition We irst consider a orm o localization that is accomplished without the use o cateory theory. This section ollows [2, ch. 12.2]. We want to take a possibly non-commutative domain R (a rin with no zero divisors), and show it to be embedded in a division rin Q. For this to be easible, we need R to ulill the ollowin condition, called the Ore condition. Deinition 1.1. Let R be a domain. We call R a riht Ore domain i or any non-zero a, b R there exist nonzero x, y R so that ax = by. Note that we only consider rins with multiplicative identity. An equivalent deinition is iven by the ollowin lemma; we will need both deinitions. Lemma 1.2. A domain R is a riht Ore domain i and only i the intersection o any two nonzero riht ideals is nonzero. Proo. Suppose R is a riht Ore domain and let I and J be two nonzero riht ideals in R. Let a I and b J be nonzero; then we know that there exist nonzero x, y R so that ax = by. As R is a domain, we know that ax. However, as I and J are riht ideals, it ollows that ax I and by J, and thus ax = by I J, which must be nonzero. Conversely, suppose that the domain R is such that no two nonzero riht ideals have a zero intersection. Let a, b R be nonzero. There exist nonzero ideals ar and br 1. 1 ar = {ar r R}

12 2 1 INTRODUCTION Their intersection must be nonzero, so there are elements x, y R such that ax = by. For the remainder o this section, suppose R is a riht Ore domain. Let C be the set o nonzero riht ideals in R; by the above lemma C is closed under intersections. A map deined on a riht ideal R o R is called R-linear i (ab) = (a)b and (a + b) = (a) + (b) or all a, b R. Suppose that I C, that : I R is an R-linear mappin and let C. We take note o some acts about : 1 () is a nonempty set as () =. I a, b 1 (), then (a b) = (a) (b), so a b 1 (). I x 1 () and r R, then (rx) = r(x), so rx 1 (). By the above 1 () is an ideal. I (I) = {} or some ideal I; then (I), so I 1 (), which is non-zero and thus an element o C. On the other hand, suppose (I) {}. As (I) must be a nonzero riht ideal, we know that (I) is nonzero; thus there is some non-zero x I so that (x). It ollows that x 1 (), which then must be nonzero and 1 () C. For two ideals I and J we deine Hom R (I, J) to be the set o all R-linear maps between I and J. Deine the set H = I C Hom R(I, R), and deine a relation on H by settin i and only i = when restricted to some ideal I C. The relation is clearly relexive and symmetric. I = on the riht ideal I and = h on the riht ideal J, then = h on the ideal I J C and so the relation is transitive. Thus is an equivalence relation on H and we can orm the set Q = H/ with elements [] = { H }. Deine an addition operation on Q by settin [] + [] = [ + ]. It can be checked that this operation is well-deined, and that (Q, +) is an abelian roup. Next, deine multiplication by settin [] [] = []. That it is well-deined ollows rom the properties o C. It is not hard to see that with these operations Q becomes a rin. Finally we check that Q is a division rin by showin that every nonzero element o Q has an inverse. Let [] Q be non-zero. As = when restricted to Ker(), we must have Ker() / C. Thus Ker() = {} and is one-to-one. Let : Im() R be deined by ((x)) = x, this is well-deined because is one-to-one, and is also R-linear. On the domain o it holds that = 1, and so [] = [1]. Similarly = 1 on Im(), so [] 1 = []. Thus Q is a division rin. Theorem 1.3. A domain R is a riht Ore domain i and only i there exists a division rin Q such that: (i) R is a subrin o Q. (ii) All objects in Q are o the orm ab 1 or a, b R. Proo. Suppose R is a riht Ore domain, and let Q be deined as above.

13 3 (i) Consider the map φ : R Q deined by settin φ(a) = [ a ], where a (x) = ax. Note that a is R-linear. We check that φ is a rin homomorphism: From φ(1 R ) = id R we see that φ maps the unity in R to the unity in Q. We see that a+b (x) = (a + b)x = ax + bx = a (x) + b (x), thus φ(a + b) = φ(a) + φ(b). Similarly, as ab (x) = (ab)x = a(bx) = a b (x) we et that φ(ab) = φ(a)φ(b). Since φ is a rin homomorphism, Im(φ) is a subrin o Q. O course, φ is onto its imae; to see that it is one-to-one, suppose φ(a) =. This means that or some nonzero ideal I, we have ai =. Thus, there is some nonzero x I so that ax = ; since R is a domain this means that a =. Thus R is isomorphic to a subrin o Q, and can itsel be considered a subrin o Q. Due to this identiication we will denote a simply as a when there is no risk o conusion. (ii) We note that i q Q, where q : I R, then or a, b R it holds that (qa)(b) = (q a )(b) = q(ab) = q(a)b, so we see that q(a) = qa in Q. Now assume q : I R is in Q, that b I is non-zero, and set a = q(b). Then qb = a and as Q is a division rin, q = ab 1. To see the reverse implication, suppose both (i) and (ii) hold and let a, b R. Then b 1 a Q, and so there exist x, y R so that b 1 a = xy 1. However, this implies that ax = by. Thus R is a riht Ore domain. Example 1.4. Consider the set o inteers, Z. This is a commutative domain; it is easy to see that it satisies the Ore condition. Then by theorem 1.3 we know that there exists a division rin Q where every element can be written on the orm xy 1, where x, y Z and y. Moreover, or any two x, y Z, with y, we have xy 1 Q. It is by now obvious that Q = Q, the ield o rational numbers. 2 Localization o a eneral cateory Let C be a cateory and let S be a class o morphisms in C. The oal o localization is to construct a cateory C S where the morphisms in S have inverses. We ormalize this idea as ollows: Deinition 2.1. The localization o the cateory C by a class S o morphisms in C is a cateory C S with a unctor F : C C S so that: For any morphism s S, the imae F (s) is an isomorphism. The unctor F : C C S is universal; i there exists another cateory D with a unctor G : C D takin elements o S to isomorphisms in D, then there exists a

14 4 2 LOCALIZATION OF A GENERAL CATEGOR unique unctor H : C S D so that the ollowin diaram commutes. C F C S G H D 2.1 A simple proo o existence We start by showin the existence o localization in one type o cateories, known as Gabriel-Zisman localization. This section ollows [5, ch. III.2.2]. Suppose that C is a small cateory, that is to say that Ob(C) is a set. Then Mor(C) = {Hom C (, ), Ob(C)} is a set, and so is any class S o morphisms in C. To ind the localization o C by S we start by constructin a raph Γ. The vertices o Γ are the objects o C. For all, Ob(C), let there be an ede rom to in the raph Γ or each morphism in Hom C (, ). Moreover, or each morphism s S, add an arrow xs. This arrow is called the ormal inverse o s. A path in Γ is a inite sequence o arrows such that each arrow starts in the vertex where the previous arrow in the sequence ended. Let two paths be equivalent i one can be transormed into the other by applyin the ollowin elementary equivalences a inite number o times: I Hom C (, ) and Hom C (, Z), then the path Z is equivalent to (can be replaced by) Z. I s S, then s is equivalent to id. xs is equivalent to id and xs s This relation is clearly relexive, symmetric and transitive; thus it is an equivalence relation. The cateory C S is deined by settin Ob(C S ) = Ob(C) and or, Ob(C) settin Hom C (, ) to be the set o equivalence classes o paths rom to. Composition o morphisms ollows rom the joinin o paths; this can be shown to be independent o the choices o representatives o the equivalence classes o paths. The unctor F : C C S is deined by settin F () = or Ob(C) and lettin F () be the equivalence class o the path or Hom C (, ). We see that i s S, then F (s) = s has an inverse, namely its ormal inverse xs. To show the universality o C S, we assume the existence o a cateory D with a unctor G : C D so that G(s) has an inverse or any s S. We construct the required unctor H : C S D as ollows: For Ob(C S ), set H() = G() (remember that Ob(C S ) = Ob(C)).

15 2.2 Skeletally small cateories 5 For any Mor(C), set H( ) = G(). For any s S, set H( xs ) = G(s) 1. For any other path φ, the composition requirement in the deinition o a unctor ives H(φ): Suppose φ = φ 1... φ n, where each φ i denotes an arrow in Γ. Then H(φ) = H(φ 1 )... H(φ n ). To check that H is well-deined, consider two equivalent paths α = ( a A 1... a A n b n ) and β = ( b B1... B m m ). The unctor H will map these paths to respectively H(α) = H(a n )H(a n 1 )... H(a ) and H(β) = H(b m )H(b m 1 )... H(b ). We know that by a inite number o elementary operations we can turn α into β. However, each o these elementary operations can be translated to an equality in D. When in C S it also holds that in D, and similarly when it also holds that Z = Z F ()F () = F () s xs = id H(x s x) = F (x) 1 F (x) = id F (x) = F (id ) = H(id ). For each step o the process o turnin α into β by the elementary equivalences we can ind an equality in D, so H(α) = H(β). Thus H is well-deined. Moreover as any other unctor H satisyin H F = G must satisy the very requirements set orth in the deinition o H we have that H = H and H is unique. 2.2 Skeletally small cateories The proo o existence in the above section works only or small cateories. In eneral the cateories we encounter are not small; examples o non-small cateories include the cateory o sets, o rins and o modules over a rin. Luckily we can extend the proo to be valid or a wider rane o cateories. We start with a deinition. Deinition 2.2. A cateory C is skeletally small i the amily o isomorphism classes o Ob C is a set. An important example o a skeletally small cateory is the cateory o modules over a rin. Suppose that C is skeletally small, and that S is a class o morphisms in C. For each equivalence class, use the stron axiom o choice to pick an object to represent it. We denote the set o representin objects R. For each object C, choose an isomorphism c :, where is the representin object o the isomorphism class containin. I =, set c = id. We construct the raph Γ as ollows:

16 6 2 LOCALIZATION OF A GENERAL CATEGOR The vertices o Γ are the objects in R. For each Hom C (, ) where, R, add an arrow. (c 1 sc ) S For each s : S, add an arrow. We deine the equivalence classes o paths as beore, and are ready to deine the cateory C S. We set Ob(C S ) = Ob(C). For two objects, Ob(C S ), we set Hom CS (, ) = { c φ c 1 φ is an equivalence class o paths }. The composition rules deined on the paths is used in C S as well. The unctor F : C C S is deined as ollows: c F () = or Ob(C). F () = c For s S, the inverse o F (s) = (c sc 1 ) S c 1. c c 1 c 1 or Hom C (, ). c c sc 1 c 1 is the morphism It remains to prove universality or the cateory C S. Suppose that D is a cateory with a unctor G : C D, so that G(s) has an inverse or each s S. Construct the unctor H as ollows: H() = G() or Ob(C S ) = Ob(C). H(G()) = F () or Mor(C). H(G(x) 1 ) = F (x) 1. By a proo analoous to that in the previous section, this makes H well-deined and unique. 2.3 Multiplicative systems and roos The two previous sections have shown procedures o localization that are restricted by the type o cateory they work in. We now ive a method o localization where the restriction is put on the class o morphisms S and not on the cateory C. The conditions on the morphisms will turn out to be hihly similar to the Ore condition, and so will the description o the elements in the localization. Multiplicative systems are widely described, or example in [5], [9] and [1]. Deinition 2.3. A multiplicative system in a cateory C is a class S o morphisms satisyin the ollowin restrictions: S1 id S or any Ob(C).

17 2.3 Multiplicative systems and roos 7 S2 For any pair o morphisms, S such that the composition exists, we have S. S3 For Z and s Z S, there exist morphisms U t S and U u so that the ollowin diaram commutes: U t u s Z The same holds when all arrows are reversed. S4 Suppose, Hom C (, ). The ollowin are equivalent: There exist t : S such that t = t. There exist s : S such that s = s. We note that the irst two axioms have little eect on localization; as id is an isomorphism, it will automatically have an inverse in the localized cateory C S, so addin it to S makes no dierence to the localization. Similarly, i and are in S they will have inverses 1 and 1 in C S. Then 1 1 will be an inverse to, so we miht as well add to S. The third axiom is practically the same as the Ore condition. Alon the same lines, in a domain R with s, a, b R non-zero it holds that sa = sb i and only i a = b, and in this case any r R makes ar = br. The last axiom thus makes the morphisms o S act like elements o a domain (to some extent; ater all cateories are not assumed to be additive). As we saw above, a morphism in the cateory C S is iven by a chain α 1 α 1 2 α n 1 2 n (2.1) where α i Hom C ( i, i+1 ) or α i = x s with s S Hom C ( i+1, i ). I S is multiplicative, it simpliies the description o morphisms a reat deal: Let s : and t : Z be morphisms so that s, t S; then we know that ts S as well. We et that x ts ts = id Z and consequently x ts = x s x t. Thus multiplicative systems allow us to compose the ormal inverses in C S just as we compose the morphisms directly rom C. x Suppose we have a morphism (path) in C S iven by 1 s 2 3. We know that there exist morphisms and t in C with t S such that the ollowin square commutes: 2 t 1 3 s 2

18 8 2 LOCALIZATION OF A GENERAL CATEGOR As t = s we et, by multiplyin with the ormal inverses o s and t, that x s = x t. id 1 With these properties, and rememberin that a morphism 1 2 = 1 1 x 2, we see that the eneral morphism in (2.1) is equivalent to a short chain s 1 n where s S. This short chain is also known as a let raction 2. We will use this to ive a third deinition o C S. Example 2.4. Let R be a domain, and deine the cateory R as the cateory consistin o a sinle object, morphisms iven by Hom R (, ) = R with the composition in bein multiplication in R. There is also an additive structure on R, ollowin rom the addition o elements in R. Note that R is a small cateory, so we can use Gabriel-Zisman localization on it. Deine S as the set o morphisms not equal to. We consider the axioms o the multiplicative system, and see that S1 is ulilled, as we only consider rins with identities. S2 is ulilled because R is a domain, and so is S4: i s S, then since sa = sb implies that s(a b) and s is not equal to zero, it must hold that a = b. Finally, we see that S3 is ulilled i R is a riht and let Ore domain, so suppose R is an Ore domain We know that there exists a cateory R S and a unctor F : R R S so that every non-zero element in S is turned into an isomorphism by F. Moreover, we know that any morphism in R S is o the orm rs 1 with s S and r R (and any element o this orm is a morphism in R S ). We can add any two elements as 1 and bt 1 by irst usin the act that we can ind two elements p, q S so that sp = tq. We then deine as 1 + bt 1 = app 1 s 1 + bqq 1 t 1 = (ap + bq)(sp) 1 ; we can check that with this as the addition operation, Hom RS (, ) becomes a division rin. In other words we have shown the same thin as we did in theorem 1.3. Note that while Gabriel-Zisman localization required C to be a small cateory, the below localization sets no such restriction. Deinition 2.5. Let C be a cateory and S a multiplicative system o morphisms in C. The localization C S o C by s is deined as ollows: Ob(C S ) = Ob(C). The morphisms o φ Hom CS (, ) are equivalence classes o triples φ = (Z, s, ), where Z Ob(C S ), s :, : and s S, usually drawn as a roo : s Z Two morphisms φ = (Z, s, ) and φ = (Z, s, ) between and are set equivalent i there exists a commutative diaram. 2 x It is also equivalent to a short chain 1 t n with t S, known as a riht raction.

19 2.3 Multiplicative systems and roos 9 t Z (2.2) s Z s Z We require that (Z, t, ) Hom CS (Z, Z ). We will denote this relation as (Z, s, ) (Z, s, ). Composition o a pair o morphisms (U, s, ) Hom CS (, ) and (V, t, ) Hom CS (, Z) is deined to be a morphism (W, sr, h), iven by the ollowin commutative diaram: s U r W Z We obtain W, r and h by the completion property (S3) o multiplicative systems. We will show that C S is a well-deined cateory, and then show that it ives a localization o C by S. Lemma 2.6. The relation is an equivalence relation. Proo. The relation is obviously relexive. To see that it is symmetric, suppose that (Z, s, ) (Z, s, ) as shown in diaram (2.2). There exists a completion o t h V Z into the commutative diaram W u Z with u S. s Z st r Z st We see that s u = str = s r, so by S4 there exists a morphism x : W W so that x S and ux = rx. Then we know that the ollowin diaram commutes: ux W trx Z Z s s s

20 1 2 LOCALIZATION OF A GENERAL CATEGOR As ux S, we know that (W, ux, trx) is a morphism, so (Z, s, ) (Z, s, ). Finally, suppose (Z, s, ) (Z, s, ) and (Z, s, ) (Z, s, ); in other words that the ollowin diaram commutes (with r, p S): Z s r s W h Z p W s i Z We are aimin to prove that (Z, s, ) (Z, s, ) in order to show that the relation is transitive. Consider the completion o the diaram W s p into U u k W W sr W sr We know that s hu = sru = s pk, and so there must exist a morphism x : U U such that x S and hux = pkx. Settin t = rux and = ikx we see that s p st = srux = s pkx = s ikx = s and t = rux = hux = pkx = ikx =. This means that the ollowin diaram commutes t U s Z s Z and (Z, s, ) (Z, s, ). Lemma 2.7. The composition o morphisms in C S is well-deined with respect to the equivalence relation. Proo. We divide this proo into two parts. First we show that two dierent compositions o the same two morphisms must be equivalent. Then we show that i the actors o two compositions are dierent but pairwise equivalent, then the compositions are equivalent as well.

21 2.3 Multiplicative systems and roos 11 First, suppose that we have two morphisms s U Z Furthermore, suppose that the two ollowin are both valid compositions o the morphisms: t V s U r W t h V Z Z We know that there exists an object T and morphisms x, y so that the ollowin diaram commutes s U r W t h V T y W x W r V Both r and r are in S, so we can select either x S or y S. We choose the ormer. Since rx = r y we have that thx = rx = r y = th y. Usin axiom S4 we see that there exists a morphism w : T T, w S so that hxw = h yw. Settin u = xw and v = yw we see that u S and that the ollowin diaram commutes. r Thereore the two compositions are equivalent. Next we want to show that i sr W u sr T h v W h Z U s U s

22 12 2 LOCALIZATION OF A GENERAL CATEGOR and t V Z Z then it holds that (V, t, )(U, s, ) = (W, sr, h) (W, s r, h ) = (V, t, )(U, s, ). To simpliy this task we look at one actor o the composition at a time. Assume V = V, t = t and =. By (U, s, ) (U, s, ) we know that there must exist a morphism (A, a, a ) so that the ollowin diaram commutes: s U a s A a U Usin axiom S3 several times we arrive at the ollowin commutative diaram: C y S y b B A B a S a b S x S U s r W t h t V V Z U s r x W Chasin the arrows, we et that srxy = s r x y and hxy = h x y. Thus we have conirmed the equivalence (W, sr, h) (W, s r, h ). Next suppose instead that U = U, s = s and =. By (V, t, ) (V, t, ), we know that there exists a morphism (A, a, a ) so that the ollowin diaram commutes: t V a t A t a V h V

23 2.3 Multiplicative systems and roos 13 As t a = ta S, we know that there must exist an object B with morphisms so that the ollowin diaram commutes: B u A v W t h Note that v S. As t a u = t h v and t S, we can ind a morphism B w B in S so that a uw = h vw. For the sake o clarity we set x = vw and b = uw. Let B be the completion B b A x W h where x S because a S. Let C be the completion C u B u B r x U We see that tabu = thxu = rxu = r x u = th x u = t a b u. Aain, as t a = ta S we et that there exists a morphism C v C S so that buv = b u v. Settin y = uv and y = u v we et the ollowin commutative diaram: t a a rx s U r W t h x S Z A s U t V V a a S B b b y S y C r W We see that srxy = s r x y, and also hxy = h x y. Thus (W, sr, h) (W, sr, h ). Thus it ollows that or two pairs o equivalent (and compatible) morphisms so that (U, s, ) (U, s, ) and (V, t, ) (V, t, ) it holds that (W, sr, h) = (V, t, )(U, s, ) (V, t, )(U, s, ) h x B (V, t, )(U, s, ) = (W, s r, h ).

24 14 2 LOCALIZATION OF A GENERAL CATEGOR Thus the composition o two roos is well-deined with respect to the equivalence relation on C S. Theorem 2.8. In deinition 2.5, the iven C S is a cateory, and there exists a unctor C C S so that the two orm a localization o C with respect to S. Proo. To see that C S is a cateory, we really only need to check the axioms on the morphisms. That identities exist in C S is obvious; or Ob(C S ) the morphism (, id, id ) is the identity. Next we consider associativity. Suppose that we have three morphisms (U, r, ) :, (V, s, ) : Z and (W, t, h) : Z Z. We need to show that ((W, t, h)(v, s, ))(U, r, ) = (W, t, h)((v, s, )(U, r, )), which means that or the two diarams r U a A b B b s a V Z Z t W h and D r U d s V Z Z which ive the two possible compositions o the morphisms, it holds that c d C t c (B, rab, hb ) (D, rd, hc d ). As both (D, rd, cd ) and (A, ra, a ) are compositions o (U, r, ) and (V, s, ), we see that (D, rd, cd ) (A, ra, a ). Thus there must exist a morphism (E, e, e ) : A D so that a e = cd e = tc d e. Usin axiom S3 on the morphisms B ab U and D d U, we W h

25 2.3 Multiplicative systems and roos 15 see that there exists an object and morphisms u and u so that the ollowin diaram commutes: u D u d B U ab We choose u S (as both ab and d are in S, we are ree to chose). We see that abu = du abu = du sa bu = scd u a bv = cdv a bv = cdv tb v = tc d v b w = c d w. The morphisms v, v, w and w are deined usin axiom S4. We see that rabw = rdw and hb w = hc d w. Thus the morphisms (B, rab, hb ) and (D, rd, hc d ) are equivalent, composition o morphisms is an associative operation and C S is a cateory. We deine the unctor F : C C S on objects by lettin F () = or Ob(C). For a morphism Hom C (, ) we set F () = (, id, ). Then F (id ) = (, id, id ), which is the identity on F () and or two morphisms : and : Z it holds that F () = (, id, ) = (, id, )(, id, ) = F ()F (). Thus F is a unctor. Finally, we need to veriy that C S and F is a localization o C with respect to S. I s S, then the morphism (, s, id ) is a two-sided inverse o F (s) = (, id, s), so F (s) is an isomorphism. It remains to check that F is universal. Suppose that G : C D is a unctor so that G(s) is an isomorphism or any s S. We need a unctor H : C S D so that G = HF, and we deine it by lookin at the requirements set or it. For Ob(C) we must have G() = HF () = H(), as F () =. For (Z, s, ) Hom CS (, ), we have that (Z, s, )(Z, id Z, s) = (Z, id Z, ) H((Z, s, )(Z, id Z, s)) = H(Z, id Z, ) H(Z, s, )G(s) = H(Z, s, )HF (s) = HF () = G() We see that we must have H(Z, s, ) = G()G(s) 1.

26 16 2 LOCALIZATION OF A GENERAL CATEGOR The iven H is a well-deined unctor rom C S to D, and we also see that it is unique (the deinition o H ollows rom the requirements on any such unctor). It remains to check that H is a unctor. That H maps identities to identities is easy to see. Lettin (, t, )(, s, ) = (Z, sr, h) be a composition o morphism in C S, we see that H((, t, )(, s, )) = H(Z, sr, h) = G(h)G(sr) 1 = G()G(h)G(r) 1 G(s) 1 = G()G(t) 1 G(t)G(h)G(r) 1 G(s) 1 = G()G(t) 1 G()G(r)G(r) 1 G(s) 1 = G()G(t) 1 G()G(s) 1 = H(, t, )H(, s, ). Thus H respects composition, and is a unctor. We show that this localization by multiplicative systems preserves addition. This is an important property o localization by a multiplicative systems; in contrast to Gabriel- Zisman localization which enerally does not preserve addition. Theorem 2.9. [5, ch. III.4.5] Let C be an additive cateory and let S be a multiplicative system. Then C S is additive, and the localization unctor F : C C S is an additive unctor 3. Proo. We prove the theorem by construction; we will irst iner a reasonable deinition o the addition operation on C S and then prove that this operation makes C S additive and F an additive unctor Since we want F to be additive, we start by notin that or two morphisms, Hom C (, ) we need (, id, ) + (, id, ) = F () + F () = F ( + ) = (, id, + ). We extend this or two morphisms (U, s, ), (U, s, ) Hom CS (, ) by deinin (U, s, ) + (U, s, ) = (U, s, + ). To see that this is well-deined, suppose (U, s, ) (V, t, ) by the roo (W, u, u ) and (U, s, ) (V, t, ) by the roo (Z, v, v ). We can ind the ollowin square completions T w Z S x Z R y S w W u U v x W u U v T y w W x 3 A unctor that acts as a roup homomorphism on the Hom-sets; see deinition 3.2

27 17 where w, x, y S. Obviously, and moreover, suwy = tu wy = tu xy = tv x y ( + )uwy = ( + )uxy = u xy + vx y = ( + )v x y. Thus (U, s, + ) (V, t, + ) and addition is well-deined. Suppose we have (U, s, ), (V, r, ) Hom CS (, ), and let (W, p, q) be the roo deined by the ollowin completion: W q V U p s Deine r = sp = tq; then r S. Moreover, deine = p and = q. It ollows that (U, s, ) (W, r, ) and (V, t, ) (W, r, ), so we can deine (U, s, ) + (V, t, ) = (W, r, ) + (W, r, ) = (W, r, + ). For C S to be additive under this operation, we need to show that Hom CS (, ) is an abelian roup. We start by showin that addition is associative: ((U, s, ) + (V, t, )) + (W, r, h) = ((R, x, ) + (R, x, )) + (R, x, h ) t = (R, x, ( + ) + h ) = (R, x, + ( + h )) = (U, s, ) + ((V, t, ) + (W, r, h)) We see that (, id, ) is the zero element or the operation, and that (U, s, ) Hom CS (, ) has an inverse (U, s, ) Hom CS (, ). Commutativity ollows rom commutativity o + in C. Thus Hom CS (, ) is an abelian roup. That composition is bilinear and that inite direct sums exist ollows rom the additivity o C in a similar manner. Thus C S is an additive cateory. Moreover, the unction F : Hom C (, ) Hom CS (, ), obtained rom the localization unctor F is a roup homomorphism, so the localization unctor is an additive unctor. 3 Localization o a trianulated cateory Many convenient structures arise when a cateory is abelian; the most important is perhaps the exact sequence. Sometimes when we are workin in an additive cateory, we can mimic the exact sequence even i our cateory is not abelian; namely by deinin distinuished trianles. We will use it in section 4 when studyin the homotopy cateory in order to ind the derived cateory. More speciically we will use the act, shown in section 3.2, that the structure o distinuished trianles makes it possible to deine a multiplicative system rom a collection o objects. More literature on trianulated cateories can be ound in [9] and [1].

28 18 3 LOCALIZATION OF A TRIANGULATED CATEGOR 3.1 Trianulated cateories Let C be a cateory, and let Σ be a unctor C Σ C which is an equivalence o cateories (or simply an equivalence ). A trianle in C is deined as a sequence o maps α β Z γ Σ(). A morphism between two trianles α β Z γ Σ() and α β Z γ Σ( ) is a triple o maps (φ, φ, φ Z ) so that the ollowin diaram commutes: α β Z γ Σ() φ φ φ Z Σφ α β Z γ Σ( ) I all three morphisms φ, φ and φ Z are isomorphisms, this is known as an isomorphism o trianles. We are now ready to ive the deinition o a trianulated cateory. Deinition 3.1. Let C be an additive cateory, and let Σ be an equivalence C Σ C. Then C is know as a trianulated cateory i there exists a amily o trianles, called distinuished trianles, satisyin the ollowin conditions: T1 Any trianle isomorphic to a distinuished trianle is distinuished. T2 For any Ob(C), the trianle id Σ() is distinuished. T3 For any, Ob(C) and any morphism Hom(, ), there exists a distinuished trianle Z Σ(). T4 The trianle α β Z γ Σ() is distinuished i and only i β Z γ Σ() Σα Σ( ) is distinuished. T5 A commutative diaram u can be embedded in a morphism o distinuished trianles as ollows 4 : v Z Σ() u v Σ(u) Z Σ( ) 4 The existence o the distinuished trianles themselves is iven by axiom T3

29 3.1 Trianulated cateories 19 T6 Given the ollowin distinuished trianles: Z Σ() Z Σ( ) Z Σ() there exists a distinuished trianle Z Σ(Z ) so that the ollowin diaram commutes: id Z Z Σ() Σ() id Σ() Z id Z Σ Σ( ) id Z Σ(Z ) In a trianulated cateory, the unctor Σ is oten called the translation or suspension unctor. In the ollowin discussion, we will need some restrictions on the unctors we use. The irst one is to preserve additivity. Deinition 3.2. A unctor between additive cateories, F : C C is called additive i it acts as a roup homomorphism rom Hom C (, ) to Hom C (F (), F ( )). One example o an additive unctor is the covariant hom-unctor Hom C (, ) (the contravariant Hom-unctor, Hom C (, ), is additive as well). Note that i the unctor F is additive, then F ( ) = F () F ( ). The second type o unctors we deine preserves distinuished trianles. Deinition 3.3. Let C and C be two trianulated cateories with Σ and Σ their respective equivalences. An additive unctor F : C C is called a unctor o trianulated cateories i F Σ = Σ F and F maps distinuished trianles to distinuished trianles. The last type o unctors does not preserve trianles, but transorms them into exact sequences. However, as it can be arued that the distinuished trianles in a trianulated cateory serve a unction analoous to that o exact sequences in abelian cateories, we could see these unctors as preservin this role.

30 2 3 LOCALIZATION OF A TRIANGULATED CATEGOR Deinition 3.4. Let C be a trianulated cateory, and let A be an abelian cateory. A unctor F : C A is called a cohomoloical unctor i or any distinuished trianle in C, the sequence is exact in A. Z Σ() F () F () F ( ) F () F (Z) Note that due to T4, we actually et a lon exact sequence F (Z[ 1]) F () F ( ) F (Z) F ([1]) We now ive a classic example o a cohomoloical unctor, which leads to a rather useul theorem. Example 3.5. Let C be a trianulated cateory, let W Ob(C), and consider the unctor Hom C (W, ). As C is trianulated, it is also additive, so its Hom-sets are abelian roups. Thus Hom C (W, ) is a unctor into the cateory o abelian roups, Ab, which is certainly an abelian cateory. Let Z Σ() be a distinuished trianle in C. We need to study the ollowin sequence: As the square Hom C (W, ) Hom C(W,) Hom C (W, ) Hom C(W,) Hom C (W, Z) id id commutes, it can be embedded in a morphism o trianles as ollows: id Σ() id Z Σ() id Σ() Thus =, and also Hom C (W, ) Hom C (W, ) =, so Im(Hom C (W, )) Ker(Hom C (W, )). Since we are workin in Ab, we can prove the reverse inclusion just by lookin at elements. Let φ Ker(Hom C (W, )) Hom C (W, ); then φ =. Since the ollowin square commutes, W φ

31 3.1 Trianulated cateories 21 it can be embedded in a morphism o trianles as ollows: W id W W Σ(W ) ψ φ Z Σ(ψ) Σ() As φ = ψ Im(Hom C (W, )), it ollows that Ker(Hom C (W, )) Im(Hom C (W, )), and Hom C (W, ) is a cohomoloical unctor. It can be shown that Hom C (, W ) is a cohomoloical unctor as well. The ollowin theorem turns our attention back to the internal workins o the trianulated cateory. Theorem 3.6. Let Z Σ() φ ψ θ Σ(φ) Z Σ( ) be a morphism o trianles in a trianulated cateory C. I φ and ψ are isomorphisms, then so is θ. Proo. As we know that Hom(Z, ) is cohomoloical, we also know that the ollowin commutative diaram has exact rows: Hom(Z, ) Hom(Z, ) Hom(Z, Z) Hom(Z, Σ()) Hom(Z, Σ( )) Hom(Z,φ) Hom(Z,ψ) Hom(Z,θ) Hom(Z,Σ(φ)) Hom(Z,Σ(ψ)) Hom(Z, ) Hom(Z, ) Hom(Z, Z ) Hom(Z, Σ( )) Hom(Z, Σ( )) By the ive lemma (lemma A.1), the act that Hom(Z, φ), Hom(Z, ψ), Hom(Z, Σ(φ)) and Hom(Z, Σ(ψ)) are isomorphisms implies that Hom(Z, θ) is an isomorphism. Particularly, is is surjective (and Hom C (, ) is a set), so there exists a morphism u Hom(Z, Z) so that θu = id Z Hom(Z, Z ). Unortunately, so ar we only know that u is a riht inverse o θ. However, i we redo the above proo with the ollowin morphism o trianles Z Σ( ) φ 1 ψ 1 Z Σ() u Σ(φ) 1 we see that u has a riht inverse v so that uv = id Z. Since θ is a let inverse o u it ollows that u is an isomorphism, which means that v = θ, and so θ is an isomorphism.

32 22 3 LOCALIZATION OF A TRIANGULATED CATEGOR It may seem cumbersome to use cohomoloical unctors to prove a property that is intrinsic to trianulated cateories, but the only thin it is used or is to show a sequence o homomorphism roups to be exact. We could have shown the result without usin the cohomoloical property but the proo would not have been as eleant. 3.2 Localization by a null system So ar, when we have studied localization it has been done by a class o morphisms (typically called S). However, we miht preer to consider a class o objects instead, and by the process o localization set the objects isomorphic to the zero object. The name or such a class is a null system. Deinition 3.7. Let C be a cateory trianulated with respect to the equivalence Σ : C C. A subclass N o Ob(C) is called a null system i: N1 N. N2 N i and only i Σ() N. N3 I, N and Z Σ() is a distinuished trianle, then Z N. Havin a null system, we can deine a multiplicative system. Theorem 3.8. Let N be a null system in the trianulated cateory C. Deine the class o morphisms S(N) by S(N) = { : a distinuished trianle Z Σ() with Z S}. Then S(N) is a multiplicative system. Proo. We will prove the theorem by checkin the conditions o deinition 2.3 in order. S1 For any Ob C we know that id can be embedded in a distinuished trianle id Σ(). As N, it ollows that id S(N) S2 Suppose : and : Z are both elements in S(N), in other words that there exists distinuished trianles Z Σ() and Z Σ( ) with, Z N. From axiom T3 o trianulated cateories we know that there exists a distinuished trianle Z Σ(). Then by T6 there must exist a distinuished trianle Z Σ(Z ). Applyin T4 twice we see that the trianle Σ(Z ) Σ( ) Σ( ) is distinuished. Then we know that Σ( ) N and thus we have N. It ollows that S(N).

33 3.2 Localization by a null system 23 S3 Suppose there exist morphisms : and x : Z with S(N). Then we know that there exists a distinuished trianle k Z Σ(), with Z N. From T3 and T4 it ollows that there exists a distinuished trianle W Z kx Z Σ(W ). As the ollowin diaram is commutative Z kx Z x k Z id Z it can be embedded in a morphism o distinuished trianles in the ollowin manner: Z kx Z Σ(W ) Σ() Σ(Z) x id Z Σ(y) Σ(x) k Z Σ() Σ() Σ( ) Usin axiom T4 to shit the diaram, we see that this ensures that the ollowin is a morphism o distinuished trianles: W Z kx Z Σ(W ) y x k Z Σ() Since Z N we have that S(N), and the axiom is satisied. S4 First note that in an additive cateory this axiom is equivalent to sayin that i s = with s S, then there exists t S such that t = (and vice versa). Thereore we suppose that or a morphism : Z there exists an s S(N) such that s =. Then, by the deinition o S(N) and T4 there exists a distinuished trianle Z s Σ(Z), where Z N. As the diaram s commutes, it can be embedded in a morphism o distinuished trianles as ollows: id Σ() h Z s Σ(Z)

34 24 3 LOCALIZATION OF A TRIANGULATED CATEGOR By T3 and T4, there exists a distinuished trianle W t h Z Σ(W ). Note that since Z N we must have t S(N). Knowin that the diaram h Z id commutes, we embed it in the ollowin morphism o trianles: W t h Z Σ(U) id It ollows that t =. The reverse implication is proved in a hihly similar manner. To know that we can deine a multiplicative system S(N) rom the null system N is not as interestin as to know that we have ound the riht one; in other words that localization by S(N) preserves trianulation (since C is trianulated) and takes elements o N to zero. Perhaps most importantly, we want to check that the localization is universal. Theorem 3.9. Let C be a trianulated cateory and N a null system in C. Then we know that (i) The localization o C by S(N), written C/N, is trianulated. Moreover the localization unctor F : C C/N is a unctor between trianulated cateories. (ii) F () = or any N. (iii) The localization unctor is universal amon trianulated unctors that have the above property. Proo. (i) Suppose C is trianulated with respect to the equivalence unctor Σ. By theorem 2.9, the cateory C/N must be additive. We deine the unctor Σ C/N : C/N C/N by settin Σ C/N () = Σ() or an object Ob(C/N) = Ob(C), and settin Σ C/N (, s, ) = (Σ(), Σ(s), Σ()) or a morphism (, s, ) : Z. To show that the latter is well-deined note irst that since Σ is a unctor we know that Σ C/N (, s, ) is a morphism Σ C/N ( ) Σ C/N (Z). It remains to check that Σ C/N will map two equivalent morphisms to the same equivalence class. Since Σ

35 3.2 Localization by a null system 25 is a unctor, i this diaram commutes U u t s v Z then so does this: Σ(u) Σ(U) Σ(v) Σ() Σ() Σ(t) Σ(s) Σ( ) Σ() Σ( ) Σ(Z) The latter ives an equivalence between the imae o two equivalent morphisms. Thus Σ C/N is well-deined as a map. That it is a unctor and urthermore an equivalence o cateories ollows directly rom the that act Σ is. Moreover we see that F Σ = Σ C/N F. We will denote Σ C/N as simply Σ when there is no risk o conusion. Let a trianle in C/N be distinuished i and only i it is isomorphic to the imae o a distinuished trianle in C. We check that the trianle axioms are satisied: T1 Follows rom the deinition o distinuished trianles in C/N. T2 Suppose Ob(C/N) = Ob(C). The trianle (,id,id ) Σ() is isomorphic (by the identity isomorphism) to F () F (id ) F () Σ() and is thus distinuished. T3 Suppose (U, s, ) : is a morphism in C/N. We know that (U, s, id U ) is an isomorphism (because (U, id U, s) is its inverse). Consider the ollowin morphism o trianles: (U,s,) Z Σ() (U,s,id U ) U (U,idU,) id id Z Z Σ(U) Σ(U,s,id U ) As the diaram shows an isomorphism o trianles, and the lower trianle is the imae o a distinuished trianle (the embeddin o in a trianle), the upper trianle is distinuished as well.

36 26 3 LOCALIZATION OF A TRIANGULATED CATEGOR T4 Suppose φ ψ Z θ Σ() is a distinuished trianle in C/N. This is true i and only i it is isomorphic to the imae o a distinuished trianle φ ψ Z θ Σ( ) in C. This trianle is distinuished i and only i ψ Z θ Σ( ) Σ(φ ) Σ( ) is distinuished. However, the imae o this trianle is isomorphic to ψ Z θ Σ() Σ(φ) Σ( ), which must be distinuished. T5 and T6 ollows rom imae chasin proos similar to the one or T4. We have already seen that F Σ = Σ C/N F, and the imae o a distinuished trianle in C is obviously distinuished, so F is a unctor between trianulated cateories. (ii) Consider N. The trianle id Σ() is distinuished in C. So must Σ() Σ() be. Thus the morphism is in S(N). Its imae will be an isomorphism in C/N and thus F () =. (iii) Suppose G : C C is a unctor between trianulated cateories so that G() = or all N. Suppose S(N), then there exists a distinuished trianle Z Σ() in C where Z N. The unctor G maps this to the distinuished trianle G() G() G( ) Σ(G()) (we have used the act that G(Z) = and GΣ = Σ G). It is trivial to see that the ollowin square commutes: G( ) id G( ) G( ) Furthermore we can embed it in a morphism o distinuished trianles in C as ollows: G( ) id G( ) G( ) Σ G( ) id G( ) G() G() G( ) Σ G() G() = id, so is a let-sided inverse o G(). Symmetrically we can ind the ollowin morphism o distinuished trianles: G() G() G( ) Σ G() id G() G() id G() G() Σ G( ) which ives, a riht sided inverse o G(). Thus G() is an isomorphism.

37 27 Since we have shown that G takes elements o S(N) to isomorphisms we know, by the universality o F, that there exists a unctor H : C/N C so that G = HF. 4 Derived cateories In [8], we started out studyin abelian cateories, and moved on to cateories o complexes and the homotopy cateory. This was done in order to deine the derived cateory, which we will do in this section. The plan or this construction can be summarized as ollows: C C(C) K(C) D(C) We start in an abelian cateory C, and orm the cateory o complexes, C(C), where the objects are lon sequences o the orm : n 1 dn 1 n dn n+1 with d n dn 1 = or all n. We constructed the homotopy cateory K(C) by creatin an equivalence relation on the Hom-sets o C(C) where i and only i was homotopic to. Finally, we studied the homotopy unctor H n : K(C) C, deined by settin H n () = Ker(d n )/ Im(dn ). Now we will pick up this thread and use H n to deine a multiplicative system in K(C). Localizin by this system will ive us D(C). However, we would like to use a null system or this purpose, so we have to start by showin the homotopy cateory to be trianulated. 4.1 Trianulation o K(C) Let C be an abelian cateory. We will now show that the homotopy cateory K(C) (which is additive) is trianulated. Let Σ : K(C) K(C) be the map takin the complex K(C) to the complex Σ() where Σ() n = n 1 and d n Σ() = dn 1. The map works on morphisms o complexes by shitin their respective elements in the same manner; i φ Hom K(C) (, ) then we set {Σ() n Σ(φ)n Σ( ) n } = { n+1 Σ(φ)n n+1 } = { n+1 φn+1 n+1 }. To check that this is well-deined in the homotopy cateory, it is enouh to check that i a map φ : is homotopic to zero, say by the collection (p n ) n Z, then Σ(φ) is homotopic to zero as well. This is shown by the act that Σ(φ) n = φ n+1 = d n p n+1 + p n+2 d n+1 = dn 1 Σ( ) qn + q n+1 d n Σ(), where q n = p n+1. It is easy to see that Σ is a unctor and an equivalence o cateories. We call this unctor the shit unctor, and typically write Σ() as [1]. Extendin the notation, we set [n] = Σ n [] or a natural number n.

38 28 4 DERIVED CATEGORIES This ives us the required suspension unctor on K(C), but to proceed with deinin its trianles we also need the notion o a mappin cone. Let φ : be a morphism o complexes; ( its mappin cone is the complex M() where M() n = n+1 n and d n M() = d n+1 ). As φ n+1 d n d n+1 Z dn Z = ( = ( d n+2 φ n+2 d n+1 φ n+2 d n+1 ) ( d n+1 φ n+1 d n ) d n+2 dn+1 + dn+1 φ n+1 d n+1 d n ) = ( ), the mappin cone M() is a complex. We deine maps α M() and M() β [1] by settin α n = ( ) id n and β n = ( id n+1 ). These maps are well-deined in K(C). I is homotopic to, it has the same domain and rane as, so M() = M(), α = α and β = β. Theorem 4.1. [9, 1.4.4] Let the distinuished trianles o K(C) be the trianles isomorphic to a trianle α M() β [1] or some, Ob(K(C)) and some Hom K(C) (, ). The cateory K(C) is trianulated with respect to this amily o distinuished trianles. Proo. We consider the conditions o deinition 3.1, but out o order. T1 Follows rom the deinition o the distinuished trianles in K(C). T3 Follows rom the deinition o the distinuished trianles in K(C). T4 We will only show one direction. Suppose that the trianle Z h [1] is distinuished. We want to show that the trianle Z h [1] [1] [1] is distinuished as well. We can assume that Z n = M n () as the trianle is isomorphic to a trianle where this is true. We know that there exists a distinuished trianle Z α M() β [1], and that by deinition M() n = n+1 Z n = n+1 n+1 n. We deine the map φ : [1] M() by φ n = ( n+1 id n+1 ), and the map ψ : M() [1] by ψ n = ( id n+1 ). It can be checked that φ and ψ are morphisms o complexes. We see that ψ n φ n = ( id n+1 ) n+1 id n+1 = id n+1, so the morphisms ψ and φ are at the very least one-sided inverses o each other, but n+1 φ n ψ n = id n+1 ( id n+1 ) n+1 = id n+1

39 4.1 Trianulation o K(C) 29 tells us that we have ot to do work to in order to show that they are two-sided inverses. We know that id n+1 id n M() = id n+1. id n Settin s n = ( id n ), we et s n+1 d n M() + dn 1 id n+1 d n+1 d n = d n+1 id n + d n id n+1 n+1 d n id n n d n 1 id n+1 n+1 d n d n = id n id n+1 n+1 = id n = id n M() φn ψ n, M() sn = which means that φψ is homotopic to id M(), and so φψ = id M() in K(C). This means that φ is an isomorphism (and so is ψ). Consider the ollowin diaram: Z h [1] [1] [1] id Z id Z φ α M() β [1] id [1] It can be checked that the diaram commutes. As each o the downward maps is an isomorphism we see that (id, id Z, φ) is an isomorphism o trianles, and so Z h [1] [1] [1] is distinuished. T2 For K(C), there exists a map. Given that M()=, the trianle id is distinuished, and then by T4 the trianle id [1] is distinuished. T5 As all distinuished trianles are isomorphic to one o the orm α M() β [1], we will assume that we are dealin with two such trianles, namely α M() β [1] and Ȳ α β M( ) [1]. We also assume that

40 3 4 DERIVED CATEGORIES there exists morphisms u : and v : Ȳ so that the ollowin diaram commutes: u As we are workin in K(C), this means that there exist maps s n : n Ȳ n 1 so that v n n n u n = s n+1 d n + dn 1 s n. Deine w : M() M( ) by lettin ( ) Ȳ w n = u n+1 (remember that M() = n+1 n ). We see that s n+1 v n ( d d n n+1 ) ( ) M( ) wn = u n+1 n+1 d n s n+1 v n Ȳ ( d n+1 ) ( = un+1 u n+1 u n+1 + d n Ȳ sn+1 d n = n+2 d n+1 ) Ȳ vn s n+2 d n+1 + vn+1 n+1 v n+1 d n ( ) ( ) u n+2 d n+1 = s n+2 v n+1 n+1 d n = w n+1 d n M(), so w is a morphism o complexes. Moreover, ( ) ( ) ( ) w n α n u = n+1 s n+1 v n = v n and β n w n = ( id n+1 id n Thus the ollowin diaram commutes Ȳ v ( = id n Ȳ ) v n = α n v n ) ( u n+1 ) s n+1 v n = ( u n+1 ) = u n+1 ( id n+1 ) = u n β n. α M() β [1] u Ȳ v w α M( ) β [1] which is what the deinition requires. T6 We assume that there exist distinuished trianles α Z β [1] Z α β [1] Z α Ȳ β [1]

41 4.1 Trianulation o K(C) 31 and that these trianles are so that Z = M() = [1], = M() = [1] Z and Ȳ = M() = [1] Z 5. Let u : Z Ȳ be deined by u n = ( ) v n n+1 = and let w = α id Z [1]β = n commutes: ( ) id n+1 n ( id n+1, let v : Ȳ be deined by ). Then the ollowin diaram α Z β [1] id Z α Ȳ u β [1] id [1] id Z Z α v [1] β [1] α Z u Ȳ α id v w Z[1] α [1] Thus we only need to show that Z u Ȳ v w Z[1] is distinuished. We will do this by indin an isomorphism φ : M(u), because then the ollowin diaram will commute, Z u Ȳ α u M(u) βu Z[1] id Z Z u Ȳ id Ȳ v φ w Z[1] id Z[1] (id Z, id, φ) will be an isomorphism o trianles and Z Ȳ u Ȳ v w Z[1] is isomorphic to a distinuished trianle. We start by noticin that that n = n+1 Z n and M(u) n = Z n+1 Ȳ n = n+2 n+1 n+1 Z n, and deine φ by settin φ n = ( id n+1 n+1 ) id n Z We check that φ is a morphism o complexes: 5 This can once aain be assumed without loss o enerality because any distinuished trianle is isomorphic to a trianle o the mappin cone sort.