Around Quillens theorem A

AR OUND QUILLEN’S THEOR EM A BRU NO KAHN Abstract. W e int ro duce a notion of cellular functor. It allows us to give a v arian t of Quillen’s pr o of of the Solomon-Tits theorem, which does not use Theorem A. W e also use it to gener alise some exact sequences of Quillen, and to reformulate them into a rank sp ectral sequence conv erging t o the homolog y of the K ′ -theory space of an in tegral sc heme. In the memory o f Daniel Quillen Contents In tro duction 1 1. Nerv es with co efficien ts 3 2. A long homology exact sequence 7 3. Cellular functors and the So lomon-Tits theorem 13 4. The rank sp ectral sequence 15 References 18 Intr oduction Let A b e a Dedekind domain. Insp ection sho ws immediately that the exact sequences of [12, Th. 3], used b y Quillen to pro v e that the K -groups of A are finitely generated when A is a ring of S -in tegers in a global field, assem ble to define an exact couple, hence a sp ectral sequence conv erging to the homology of K A . This giv es p oten t ia lly more pow er to Quillen’s metho d, whic h as suc h yields no information on the ranks of these K -g roups. A natural wa y t o imagine suc h a sp ectral se quence is to consider the maps B Q n − 1 P ( A ) → B Q n P ( A ) Date : June 2014. 2010 Mathematics Subje ct Classific ation. 19D50, 55U99. 1 2 BRUNO KAHN used b y Quillen as homotopy c ofibr ations rather than homotop y fibra- tions. The resulting rank sp ectral sequence coincides with the ab ov e- men tioned sp ectral sequence: this is not immediately ob vious, but fol- lo ws from a simple argument of V ogel, see Remark 4.3.4. The aim of this note is to construct the rank sp ectral sequence in a w ay as functorial as p o ssible. The t wo op erativ e ingredients are Thoma- son’s theorem on the nerv e of a Grothendiec k construction (Theorem 1.4.3) and the n otion of c el lular functor (Definition 2 .3.2), whic h is w ell-adapted to the presen t con text thanks to Theorem 2.3 .6 . After the first v ersion of this pap er w a s written, F ei Sun observ ed that The- orem 2.3 .6 may also b e used to reco v er part of Quillen’s pro of of the Solomon-Tits theorem in [12]. W e complete Sun’s remark in § 3, by co vering the missing part: unlik e in Quillen’s argumen t, Theorem A is not used there. In this light, one might think of Theorem 2.3.6 as “dual” to Theorem A, p oten tially pla ying a similar rˆ ole fo r homotop y cofibrations as Theorem A plays for homotop y fibrations. The main theorem is Theorem 4.3.3: w e apply the theory in the sligh tly more g eneral case o f torsion-free sheav es ov er an integral sc heme, whic h migh t b e useful elsewhe re. The next step is to compute the d 1 differen tials of this rank sp ec- tral sequence. This has b een done by F e i Sun in his thesis, using the universal mo dular symb o l s of Ash-Rudolph [1]: see [9] and [13]. Ac kno wledgemen t s. I thank Pierre V ogel f or ex plaining me the ar- gumen t of Remark 4.3.4 in 2008, and Georges Maltsiniotis for p oin t ing out Thomason’s paper [14]. I also thank F e i Sun for sev eral dis cussions around these ques tions and Jo¨ el Riou for helpful commen ts, esp ecially for p o in ting out a wrong axiomatisation of Propo sition 3.1.2 in a pre- vious v ersion of this man uscript. I don’t think Theorem A is used explicitly an ywhere. Y et I feel its spirit is prev alen t in this text, hence the title. Notation. W e denote b y Set , Ab , sSet , Cat the category of (small) sets, ab elian groups, simplicial sets, categories. F or n ≥ 0 , [ n ] denotes the totally or dered set { 0 , . . . , n } , considered as a sm all category . W e write ∗ fo r the catego ry with one ob ject a nd one morphism (sometimes for the set with 1 elemen t). Finally , ∆ denotes the catego r y of simplices (ob jects: finite nonempt y ordinals, morphisms: non-decreasing maps). W e shall use Mac Lane’s comma notatio n [10, p. 47]: if w e hav e a diagram of functors C F − → D F ′ ← − C ′ AROUND QUILLEN’S THEOREM A 3 the comma category F ↓ F ′ has f or ob jects the diag ram F ( c ) → F ′ ( c ′ ), and for morphisms the obvious commutativ e diagrams. W e use the follo wing abbreviations: if c ∈ C , yielding the functor F c : ∗ → C , w e write F c ↓ F ′ = c ↓ F ′ ; similarly o n the r igh t . 1. Ner ves with coefficients Subsections 1.1 – 1.3 can essen tially b e fo und in Go erss-Jardine [6, Ch. IV]. 1.1. Set-v alued co efficien ts. 1.1.1 . Let D ∈ Cat b e a small category . The ne rve of D is the simplicial set N ( D ) with N n ( D ) = O b Cat ([ n ] , D ) = a d 0 →···→ d n ∗ , d i ∈ D cf. [5 , I I, 4.1]. 1.1.2 . Let D ∈ Cat and let F : D → Set b e a cov arian t functor. The nerve of D with c o effici e n ts in F is the simplic ial set N ( D , F ) with N n ( D , F ) = a d 0 →···→ d n F ( d 0 ) , d i ∈ D cf. [5, App. I I, 3.2]. F or F the constan t functor with v alue ∗ , w e reco ve r the nerv e of D . 1.1.3 . Let ( D , F ) be as in 1.1.2. W e hav e the asso ciated category [ D , F ] = { ( d, x ) | d ∈ D , x ∈ F ( d ) } where a morphism ( d, x ) → ( d ′ , x ′ ) is a morphism f ∈ D ( d , d ′ ) suc h that F ( f )( x ) = x ′ . This category has tw o other equiv alen t descriptions: (1) [5, I I, 1.1 ] Let y = D op → C at ( D , Set ) b e the “coY oneda” em b edding: then [ D , F ] ≃ y ↓ F , where F is considered as an ob ject o f Cat ( D , Set ). (2) [ D , F ] ≃ ∗ ↓ F , where ∗ ∈ Cat ( D , Set ) is the constan t functor with v alue ∗ and F is considered as a functor. The following lemma is ob vious: 1.1.4. Lemma. T her e is a c anonic al isomo rp hism: N ( D , F ) ≃ N ([ D , F ]) . 4 BRUNO KAHN 1.2. Ab elian group-v alued co efficien ts. 1.2.1 . Supp ose that F tak es its v alues in the category Ab of ab elian groups. Then N ( D , F ) is a simplicial ab elian group; it has homolo gy gr oups [5, App. I I, 3.2 , p. 153] H i ( D , F ) = H i ([ N ( D , F )]) = π i ( N ( D , F ); 0) where [ X ] is the c hain complex asso ciated to a simplicial ab elian group X by taking for differen tials the alternating sums of the f aces. 1.2.2 . Let us recall the Eilen b erg-Zilb er–Cartier t heorem as exp ounded in [7, p. 7] (see also [4, 2.9 and 2.16]). T o a bisimplicial ab elian group X , o ne ma y asso ciate a double complex [ X ] as in the simplicial case. T o a bisimplicial ob ject X o ne ma y a sso ciate the diag o nal simplicial ob ject δ X : ( δ X ) n = X n,n and to a double complex C one may asso ciate the total complex T ot C : (T ot C ) n = M p + q = n C p,q . Then, giv en the diagram of functors ( ∆ × ∆ ) op Ab [ ] − − − → C ++ ( Ab ) δ   y T ot   y ∆ op Ab [ ] − − − → C + ( Ab ) there exist tw o natural transformations sh uffle X : T ot[ X ] → [ δ X ] Alexander-Whitney X :[ δ X ] → T ot[ X ] whic h are quas i-in v erse homotop y equiv alences. 1.3. Simplicial set-v alued c o efficien ts. 1.3.1 . If X is a simplicial set, w e ma y ass o ciate to it the free sim plicial ab elian group Z X generated by X , with ( Z X ) n = Z X n . Similarly with a bisimplicial set. The homolo gy o f X is the homotop y of Z X , or equiv alen tly the homology of [ Z X ]. Similarly with co efficien ts in an ab elian group A , using Z X ⊗ A . W e shall usually write [ Z X ] =: C ∗ ( X ); if X = N ( D ) for a category D , w e abbreviate C ∗ ( X ) into C ∗ ( D ). AROUND QUILLEN’S THEOREM A 5 1.3.2 . Let D ∈ C at and let F : D → sSet b e a functor. W e ma y generalise the definition of 1 .1.2 to get a bisim p licial set N ( D , F ): N p,q ( D , F ) = a d 0 →···→ d p F q ( d 0 ) , d i ∈ D . W e ma y then t ak e the diagonal δ N ( D , F ), whic h is a simplicial set. W e define π i ( D , F ; ( d 0 , x 0 )) = π i ( δ N ( D , F ); ( d 0 , x 0 )) C ∗ ( D , F ; A ) = C ∗ ( δ N ( D , F )) ⊗ A H i ( D , F ; A ) = H i ( C ∗ ( D , F ; A )) for d 0 ∈ D 0 and x 0 ∈ F 0 ( d 0 ) a c hosen base p oint, and for A a n ab elian group of co efficien ts. 1.3.3. Lemma. a) L et X = ( X p,q ) , Y = ( Y p,q ) b e two bisimplic ial sets and ϕ : X → Y a bis i m plicial map. Supp ose that, for e ac h p ≥ 0 , ϕ p, ∗ : X p, ∗ → Y p, ∗ is a we ak e quivale nc e. Th e n δ ϕ : δ X → δ Y is a we ak e quivalenc e. b) L et F , G : D → sSet b e two functors, and let ϕ : F → G b e a morphism of functors. Supp ose that, for e ach d ∈ D , ϕ ( d ) : F ( d ) → G ( d ) is a we ak e quivalenc e. Then δ N ( D , ϕ ) : δ N ( D , F ) → δ N ( D , G ) is a we ak e q uiva l e nc e. Pr o of. a) is well-kno wn (for example, it is the sp ecial c ase of the Bous- field-F riedlander t heorem of [6, Th. 4.9 p. 2 29] where Y = W = ∗ ), and b) follows from a).  1.3.4 . F or a n ab elian group A and for q ≥ 0, w e may consider the additiv e functor H q ( F , A ) : D → Ab d 7→ H q ( F ( d ) , A ) . This giv es a meaning to: 1.3.5. Lemma. Ther e is a sp e ctr al se quenc e E 2 p,q = H p ( D , H q ( F , A )) ⇒ H p + q ( D , F ; A ) Pr o of. W e shall use 1.2 .2: it implies that [ δ Z N ( D , F )] is homotopy equiv alen t to T ot[ Z N ( D , F )]. Therefore H ∗ ( D , F ; A ) := H ∗ ([ δ Z N ( D , F )] ⊗ A ) ≃ H ∗ (T ot[ Z N ( D , F )] ⊗ A ) = H ∗ (T ot M p,q M d 0 →···→ d p [ Z F q ( d 0 )] ⊗ A ) . 6 BRUNO KAHN Consider t he first sp ectral sequence asso ciated to this double complex in Cartan-Eilen b erg [3, Ch. XV, § 6]: the formula (1) of lo c. cit, p. 3 31 sho ws that it is the desired sp ectral sequence.  1.4. Category-v alued co efficien ts. (See a lso [15, IV.3].) 1.4.1 . Let D ∈ C at and let F : D → Cat b e a functor. Comp osing F with the nerv e functor, w e get a functor N ◦ F : D → sSet , hence a bisimplicial set as in 1 .3.2 N ( D , F ) = N ( D , N ( F )) . 1.4.2 . W e no w extend the construction in 1.1.3. This yields the cate- gory D R F (Grothendiec k construction, [SGA1, Exp. VI, §§ 8,9]) : • Ob jects are pairs ( d, x ), d ∈ O b ( D ) , x ∈ O b ( F ( d )). • F or tw o ob jects ( d, x ) , ( d ′ , x ′ ), a mor phism ( d, x ) → ( d ′ , x ′ ) is a morphism f : d → d ′ and a morphism g : F ( f )( x ) → x ′ . • F or three o b jects ( d, x ) , ( d ′ , x ′ ) , ( d ′′ , x ′′ ) and tw o morphisms ( f , g ) : ( d, x ) → ( d ′ , x ′ ), ( f ′ , g ′ ) : ( d ′ , x ′ ) → ( d ′′ , x ′′ ), ( f ′ , g ′ ) ◦ ( f , g ) := ( f ′ ◦ f , g ′ ◦ F ( f ′ )( g )). The G rothendiec k construction is cov ariant in F . In particular, there is a canonical functor D R F → D induced by the morphism F → ∗ , where ∗ is the cons tan t functor with v alue the p oint catego ry . W e call this functor the augme ntation . Lemma 1.1.4 then generalise s as 1.4.3. Theorem (Thomason [14, Th. 1-2]) . Ther e i s a c a n onic al we ak e quivalenc e δ N ( D , F ) → N ( D Z F ) sending a c el l ( d 0 f 1 − → . . . f n − → d n , x 0 g 1 − → . . . g n − → x n ) ( x i ∈ F ( d 0 ) ) to the c el l  ( d 0 , y 0 ) h 1 − → . . . h n − → ( d n , y n )  ( y i ∈ F ( d i )) with y i := F ( f i . . . f 1 ) x i and h i = ( f i , F ( f i . . . f 1 ) g i ) .  1.4.4 . Let T : C → D b e a f unctor b etw een t w o small categories. T o T w e asso ciate the functor F T : D → Cat sending d to T ↓ d . The category D R F T ma y be identifie d with the comma category T ↓ I d D . Therefore there are 3 functors: p 1 : D Z F T → C , p 2 : D Z F T → D , s : C → D Z F T with p 1 ([ T ( c ) ϕ − → d ]) = c, p 2 ([ T ( c ) ϕ − → d ]) = d, s ( c ) = [ T ( c ) = T ( c )] . AROUND QUILLEN’S THEOREM A 7 W e ha ve : 1.4.5. Lemma. s i s left adjoint to p 1 . Henc e s and p 1 induc e quasi- inverse homotopy e quivalenc es N ( C ) ≃ N ( D R F T ) .  F rom Theorem 1.4.3 and Lemma 1.3.5, w e deduce 1.4.6. Corollary (cf. [5 , App. I I]) . The r e is a sp e c tr al se quenc e E 2 p,q = H p ( D , H q ( F T , A )) ⇒ H p + q ( C , A ) for a n y ab elian gr oup A .  2. A long homology exact sequence 2.1. Sp ectral sequences and exact couples. 2.1.1 . Let T b e a triangula t ed category with coun table direct sums, and let C 0 i 1 − → . . . i n − → C n i n +1 − → . . . b e a sequence of ob jects of T . Let C b e a homotopy colimit (mapping telescop e) of the C n [2]. Let H : T → A b e a (co)homological functor to some ab elian categor y A : we assume that H commutes with countable direc t sums. (Alternately , w e could refuse infinite direct sums and assume that i n is an isomorphism for n large e nough.) T o g et an asso ciated spectral sequence, the simplest is the tec hnique of exact couples [8 , pp. 152–153]: for each n , c ho ose a cone C n/n − 1 of f n , so that the exact t r ia ngles C p − 1 i p − → C p j p − → C p/p − 1 k p − → C p − 1 [1] yield long homology exact se quences . . . H n ( C p − 1 ) i p,n − → H n ( C p ) j p,n − → H n ( C p/p − 1 ) k p,n − → H n − 1 ( C p − 1 ) . . . where H n ( X ) := H ( X [ − n ]). The exact couple defined by D p,q = H p + q ( C p ) , E p,q = H p + q ( C p/p − 1 ) and the relev ant i, j, k defi ne a sp ectral sequence abutting to H p + q ( C ). The E 1 -terms of this sp ectral sequence a re simply E 1 p,q = E p,q , and d 1 p,q = j p − 1 ,p + q − 1 k p,p + q is induced b y j p − 1 [1] k p . Here is a more concrete desc ription o f this differen tial: 8 BRUNO KAHN 2.1.2. Lemma. L et C p/p − 2 b e a c one for i p i p − 1 , so that we may obtain a c om mutative diagr am C p − 2 C p − 2 i p − 1   y i p i p − 1   y C p − 1 i p − − − → C p j p − − − → C p/p − 1 k p − − − → C p − 1 [1] j p − 1   y   y =   y j p − 1 [1]   y C p − 1 /p − 2 ¯ i p − − − → C p/p − 2 ¯ j p − − − → C p/p − 1 ¯ k p − − − → C p − 1 /p − 2 [1] of exact triangles (f r om the suitable axiom of triangulate d c ate g ories). Then d 1 p,q is the b ound a ry map ¯ k p,n in the lo n g e x act se quenc e . . . H p + q ( C p − 1 /p − 2 ) ¯ i p,n − → H p + q ( C p/p − 2 ) ¯ j p,n − → H p + q ( C p/p − 1 ) ¯ k p,n − → H p + q − 1 ( C p − 1 /p − 2 ) . . . Pr o of. The diagram show s that ¯ k p = j p − 1 [1] ◦ k p .  2.1.3 . In the usual case of a filtered complex C p = F p C , we may of course c ho ose C p/p − 1 = F p C / F p − 1 C and C p/p − 2 = F p C / F p − 2 C . 2.1.4 . Let Q 0 T 1 − → Q 1 T 2 − → · · · → Q n → . . . b e a sequence of cat- egories and f unctors. Let Q = lim − → Q n . (Sinc e there are no natural transformations in v olv ed this is a na ¨ ıv e colimit, defined ob ject wise and morphism wise.) Considering the corresp onding sequence of chain com- plexes of nerv es C ∗ ( Q 0 ) i 1 − → C ∗ ( Q 1 ) . . . w e get from the y o g a of 2 .1.1 a spectral sequence abutting to H ∗ ( Q ) (p ossibly with co efficien ts). If the functors T n are faithful and injectiv e on ob jects, the maps i n are injectiv e and w e a re in the simpler situation of 2.1.3. 2.2. Unreduced and reduced homology. 2.2.1 . Let ( X , x ) b e a p ointed simplicial set. The r e duc e d homolo gy of X with co efficien ts in an ab elian group A is ˜ H i ( X , A ) = H i ( X , x ; A ) := Cok er( H i ( x, A ) → H i ( X , A )) (= H i ( X , A ) if i 6 = 0) . This definition apparen t ly dep ends on the c hoice of x : if w e don’t w ant to c ho o se a basep oint, w e may alternately define ˜ H i ( X , A ) = Ker( H i ( X , A ) → H i ( ∗ , A )) . AROUND QUILLEN’S THEOREM A 9 An y c hoice of x ∈ X 0 will split the map X → ∗ , realising ˜ H i ( X , A ) as the ab ov e-described direct summand of H i ( X , A ). A homotopy cofibre seq uence X → Y → Z is equiv alen t to a homo- top y co cartesian square X − − − → Y   y   y ∗ − − − → Z hence t he corresp o nding long exact homolo gy sequen ce may b e written (via Ma ye r-Vietoris!) as . . . H i ( X , A ) → H i ( Y , A ) → ˜ H i ( Z , A ) → H i − 1 ( X , A ) → . . . 2.2.2 . Supp ose F : D → sSet is a functor; supp ose that F ( d ) 6 = ∅ for an y d ∈ D . W e then define C ∗ ( D , ˜ F ; A ) = Ker ( C ∗ ( D , F ) → C ∗ ( D )) ⊗ A where the last map is induced b y the natural tra nsformation F → ∗ , and H i ( D , ˜ F ; A ) = H i ( C ∗ ( D , ˜ F ; A )) . (Here the map F → ∗ is not necessarily split, so w e hav e to b e more careful. W e think of ˜ F as a desuspension of the homotop y cofibre of F → ∗ .) More generally , if F → G is a morphis m of functors, w e define C ∗ ( D , F → G ; A ) = cone ( C ∗ ( D , F ) → C ∗ ( D , G )) [ − 1] . If G = ∗ , then C ∗ ( D , F → ∗ ; A ) is homotopy equiv alen t to C ∗ ( D , ˜ F ; A ) under the nonemptiness assumption on F . 2.2.3 . Supp ose F : D → Cat is a functor. W e ha ve the pro jection F → ∗ , where ∗ is the constan t functor with v alues the category with 1 onject and 1 morphism. As in 2 .2.2 w e define C ∗ ( D , ˜ F ; A ) = Ker ( C ∗ ( D , F ) → C ∗ ( C )) ⊗ A H i ( D , ˜ F ; A ) = H i ( C ∗ ( D , ˜ F ; A )) . pro vided F ( d ) 6 = ∅ for an y d ∈ D . In general, we define H ∗ ( D , ˜ F ; A ) as the homology of C ∗ ( D , F → ∗ ; A ) as b efore. 2.3. Cellular functors. 2.3.1. Lemma. L et F T b e as in 1.4.4. Supp ose T ful ly faithful. Then, for a n y c ∈ C , F T ( T ( c )) has a final obje ct. Pr o of. Suc h a final ob ject is give n by [ T ( c ) = T ( c )].  10 BRUNO KAHN 2.3.2. Definition. Let T : C → D b e a functor. W e sa y that T is c el lular if • T is fully fa ithful. • F or an y d ∈ D − C and any c ∈ C , D ( d , c ) = ∅ . If T is cellular, then it defin es a “stratification” D = C ` ( D − C ) in the follow ing sense: 2.3.3. Lemma. L et T : C → D b e a c el lular functor; let F 1 , F 2 : D → E b e two functors and let ϕ : F 1 ⇒ F 2 b e a natur al tr ansformation. Then ther e is a unique factorisation of ϕ as a c o mp osition F 1 ϕ 1 ⇒ F ϕ ϕ 2 ⇒ F 2 such that F ϕ ( d ) = F 2 ( d ) , ϕ 2 ( d ) = 1 F 2 ( d ) for d ∈ D − T ( C ) , an d F ϕ T ( c ) = F 1 T ( c ) , ϕ 1 ( T ( c )) = 1 F 1 T ( c ) for c ∈ C . Pr o of. F o r simplicit y , w e drop the no tation T in this pro o f . Let us define a functor structure on F ϕ as follo ws: if f : d → d is a morphism in D , then three cases ma y o ccur: • d, d ′ ∈ D − C . W e define F ϕ ( f ) a s F 2 ( f ). • d ∈ C , d ′ ∈ D − C . W e define F ϕ ( f ) a s ϕ d F 1 ( f ) = F 2 ( f ) ϕ c . • d, d ′ ∈ C . W e defin e F ϕ ( f ) as F 1 ( f ). Similarly , w e define ϕ 1 ( d ) as ϕ ( d ) for d ∈ D − C and ϕ 2 ( c ) = ϕ ( c ) for c ∈ C . These definitions are the only p ossible o nes if the lemma is to b e correct. Chec king t ha t F ϕ resp ects comp osition of mor phisms and that ϕ 1 , ϕ 2 do define natural tra nsformations is a matter of case-b y-case b o o kke eping.  2.3.4. Pr op osition. L e t T : C → D and ϕ : F 1 ⇒ F 2 b e as in L emma 2.3.3, with E = sSet , and let D − C b e the ful l sub c ate gory of D giv e n by the obje cts n o t in C . Assume that ϕ T ( c ) is a we ak e quivalenc e for any c ∈ C . Then the c ommutative diagr am of bisimplicial se ts N ( D − C , F 1 ) − − − → N ( D , F 1 ) ϕ   y ϕ   y N ( D − C , F 2 ) − − − → N ( D , F 2 ) is h o motopy c o c artesian , i.e. b e c omes so after applying the d i a gonal δ . AROUND QUILLEN’S THEOREM A 11 Pr o of. Applying Lemma 2.3.3, we can enlarge the ab ov e dia g ram as follo ws: (2.1) N ( D − C , F 1 ) − − − → N ( D , F 1 ) ϕ 1   y ϕ 1   y N ( D − C , F ϕ ) − − − → N ( D , F ϕ ) ϕ 2   y ϕ 2   y N ( D − C , F 2 ) − − − → N ( D , F 2 ) It suffices to sho w that the top a nd botto m squares in (2 .1) are b oth homotop y co cartesian. By h ypo thesis ϕ 2 ( d ) is a w eak equiv alence for d ∈ C , and it is tr ivially a w eak equiv alence for d ∈ D − C . Therefore, b y Lemma 1.3.3 b), the t wo vertic al maps in the b otto m square of (2 .1 ) b ecome w eak equiv- alences a f ter applying δ ; a fortiori , this b ot t om square is homoto py co cartesian. On the top right of (2.1), a ty pical term is a d 0 →···→ d p F 1 ( d 0 ) . W e split this copro duct in t w o parts: one is where all the d i are in D − C (call it A 1 ) and t he other is the rest (call it B 1 ). Similarly o n the middle righ t of (2.1 ) (call them A ϕ and B ϕ ). No w the cellularity assumption implies that all cells in B 1 and B ϕ b egin with d 0 ∈ C . But for suc h a d 0 , F 1 ( d 0 ) → F ϕ ( d 0 ) is a bijection. Th us B 1 → B ϕ is bijective . On the other hand, by definition of the cofibre, A 1 and A ϕ b ecome a p oin t in the cofibres. Thus the induced map on cofibres is bijectiv e, and the top square o f (2.1) is homotop y co cartesian as we ll.  2.3.5. Definition. A (naturally) comm utativ e square of categories and functors is homotopy c o c artesia n if it is af t er applying the nerv e func- tor. 2.3.6. Theorem. L et D − C b e the ful l sub c ate g ory of D given by the obje cts not in C . If T is c el lular, the natur al ly c ommutative diagr a m of c ate gories ( D − C ) R F T p − − − → C ε   y T   y D − C ι − − − → D 12 BRUNO KAHN is homotopy c o c artesian, wher e ε is the augmentation (se e § 1.4.2), p is induc e d by the first pr oje ction p 1 of L emma 1.4.5 and ι is the inclusion. Pr o of. Note first the nat ur a l transformation u : T ◦ p ⇒ ι ◦ ε u [ T ( c ) f − → d ] = f whic h explains “natura lly commu tativ e” (here in the w eak se nse). By Theorem 1.4 .3 and Lemma 1.4.5, it suffices to prov e that the (commu- tativ e) diagram of bisimplicial sets N ( D − C , F T ) − − − → N ( D , F T )   y   y N ( D − C ) − − − → N ( D ) is homot op y co cartesian, i.e. b ecomes so after applying the dia g onal δ . In view of Lem ma 2.3.1, this is a special case of Prop osition 2.3.4.  2.3.7. Corollary . L et T : C → D b e a c el lular functor. Then the mapping c o ne of C ∗ ( T ) : C ∗ ( C ) → C ∗ ( D ) is quasi-isom orphic to C ∗ ( D − C , ˜ F T )[1] (cf. 2 . 2 .2). In p articular, we have a long exact s e quenc e · · · → H i ( D − C , ˜ F T ; A ) → H i ( C , A ) → H i ( D , A ) → H i − 1 ( D − C , ˜ F T ; A ) → . . . for a n y ab elian gr oup A . Pr o of. This follo ws from Theorems 2.3.6 a nd 1.4.3.  2.4. Cellular filtrations. 2.4.1. Theorem. L et Q 1 → Q 2 → · · · → Q n → · · · → Q b e a se quenc e of c a te g ories. We assume: • The functors T n : Q n − 1 → Q n ar e c el lular (2.3.2). • Q = lim − → Q n . Write F n for F T n . Then, for any ab elian gr oup A , ther e is a sp e ctr al se quenc e of homolo gic al typ e E 1 p,q = H p + q − 1 ( Q p − Q p − 1 , ˜ F p ; A ) ⇒ H p + q ( Q , A ) . Pr o of. This is t he sp ectral sequenc e of 2.1 .4 , taking Coro lla ry 2.3 .7 in to accoun t.  AROUND QUILLEN’S THEOREM A 13 3. Cellular functors and the Solomon-Tits t he orem 3.1. An abstract v ersion of the Solomon-Tits theorem (for GL n ). This section w as catalysed b y F ei Sun’s insigh t that one can use Theorem 2 .3.6 to understand part of Quillen’s pro o f of the Solomon- Tits theorem in [12]. W e start with an abstraction of his argumen t. 3.1.1. P rop osition ( Sun, essen tially) . L et V b e an p ose t (c onsider e d as a c ate gory), H a subset of maximal elements in V and Y = V − H . F or any W ∈ V , write V W = { X ∈ V | X < W } . The n the natur al ly c ommutative d iagr am a H ∈ H V H j − − − → Y k   y i   y a H ∈ H ∗ l − − − → V is hom otopy c o c artesian. Her e i is the inclusion, j is c omp on e ntwise the in c lusion, k is the tautolo gic al pr oje ction and l is the inclusion of the di s cr ete set ` H ∈ H ∗ = H into V ; a natur al tr ansformation ij ⇒ lk is d e fine d by the ine quality W ≤ H for W ∈ V H . Pr o of. The h yp otheses imply that i is a cellular functor, so it suffices to compute H R F i = ` H ∈ H i ↓ H . Clearly , i ↓ H = V H .  Sun’s second insigh t was that one can replace the simplicial complex Y in the pro of of Quillen’s Claim in [12, § 2] b y t he p o set of it s ve rtices. W e make use of this observ atio n no w. 3.1.2. P rop osition. K e e p the notation of Pr op osi tion 3.1.1. Assume that ther e exists L ∈ V such that (i) H = { W ∈ V | W maximal and L  W } . (ii) F or any W ∈ Y , the supr emum L ∨ W exists in V . Then Y is c ontr actible. Pr o of. It is direc tily inspired b y Quillen’s pro of of his claim ( lo c. cit. ), but a voids the use of Theorem A. F or W ∈ Y , L ∨ W cannot b e in H b y (i), hence ϕ ( X ) = L ∨ X defines a n endofunctor ϕ : Y → Y , and the inequalit y X ≤ L ∨ X defines a natural tra nsformation I d Y ⇒ ϕ . Now ϕ ( Y ) = { X ∈ Y | L ≤ X } 14 BRUNO KAHN has the minimal elemen t L , hence is contractible. Th us ϕ factors through a contractible p oset; since ϕ is homotopic to I d Y , this prov es the claim.  3.1.3. Corollary . Under the assumptions of Pr op osition 3.1.2, ther e is a zi g zag of hom o topy e quivalen c es _ H ∈ H Σ N ( V H ) ∼ − → N ( V ) / N ( Y ) ∼ ← − N ( V ) . Pr o of. F o r an y p oset X , write ¯ X for the p oset X ∪ { + } , where + is a new elemen t larger than all elemen ts in X : th us ¯ X is con tractible. Consider the strictly comm utativ e diagram of functors a H ∈ H V H j − − − → Y i ′   y i   y a H ∈ H ¯ V H ¯  − − − → V ε x   || a H ∈ H ∗ l − − − → V . Here i ′ is comp onent wise the natura l inclusion, ¯  sends V H to itse lf and + H to H while ε sends ∗ H to + H . The latter f unctor has a re- traction ¯ k which extends the functor k of Prop osition 3.1.1. F rom this and the latter proposition, w e deduce that the top sq uare is homot o p y co cartesian. W e ded uce a homotopy equiv alence _ H ∈ H N ( ¯ V H ) / N ( V H ) ∼ − → N ( V ) / N ( Y ) induced b y ¯  . But N ( ¯ V H ) / N ( V H ) is canonically equiv alen t to Σ N ( V H ); the conclusion then fo llows from Prop osition 3.1.2.  3.2. F rom abstract to concrete. G iv en a p oset V , write Simpl( V ) for the abstract simplicial complex asso ciated to V : t he simplices of Simpl( V ) a re b y definition the finite totally ordered subsets of V . Let V b e a finite-dimensional v ector space ov er a [p ossibly sk ew] field k . Its Tits building T ( V ) is [canonically isomorphic to] the flag complex o f V : elem en ts of T ( V ) are flags o f prop er subspaces of V [12, § 2]. It follow s that T ( V ) = Simpl( V ), where V is the p o set of prop er subs paces of V . Thus T ( V ) = ∅ if dim V ≤ 1; w e now assume AROUND QUILLEN’S THEOREM A 15 dim V ≥ 2 . Cho osing a line L ⊂ V , w e get f r o m Corolla ry 3.1.3 Quillen’s decomp osition T ( V ) ∼ _ H ∈ H Σ T ( H ) where H is the set of hyperplanes in V whic h do not con tain L . It follo ws inductiv ely that T ( V ) has the homotop y t yp e of a b ouquet of ( n − 2)-spheres. 4. The rank s pectral seque nce 4.1. K -theory of sc hemes. The first example of application of Theo- rem 2.4.1 is to Quillen’s Q -construction Q ( X ) o n the e xact category of lo cally free shea v es of finite rank ov er a sc heme X . Let Q n = Q n ( X ) b e the full su b category of Q ( X ) consisting o f lo cally free shea v es of r a nk ≤ n . Then the assumptions of Theorem 2 .4.1 are satisfied b ecause, in Q ( X ), there are no morphisms from a lo cally free sheaf of rank n to a lo cally free sheaf of rank < n . The resulting sp ectral sequence ma y b e called the r ank sp e ctr al se quenc e (for the ho mo lo gy of Q ( X )) . Note that Q n − Q n − 1 is a group oid, hence w e get (4.1) E 1 p,q = M E α H p + q − 1 (Aut( E α ) , ˜ F p ) where E α runs thro ugh the set o f isomorphism classes of lo cally free shea v es o f rank p . W e to ok co efficien ts Z , for simplicit y . 4.2. K ′ -theory of in tegral sc hemes. Let X b e an in tegral sc heme, with function field K . If η = Sp ec K is the generic p oint of X , w e ha v e the inclusion j = η → X . If E is a sheaf o f O X -mo dules, w e write E K for j ∗ E . 4.2.1. Definition. a) A coheren t sheaf E on X is torsion-fr e e if the map E → j ∗ E K is a monomorphism. b) A subsheaf E ′ of a coheren t sheaf E is pur e if E /E ′ is torsion-free. 4.2.2 . Inside the exact category of coheren t shea ve s, the full sub cate- gory of torsion-free shea v es is closed under extensions and subob jects. A monomo r phism E ′ → E b et w een t o rsion-free sheav es is admissible (within the exact category of torsion-free shea v es) if and only if E ′ is pure in E . 4.2.3. Lemma. L et Q coh ( X ) b e Quil len ’s Q -c onstruction on the c ate- gory of c oher ent sh e a ves of O X -Mo dules, and let Q tf ( X ) b e the ful l sub- c ate gory of torsion-fr e e she aves. Th e n the inclusion Q tf ( X ) → Q coh ( X ) is a we ak e q uiva l e nc e. 16 BRUNO KAHN Pr o of. The conditions o f the resolution theorem [11, T h. 3] are v erified since an y lo cally free sheaf is torsion-free.  4.2.4. Prop osition. L et E b e a (c oh e r ent) torsion-fr e e she af on X , with ge neric fibr e E K . Then the map F 7→ F K defines a bije ction fr om the set Gr ( E ) of pur e subshe aves of E to the set Gr ( E K ) of subve ctor sp ac es of E K . Pr o of. Let V b e a sub -v ector space of j ∗ E = E K . Define E ∩ V = E × j ∗ j ∗ E j ∗ V . Then E ∩ V ∈ Gr ( E ), b ecause the map E / ( E ∩ V ) → j ∗ j ∗ ( E / ( E ∩ V )) is a monomorphism (by definition of E ∩ V ). So w e ha v e tw o maps: j ∗ : Gr ( E ) → Gr ( E K ); E ∩ − : Gr ( E K ) → Gr ( E ) . W e hav e ( E ∩ V ) K = j ∗ ( E × j ∗ j ∗ E j ∗ V ) = j ∗ E × j ∗ j ∗ j ∗ E j ∗ j ∗ V = j ∗ E × j ∗ E V = V so that j ∗ ◦ ( E ∩ − ) = I d . On the o t her hand, if F is a pure subsheaf of E , then F ⊆ E ∩ j ∗ F , and j ∗ E = j ∗ ( E ∩ j ∗ F ), hence (b y exactness of j ∗ ), j ∗ ( E ∩ j ∗ F /F ) = 0. Th us ( E ∩ j ∗ F ) /F is a torsion sub sheaf of the tor sion- free sheaf E /F , hence is 0, and our tw o maps are inv erse to eac h other.  4.2.5 . W e write Q tf n ( X ) for the full sub catego ry of Q tf ( X ) of torsion- free shea v es E suc h that dim K E K ≤ n . W e g et another rank spectral sequence (4.2) E 1 p,q = M E α H p + q − 1 (Aut( E α ) , ˜ F p ) ⇒ H p + q ( Q coh ( X )) cf. L emma 4.2.3. 4.2.6. Corollary . L et E ∈ Q tf n ( X ) , wi th generic fibr e E K ∈ Q n ( K ) . Then the functor j ∗ : Q tf n − 1 ( X ) ↓ E → Q n − 1 ( K ) ↓ E K is a n e quivalenc e of c a te gori e s . Pr o of. These categories are equiv alen t to the ordered sets of prop er la y ers of torsion-f ree subshea v es of E and j ∗ E (compare [11, t o p p. 102]). Th us the result directly follow s from Prop o sition 4.2.4.  AROUND QUILLEN’S THEOREM A 17 4.2.7. Example. Let X b e an inte gral Dedekind sc heme (= no etherian, regular of Krull dimension ≤ 1), with function field K . As is w ell- kno wn, a coherent sheaf F ov er a Dedekind sch eme is torsion- f ree if and only if it is lo cally free. Th us the a b o v e generalise s the remark of [12, pp. 191–192 ]. 4.3. The Tit s building. 4.3.1 . In Corollary 4.2.6, supp o se n ≥ 2. By [12, Prop. p. 1 8 8], the classifying space of the p oset Q n − 1 ( K ) ↓ E K = J ( E K ) is GL ( E K )- w eakly equiv alent to the susp ension o f nerv e of the Tits building of E K , whic h in turn is w eakly equiv alen t to a wedge of ( n − 2)-spheres b y the Solomon-Tits theorem ([12, Th. 2 p. 180], see § 3.2). Hence ( F n ) | E is Aut( E )- weakly equiv alen t to a w edge of ( n − 1 ) -spheres. 4.3.2 . In Corollary 4.2.6, supp ose n = 1. Then Q n − 1 ( K ) ↓ E K has t w o elemen ts: 0 → E K and E K → 0. Hence the conclusion of 4.3.1 is still true. 4.3.3. Theorem. If X is an inte gr al scheme, then the E 1 -terms o f the r ank s p e ctr a l se q uen c e (4.2) (with Z -c o efficients) ar e E 1 p,q = M E α H q (Aut( E α ) , st ( E α )) wher e E α runs thr ough the isomorphism classes of torsion-fr e e she aves of r ank p , and st ( E α ) = ˜ H p − 1 (( F p ) | E α ) is the [r e duc e d] Steinb er g mo dule of j ∗ E α . Pr o of. This follows from C orollary 4.2 .6, 4.3.1, 4.3.2 and Le mma 1.3 .5.  4.3.4. R emark. By an argument of V ogel, the exact sequence s from Corollary 2.3.7 then coincide with those of Q uillen in [12, Th. 3 p. 181]. In general, consider a map f : E → B whose homotopy fibre F has the homotop y ty p e of a b ouquet of n -spheres, with n > 0. So the Lera y- Serre spectral se quence yields a long exact seq uence (4.3) · · · → H p ( E ) → H p ( B ) → H p − n − 1 ( B , H n ( F )) → H p − 1 ( E ) → . . . If C is the homot op y cofibre of f , we hav e another long exact se- quence (4.4) · · · → H p ( E ) → H p ( B ) → ˜ H p ( C ) → H p − 1 ( E ) → . . . and w e w an t to know that the t w o sequences coincide. 18 BRUNO KAHN Here is V ogel’s argumen t. W e may assume that f is a Serre fibtrat io n. Let E ′ b e its mapping cylinder and C F the cone ov er F , so that we ha ve a fibration of pairs ( C F , F ) − − − → ( E ′ , E )   y ( B , B ) . Since H q ( C F , F ) = ( H n ( F ) for q = n + 1 0 else , the Lera y-Serre sp ec- tral sequence for the pair H p ( B , H q ( C F , F )) ⇒ H p + q ( E ′ , E ) yields isomorphisms ˜ H p ( C ) ≃ H p ( E ′ , E ) ≃ H p − n − 1 ( B , H n +1 ( C F , F )) ≃ H p − n − 1 ( B , H n ( F )) whic h comm ute with the differen tials of (4.3) a nd (4 .4) b y functoriality . Reference s [1] A. Ash, L. Rudolph The Mo dular Symb ol and Continu e d F ra ctions in Higher D imensions , lnven t. ma th. 55 (19 79), 2 41–2 50. [2] M. B¨ ok stedt, A. Neeman Homotopy limits in triangulate d c ate gories , Com- po sitio Math. 8 6 (19 93), 209–2 3 4. [3] H. Cartan, S. Eilenberg Homo logical a lgebra, Princeton, 19 50. [4] A. Dold, D. P upp e Homolo gie nicht-additiver F u nktor en. Anwendungen , Ann. Inst. F o urier 11 1961 201 –312 . [5] P . Gabriel, M. Zisman Calculus of f ractions and homotop y theo ry , E rgeb. Math. 35 , Spring er, 1967. [6] P . Go e rss, J. F. Jardine Simplicial homotopy theory , Prog ress in Math. 174 , B irkh¨ auser, 1999. [7] L. Illusie Complexe cotang ent et d´ efor mations, I, Lect. Notes in Math. 239 , Springer, 1971 . [8] B. K ahn Motivic c ohomol o gy of smo oth ge ometric al ly c el lular varieties , Pro c. Symp. Pure Math. 67 , Amer. Math. So c., 1999, 1 49–17 4. [9] B. Kahn and F. Sun On the universal mo dular symb ols , preprint, 2 014. [10] S. Mac Lane Ca tegories for the working mathematician, Grad. T exts in Math. 5 , Springer (2nd ed.), 19 9 8. [11] D. Quillen Higher algebr aic K - t he ory, I , Lect. No tes in Ma th. 341 , Springer, 1972 , 85–1 47. [12] D. Quillen Finite gener ation of the gr oups K i of rings of algebr aic inte gers (notes by H. Ba ss), Lec t. Notes in Math. 341 , Springer, 1 9 72, 17 9–19 8. [13] F. Sun Th e d 1 -differ ential of t he r ank sp e ct r al se qu enc e for algebr aic K - the ory , thesis, to app ear. [14] R. W. Thomason Homotopy c olimits in the c ate gory of smal l c ate gories , Math. Pro c. Cambridge Philos. So c. 85 (1979 ), 91 –109 . AROUND QUILLEN’S THEOREM A 19 [15] C. W eib el The K-b o ok: An introductio n to algebraic K-theory , Grad. Stud- ies in Math. 145 , AMS, 2013 . [SGA1] Revˆ etements ´ etales et gro up e fondament al (SGA1), dirig´ e pa r A. Grothendieck, Lect. Notes in Ma th 224 , Springer, 197 1. Recompo sed and annotated edition: Do c. Math. 3 , Soc . Math. F r ance, 2003. Institut de Ma th ´ ema tiques de Jussieu, Case 247, 4 place Jussieu, 75252 P aris Cedex 05, FRANCE E-mail addr ess : kah n@math .juss ieu.fr