On the K-theory of hermitian categories

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Jens Hornbostel

To cite this version:

Jens Hornbostel. On the K-theory of hermitian categories. Mathematics [math]. Université

Paris-Diderot - Paris VII, 2001. English. �tel-00006013�

TH  ESE

Spe ialite: mathematiques

presentee par

Jens HORNBOSTEL

pour obtenir leDipl^ome de

DOCTEUR de L'UNIVERSIT 

E PARIS 7

Sur la K-theorie des ategories hermitiennes

Dire teur de these: Max KAROUBI

Rapporteurs : Daniel R. GRAYSON et John F. JARDINE

Soutenue le 19 juin 2001 devant le jury ompose de :

Jean BARGE John F. JARDINE Bruno KAHN Max KAROUBI Fabien MOREL Christophe SOUL  E

Remer iements

Entoutpremierlieu,mestressin eresremer iementsvontaMaxKaroubi. Apres m'avoir initiea laK-theorie etm'avoir propose le sujetil y a plus de trois ans, iln'a pas essede meguideretde m'en ourager. Jeluisuis re on-naissant pour notre groupe de travail ou j'ai pu exposer regulierement mon travailainsi que pour ses nombreuses remarques etnos dis ussions.

Je voudrais aussi remer ier Mar o S hli hting, ave qui j'ai souvent dis- utesurlaK-theoriehermitienne. Ila onsa rebeau oupdetempsadis uter ertaines hoses tres en detail; et il a ni par tellement s'interesser ausujet que nous avons attaquequelques questionsensemble.

MagratitudevaegalementaBrunoKahnpour l'inter^etqu'ilamanifeste pour mon travail et surtout pour plusieures remarques et dis ussions tres stimulantes.

Pour d'autres dis ussions et remarques, je remer ie Paul Balmer, Bern-hard Keller etFabien Morel.

John F. Jardine et Daniel R. Grayson me font un tres grand honneur d'avoir bien voulu juger e travail, et je les remer ie pour leurs remarques nombreusesetdetailleesquiontpermisd'ameliorerla larteetlapresentation de mathese.

Je suis egalement tres honore que Jean Barge, John F. Jardine, Bruno Kahn,MaxKaroubi,FabienMoreletChristopheSouleaienta eptedefaire partie du jury.

Enn,je remer ie James Martin,Mal olm S honeld etWolfgangPits h pour leur soutien on ernant l'anglais et le fran ais. Les erreurs qui restent relevent entierement de maresponsabilite.

Contents

1 Introdu tion (en fran ais) 3

2 Introdu tion 6

3 The hermitian K-theory of rings 9

4 The hermitian K-theory of additive ategories 18

5 The hermitian K-theory of exa t ategories 22

6 Lo alization I: Negative hermitian K-theory of additive

at-egories 30

7 A Waldhausen-like model for U-theory 35

8 Lo alization II: HermitianK-theory of hereditary rings 38

9 Devissage and al ulations 45

10 Some further remarks and open problems 54

11 The proof of the Lo alization Theorem 60

1 Introdu tion (en fran ais)

Apres l'introdu tion de K 0

par Grothendie k [BS℄, la K-theorie algebrique s'est developpee d'abord pourla ategoriedes modulesproje tifsde typeni sur un anneau A. (Tous nos anneaux sont asso iatifs et unitaires, et tous nos modules sont des modules a droite.) Ce i est une ategorie additive. Une ategorie additive peut ^etre onsideree ou omme ategorie monodale symetrique (parfois on appelle une telle ategorie une ACU- ategorie ten-sorielle) ou omme ategorie exa te (ou la stru ture exa te est donnee par les suites exa tes ourtes s indees). Quillen [Q1, Q2℄ a deni la K-theorie pour es deux lasses de ategories via les onstru tions S

1

S et Q. Il a demontre queles deux denitions on identpour les ategoriesadditiveset qu'elles generalisent la onstru tion Plus [Lo1℄. La onstru tion Q permet de demontrer des theoremesimportants,parexemplelestheoremes de lo al-ization etde resolution.

Supposons maintenant que nos modules proje tifs sur l'anneau A sont munis d'une formebilineaire symetrique (ou de maniere plus generale d'une forme -hermitienne). Notre obje tif est de de rire la K-theorie de ette ategorie que nous baptiserons la \K-theorie hermitienne". La K-theorie hermitiennedevraitegalementexisterdans dessituationsplusgenerales, par exemple pour la ategorie des bres ve toriels sur un s hema munis d'une formebilineaire symetrique.

En omparant la K-theorie lassique ave la K-theorie hermitienne, on on-state des dieren es assez importantes: Ainsi, les theoremes ites i-dessus n'ontpasd'analogueenK-theoriehermitienne. Pourune ategorieexa te,la K-theoriehermitiennen'estm^emepasdenie. Ilyaune denitionseulement pour les ategories additives ainsi que quelques resultats dans le as parti- ulier des modules proje tifs munis d'une forme hermitienne. Dans ette these, nous allonsdenirlaK-theorie hermitiennepour une ategorieexa te C ommeetantlesgroupesd'homotopiede l'espa e lassiantd'une ertaine ategorieL(C

h

)(Denition5.10). Nousallonsegalementdenirla U-theorie de C

h

ommeetantlesgroupesd'homotopiedel'espa edes la etsdel'espa e lassiantd'une ategorieW(C

h

). Ensuitenouspouvonsetablirunebration homotopique induite par lefon teur hyperboliqueH

BQ(C) H !BL(C h )!BW(C h )

qui generalise le as additif (Theoreme 5.7). Comme appli ation prin ipale, nous demontrons un Theoreme de Lo alisation pour la K-theorie hermiti-enne d'un anneau de Dedekind A, par rapport a un systeme multipli atifS (Theoreme 8.7). Plus pre isement, nous demontrons l'existen e d'une

bra-tion homotopique BL(P(A) h )!BL(P(S 1 A) h )!BW((T S ) h ); ou T S

est la ategorie des A-modules de S-torsion. Par onsequent, nous avons une suite exa te longue

:::!U n ((T S ) h )!K n (P(A) h )!K n (P(S 1 A) h )!:::

En plus, nous demontrons un Theoreme de Devissage (Theoreme 9.5) qui nous permet de rempla er U

n ((T S ) h ) par U n (P(A=}) h ) ou on prend la somme sur tous lesideaux maximaux.

Pour A l'anneau des entiers dansun orps de nombres F,nous pouvons al- ulerlaU-theoriedeses orpsresiduels(Corollaire9.9)enutilisantles al uls de QuillenetFriedlander,etnousendeduisons(Theoreme9.11)parexemple

1 K h n (A)  = ! 1 K h n (F) 8n 3;4mod8; 1 K h n (A)  = ! 1 K h n (F) 8n 0;7mod8:

Ce texte est stru ture de la maniere suivante:

Danslase tion3,nousrappelonsquelques faitssurlaK-theoriehermitienne des modules, tout en prenant un point de vue qui nous permet aisement de generaliseraux ategories ave dualite. En suivant [CL2℄,[S h3 ℄, nous intro-duisons une ategorie W(

 P(A)

h

) qui donne un dela age de la bre homo-topique du fon teur hyperboliquede la K-theorie lassiquevers laK-theorie hermitienne(Theoreme3.15) du moinssi2est uneunitedansnotre anneau. La se tion 4traite de la K-theorie hermitienne des ategories additivesave dualite. La philosophie est que laplupart des hoses qui sontvraies pour la K-theoriehermitiennedesanneauxest en orevraiepour laK-theorie hermi-tienne des ategoriesadditives.

LaK-theoriehermitienned'une ategorieexa teave dualiteetsesproprietes fondamentales mentionnees i-dessus sont de rites dans la se tion5.

Dans la se tion 6,nous etablissons un Theoreme de Lo alisationpour la K-theoriehermitiennedes ategoriesadditives(Corollaire6.7) quinous permet de denir leur K-theoriehermitienne negative.

Dans la se tion 7, nous onstruisons une ategorie additive simpli ialeave dualitequidonneun modelealaWaldhausenjiR

h 

Cjpour laU-theoried'une ategorieexa te C ave dualite.

La se tion 8 ontient le Theoreme de Lo alisation pour la K-theorie her-mitienne des anneaux de Dedekind (et de maniere plus generale pour les anneaux hereditaires, en parti ulier pour un anneau de groupe AG ave G un groupe ni et A un anneau de Dedekind dans lequel l'ordre du groupe est une unite). Plusieurs strategies et diÆ ultes pour le demontrer sont

es-Danslase tion9,nousdemontronsunTheoreme deDevissagequinous per-met desimplierlabrationhomotopique i-dessus enrempla antW((T

S ) h ) parW(P(A=}) h

)(Theoreme9.5). Commenouspouvons al ulerlesgroupes d'homotopie de ette ategorie si A=} est ni (prenons les entiers dans un orpsdenombres,parexemple),notreresultatde lo alisationplut^otabstrait devient tres on ret et nous permet de voir dans quels degres l'in lusion A !Quot(A) induitun isomorphisme en K-theorie hermitienne (Theoreme 9.11).

Danslase tion10nousevoquons d'autres problemes,enparti ulier eux qui apparaissentsinousrempla onsnotre ategorieexa te C par la ategoriedes omplexes de haines Ch

b (C).

La se tion 11 ontient la demonstration du Theoreme de Lo alisation8.7. Lesresultats prin ipaux de lase tion 5 ont ete annon es dans [Ho℄. Les se tions 6,7 et 11 sont un travail en ommun ave Mar o S hli hting, bien que jesoisentierementresponsablede lapresentation etde tous ses defauts.

2 Introdu tion

After K 0

was introdu ed by Grothendie k [BS ℄, algebrai K-theory rst de-veloped for the ategory of nitely generated proje tive modules overa ring A. (All rings we are dealingwith are supposed tobe asso iative and with a unit, and all modules are right modules.) This is an additive ategory. An additive ategory an be regarded as a symmetri monoidal ategory (also alledaACU-tensor ategory)orasanexa t ategory(wheretheexa t stru -tureisgivenbythesplitexa tsequen es). Quillen[Q1,Q2℄denedK-theory for both kinds of ategories via the onstru tions S

1

S and Q. He proved that the two denitions oin ide for additive ategories and that they gen-eralize the Plus onstru tion [Lo1℄. The Q- onstru tion allows us to prove importanttheorems su h as thoselinked to lo alizationand resolution et .

Assumenowthatourproje tivemodulesoveragivenringAareequipped with asymmetri bilinear(ormoregenerally withan-hermitian) form. We want todes ribe the K-theoryof this ategorywhi hwewill all\hermitian K-theory". Hermitian K-theory should also over more general situations, for example the ategory of ve tor bundles over a s heme equipped with a bilinear symmetri form.

Comparing lassi alK-theory andhermitianK-theory, the situationis quite dierent. We have no hermitian analogues of the theorems in lassi al K-theory itedabove. Foranexa t ategoryingeneral,thehermitianK-theory was not even dened. We only had a denition for additive ategories and very few results in the spe ial ase of proje tive modules equipped with a hermitian form. In this thesis, we will give a general denition of the hermitian K-theory of an exa t ategory C as the homotopy groups of the lassifying spa e of a ertain ategory L(C

h

) (see Denition 5.10). We an alsodenetheU-theoryofC

h

asthehomotopygroupsoftheloopspa eofthe lassifying spa e of the ategory W(C

h

). Then we an establisha homotopy brationindu ed by the hyperboli fun tor H

BQ(C) H  !BL(C h )!BW(C h )

whi h generalizes the additive ase (see Theorem 5.7). As an important appli ation, we prove a Lo alizationTheorem for the hermitianK-theory of a Dedekind ring A and a multipli ative subset S (see Theorem 8.7). We prove the existen e of ahomotopy bration

BL(P(A) h )!BL(P(S 1 A) h )!BW((T S ) h ) where T S

is the ategory of S-torsion modules. Consequently, we get a long exa t sequen e

:::!U ((T ) )!K (P(A) )!K (P(S 1

Moreover, we anprovea Devissage theorem (Theorem 9.5) whi hallowsus torepla eU n ((T S ) h )byU n (P(A=}) h

)wherethesumistakenoverallprime ideals } dierent from(0).

IfAisthe ringofintegersinanumbereld F, we an omputethe U-theory of these niteresidue elds (Corollary9.9) using al ulations of Quillenand Friedlander, and we obtain amongother things

1 K h n (A)  = ! 1 K h n (F) 8n3;4mod8 1 K h n (A)  = ! 1 K h n (F) 8n 0;7mod8 (see Theorem 9.11).

This textis organized as follows:

In se tion 3, we re all some fa ts about the K-theory of hermitian mod-ules, taking a point of view whi h allows us to immediatelygeneralize to a ategori al viewpoint. In parti ular, following [CL2 ℄,[S h3 ℄, we introdu e a ategory W(

 P(A)

h

) whi h gives a delooping of the homotopy ber of the hyperboli fun tor from lassi al K-theory tohermitianK-theory (Theorem 3.15) at least if 2is a unit inour ring.

Se tion 4 deals with the hermitianK-theory of additive ategories with du-ality. The philosophy is that most what is true for the hermitianK-theory of rings is still true for the hermitian K-theory of additive ategories with duality.

ThehermitianK-theoryofanexa t ategorywithdualityanditsbasi prop-erties as mentioned above are des ribed inse tion 5.

In se tion6, westate the lo alizationtheorem forthe K-theoryof hermitian additive ategories (Corollary 6.7) whi h allows us to dene their negative K-theory.

In se tion 7, we onstru t a simpli ialadditive ategory with duality whi h yieldsa Waldhausen-likemodeljiR

h 

Cjforthe U-theory ofanexa t ategory C with duality.

Se tion 8 ontains the Lo alization Theorem for the hermitian K-theory of Dedekind rings (and more generally for hereditary rings, in parti ular for group ringsAG whereG isa nitegroup andA isa Dedekind ring inwhi h the orderof the group isa unit). Some strategies and diÆ ulties of possible proofs are sket hed. The proof willbe the subje t of se tion 11.

In se tion 9, we prove a Devissage theorem whi h allows to simplify the above homotopy bration repla ing W((T

S ) h ) by W(P(A=}) h ) (Theorem 9.5). As we an al ulate the homotopy groups of this ategory if A=} is nite ( onsider the integers in a number eld, for example), our somewhat abstra t lo alization result be omes very on rete and allows us to see in whi hdegrees the in lusionA !Quot(A)indu esanisomorphismin hermi-tian K-theory (Theorem 9.11).

In se tion 10 we sket h some further questions, in parti ular those arising when we repla e an exa t ategory C by the ategory of hain omplexes Ch

b (C).

Se tion 11 ontains the proof of the Lo alizationTheoerem 8.7.

The main results of se tion 5have been announ ed in[Ho℄. Se tions 6,7 and 11are joint work with Mar o S hli hting, but I am entirely responsible for the presentation and allthe defe ts it has.

3 The hermitian K-theory of rings

Before we start with the K-theory of hermitian modules, we re all some basi fa ts (without proofs) about simpli ial sets and lassifying spa es. A topologist mightskip this part.

A simpli ial set is by denition a fun tor F :
op

! Set . Re all that
is the ategory whose obje ts are the non-negative integersn, onsidered as an ordered set n = f0 < 1 < ::: < ng, and a morphism f : m ! n is a monotoni map, i.e. i  j implies f(i)  f(j). In parti ular, we have the fa e maps Æ

i

: n ! n+1 (the i tells us whi h element is not in the image) and the degenera y maps 

j

:n !n 1 (where the j indi atesthe element onto whi htwo elements are mapped). In fa t, up to permutation any map is a unique omposition of these maps. We write

op

Set for the ategory of simpli ialsets. Thenwe have apair of adjointfun tors

j j:
op

Set !Top:Sing:

Here j jis the so- alled geometri realizationfun tor and Sing = Hom Top (
 top ; )where
n top

isthetopologi alstandardn-simplex. Moreover, both ategories anbeequippedwiththestru tureofa losedmodel ategory, and the abovepair of adjointfun tors respe ts enoughof this stru ture that byatheoremofQuillenitbe omesanequivalen eofthehomotopy ategories. Next, we havethe nervefun tor

N :small at!
op

Set : Forasmall ategoryC,thesetNC

m

ontainsallthediagramsof ompositions of m omposable morphismof C. The fa emaps orrespond tothe omposi-tion of maps and the degenera y maps are given by insertingidentity maps. The lassifying spa e BC of a small ategory C is dened by BC := jNCj. Observe that this generalizes the lassi al denition of the lassifying spa e of a dis rete group when we onsider it as a ategory with a single obje t. Fromnowon, whenwetalkaboutrealizations,weassumethatthe ategories wearedealingwitharesmall. Tosimplifyournotations,wewillusethesame symbolfor a ategory, itsnerve and its lassifying spa e.

LetAbearingequippedwithananti-involution,i.e.,amorphismofrings : A ! A

op

su h that  

a = a 8a 2 A. Furthermore, hoose  2 enter(A) with  =1. Consider anitely generated proje tive A-module M.

Denition 3.1 If M is equipped with a sesquilinear non-degenerateform  (anti-linear in the rst omponent) su h that (m;n) = (n;m)  8 m;n 2 M, we say that (M;) is an -hermitian module. The dual of M is dened by t M = Hom A;anti (M;A), i.e., f 2 t M , f(ma) = af(m). This is an

With  is asso iated  : M ! t

M via (m) = ( ;m). One observes that  is an isomorphism and that

t

 =  ; more pre isely, the triangle M  

!!C

C

C

C

C

C

C

C

ev

//

t 2 M t 

||xx

xx

xx

xx

() t M

ommutes(withthe anoni alisomorphismev(m)(f)=f(m)8m2M; f 2 t

M). It follows that the ategory of -hermitian modules over A and the unitarymorphisms (i.e. thoserespe tingtheform-itwould bebetterto all them metri ) is anoni allyisomorphi to the following ategory

 P(A) h : Denition 3.2 Let  P(A) h

be the ategory of hermitian modules relative to (A; t ;;). An obje t is an isomorphism  : M  = ! t M su h that (*) ommutes. A morphism  : (M;) ! (N; ) is a ommutative square

M 

//

 = 



N  =



t M t N t 

oo

Remark: We an also drop the ondition that the morphisms  and are isomorphisms. Then weobtainthe ategoryP(A)

hd

of allhermitianobje ts, in luding the degenerate ones. Of ourse, P(A)

hd

ontains P(A) h

as a full sub ategory.

Examples: For A ommutative,= Id and  =1 (resp.-1) we obtain the theory of(anti-)symmetri forms. WhenA=Cisthe omplexnumbers, the omplex onjugation is an anti-involution. If A is a ommutative ring and G is a group, then the group ring AG is equipped with an anti-involution sending ag toag 1 . (  P(A) h

;)is a symmetri monoidal ategory via (M;)(N; )=(M  N; ),soitsK-theoryisdened byQuillen'sS

1 S- onstru tion(see[Q2℄ or p.16): Denition 3.3 K n (  P(A) h ):= n (Iso(  P(A) h ) 1 Iso (  P(A) h ))8n 0

One often writes  K h n (P(A)) oreven  K h n (A) insteadof K n (  P(A) h ). If the  is understood,itis sometimes dropped. Many peoplewrite (M;)?(N; ) instead of (M;)(N; ).

Let us re all some fa ts from the lassi al K-theory of nitely generated proje tive modules P(A). Any proje tive module is a dire t summand of a

is onal inP(A). This onalityimplies, amongother things, that: K 1 (P(A))  = H 1 ( olim Aut(A n

);Z ) (in the lassi al notation, one writes GL(A) insteadof olim Aut(A

n ) ) K 2 (P(A))  = H 2 ([GL(A);GL(A)℄;Z ) K n (P(A))  =  n (BGL(A) +  K 0

(P(A))) (in the lassi al notation, one writes K n (A)instead of K n (P(A))) BGL(A) +  K 0 (A) ' (BGL(A) + K 0

(A)) (see [Wag℄, A is the algebrai suspension of the ring A: Its elements are the innite matri es over Ahaving onlyanite numberof non-zero elements inanyrowand any olumn, divided by those matri es having only a nite number of non-zero elements. ).

It also implies the existen e of a \Volodin model" [Su ℄. Furthermore, the des riptionusingthePlus onstru tion anbeusedto al ulatetheK-theory of aniteeld[Q3℄and thefreepart ofthe K-theoryofanalgebrai number eld [Bo℄.

Inthe ategory 

P(A) h

,wewouldliketohavea onalsub ategorywhi h behaveslikethefreemodulesinP(A)andenablesustoprovesimilarresults. For this, itis ne essary tointrodu ehyperboli and metaboli modules: Denition 3.4 A module (P;) ishyperboli ifthere isamoduleM andan isomorphism (P;)  = (M  t M; 0 1  0
)=:(H(M); M ). It is metaboli if thereisa(possiblydegenerate)hermitian module(M;)andanisomorphism (P;)  = (M  t M;  1  0
)

Lemma 3.5 a) Any hermitian module (M;) is a dire t summand of a metaboli module: (M;)(M; )  = (M  t M;  1  0
) b) If there exists  2 enter(A) su h that +



 = 1 , then any hermitian module is a dire t summand of a hyperboli module: (M;)(M; )

 = (H(M); M ) Proof: a) M M  1 1 0  

//

  0 0  



M  t M   1  0 



t M  t M t M M  1 0 1 t  

oo

ommutes. b) M  t M  1 0  1 

//

  1  0 



M  t M  0 1  0 



t M M t MM  1  0 1 

oo

We willalways say \if 2is invertible" even if weaker onditions (i.e. the existen e of a  su h that +



 = 1 or the fa t that the hyperboli mod-ules form a onal sub ategory of

 P(A)

h

) are suÆ ient in many proofs. In this ase, we an on lude, as for lassi al K-theory (repla ing A

n by (H(A n ); A n)),that: K 1 (  P(A) h )  = H 1 ( olim Aut(H(A n ); A

n);Z ) (one often writes 

O(A) instead of olim Aut(H(A

n ); A n) ) K 2 (  P(A) h )  = H 2 ([  O(A);  O(A)℄;Z) K n (  P(A) h )  =  n (B  O(A) + K 0 (  P(A) h

)) In the notation of [Ka2℄, one sets  L n (A):= n (B  O(A) + K 0 (  P(A) h

)independent ofthe fa t whether 2isinvertibleornot. Thisshouldnotbe onfusedwiththe L-theoryofWall, Rani ki and Mis henko, see the rst remark below.

B  O(A) + K 0 (  P(A) h )'(B  O(A) + K 0 (  P(A) h ))

and that a\Volodinmodel" an be onstru ted [So℄. Furthermore,itis pos-sible to al ulatethe K-theory of a nite eld [Fr℄ and the free part of the K-theory of analgebrai numbereld [Bo℄.

Remark: The idea of dening the L-theory (agoodreferen e is[Ra2℄) of a ringAisasfollows: Consider theabelianmonoidofn-dimensional\Poin are omplexes" of proje tive modules of nite type over A together with an n- y leplayingtheroleofthe-hermitianform. Thenwedivideoutbya ertain \ obordism" relation to get the group L

n

(A;). The geometri interest in these L-groupsis thatthey ontain obstru tion lasses inthe surgery theory of a Poin are spa e X when A = Z

1

(X). We have periodi ity in L-theory [Ra1℄,i.eL n (A;)  = L n+2

(A; ). This shouldbe ompared with the higher Witt groups (see Denition 3.13) where the same is true up to 2-torsion [Ka6℄, i.e.  W n (P(A))Z[1=2℄  =  W n+2

(P(A))Z [1=2℄. Moreover, these twotheories oin ideupto2-torsion[Lo2℄: L

n (A;)Z [1=2℄  =  W n (P(A)) Z[1=2℄.

Remark: Instead of looking at hermitian modules, we an deal with the (symmetri monoidal) ategoryofquadrati formsoveragivenring: onsider the sesquilinear non-degenerate forms Sesq(M) on a nitely generated pro-je tive module M. We have a morphism T



:Sesq(M)!Sesq(M) indu ed byT



()(m;n) =(n;m) . Thenthequadrati formsonM (morepre isely, the lassesoftheirasso iatedbilinearforms)are equalto oker(T



1)while the -hermitian forms are equal to ker(1 T



). The bilinearization T 

+1 indu es a morphism oker(1 T



) ! ker(1 T 

) whi h is an isomorphism if 2 is invertible [Wall℄. In the ategory of quadrati forms, the hyperboli obje ts always form a onal sub ategory. Therefore the Plus onstru tion always gives a modelfor its K-theory. To study the dieren es between the ategories of hermitian modules and that of quadrati modules and their K-theories, one introdu esform parameters [Bak℄.

the forgetfulfun tor and the hyperboli fun tor F :  P(A) h !P(A) and

H :Iso(P(A))!Iso(  P(A) h ) given by H(M) = (H(M); M

) on the obje ts and H(f) = (

f 0 o t f 1
) on the morphisms. These fun tors indu e fun tors F



and H 

between the lassifying spa es for the K-theories.

Denition 3.6 U(  P(A) h ):=hob(H  ;0) V(  P(A) h ):=hob(F  ;0)  U n (A):= n (U(  P(A) h ))

Here, hob(f;y)isthe homotopyberoff :X!Y overapointy 2Y,i.e., thehomotopylimitofthe diagramfyg!Y

f

X . Consequently,thesefour spa esformahomotopy artesian square,i.e.,hob(f;y)'fyg

Y Y I  Y X. Itfollowsthatwegetalongexa thomotopysequen e. Ingeneral,hob (f;y) depends on y, but any path onne ting y and y

0

in Y gives a homotopy equivalen e of the homotopy bers over these two points. That's why we often suppress y in hob (f;y) if Y is onne ted or if f is a morphism of H-groups.

Re all that a homotopy artesian square remains homotopy artesian if we repla e one of the four spa es by a spa e having the same homotopy type. We say that F

g ! X

f

! Y is a homotopy bration if it ts in a homotopy artesian square wherethe fourth spa eis ontra tible. Thisis equivalent to saying that f Æg is homotopi to a onstant map y~for some y 2Y and the indu ed map (depending on the hoi e of the ontra tion) F ! hob (f;y) is a homotopyequivalen e.

Theorem 3.7 (\Fundamental Theorem") There is a homotopy equivalen e

V(  P(A) h )'(U(  P(A) h )) if 2 isinvertible.

Proof: See [Ka7℄. The main idea is that the homotopy groups of both spa es are equipped with a K

0 (

1

(P(Z[1=2℄) h

))-module stru ture and the isomorphism between the homotopy groups is given by the multipli ation with an element of this ring.

Karoubi and Gien [CL2℄ proposed a ategory W( 

P(A) h

) whose loop spa e is a modelfor U( P(A) ) if 2is invertible:

Denition 3.8 Let W( 

P(A) h

) be the ategory whose obje ts are the same asthoseof

 P(A)

h

. Forthemorphisms,wesetW( 

P(A) h

)((P;);(M; ))= fequivalen e lassesof(q;(L;);i)g,wherearepresentativeisgivenbya om-mutative diagram (in P(A))

P  = 



L q

oooo

//

i

//





M  =



t P

//

t q

//

t L t M t i

oooo

with q a split epimorphism and i a split monomorphism. We also demand that the in lusion of L

? M

:=ker( t

iÆ ) in M fa tors over L; in other words we have a ommutative triangle

L ? M

||

||y

y

y

y

""

""E

E

E

E

E

E

E

E

L

//

i

//

M Consequently, L 0 := ker  = kerq  = L ? M

. We say that L is a \sur-lagrangian" (and we often write L

? instead of L ? M ). (q;(L;);i) and (q 0 ;(L 0 ; 0 );i 0

) are equivalent if there is an isomorphism in P(A)  :L!L 0 su h thatq=q

0

Æ andi=i 0

Æ . The ompositionof morphismisexplained in the followinglemma.

In fa t, the ondition that L ?

is in luded in L is equivalent to saying that the outer re tangleisa bi artesiansquare, and thiswillbe usedinthe proof of the following lemma:

Lemma 3.9 W( 

P(A) h

) is a ategory with the omposition law spe ied below.

Proof: Given two morphisms [v;(L 1 ; 1 );i℄ 2 W(  P(A) h )((M; );(N;)) and [u;(L 2 ; 2 );j℄2W(  P(A) h )((N;);(P;)),weset L:=L 1  N L 2 . Then their ompositionis [(vÆu);~ (L;

t ~ iÆ 2 Æ ~ i);(jÆ ~

i)℄ dened by the diagram L

//

~ i

//

~ u



art L 2

//

j

//

u



P L 1 v



//

i

//

N (+) M

and one he ks that this is well-dened. It remains toshow that L ? P

fa tors overL. Considerthe following ommutativediagram(inwhi hwehave sup-pressed the isomorphims ; and ):

L

//

~ i

//

~ u



(1) L 2

//

j

//

u



(2) P t j



L 1

//

i

//

v



(3) N

//

t u

//

t i



(4) t L 2 t~ i



M

//

t v

//

t L 1

//

t ~ u

//

t L

The square (1) is not only artesian, but also o artesian (see the appendix of [Ke1℄), so the square (4) is bi artesian. Similary, one an show that the squares (2)and (3) are alsobi artesian. It follows that the big square om-posedoffourbi artesiansquaresisbi artesian. Thisimpliesthatker(

t ~ iÆ t j) fa tors over L.

Instead of looking at the morphisms indu ed by \sur-lagrangians", we an onsider \sub-lagrangians"(also alled \totally isotropi "modules):

Denition 3.10 LetW 0 (  P(A) h

)bethe ategorywhoseobje tsarethesame as those of

 P(A)

h

. For the morphisms, we set W 0 (  P(A) h )((P;);(M; )) = fequivalen e lasses of (p;(T ? ;);j)g where T ?

is the orthogonal of some obje t T and a representative is given by a ommutative diagram (in P(A))

P  = 



T ? p

oooo





//

j

//

M  =



t P

//

t p

//

t T ? t M t j

oooo

with p a split epimorphism and j a split monomorphism. We also demand that T isan isotropi sub-module of M; in other words we have a ommuta-tive triangle: T

||

||z

z

z

z

!!

!!B

B

B

B

B

B

B

B

T ? M

//

j

//

M

Lemma 3.11 We have an isomorphism of ategories F : W(  P(A) h )  = ! W 0 (  P(A) h ) given by F([q;(L;);i℄) = [qÆ ;((L ? ) ? ; ÆÆ );iÆ ℄ with :(L ? ) ?  = !L. Proof: Be ause L ? 

//

//

M t iÆ

// //

t L is exa t, L i

//

//

M t Æ

// //

t L ?

isalsoexa t. Itfollowsthatthereisanisomorphism :(L ? ) ?  = !Lover M and that F([q;(L;);i℄) is a morphism in W

0 (  P(A) h ). We an dene G : W 0 (  P(A) h ) ! W(  P(A) h

) in a dual way setting G([p;(T ?

;);j℄) := [p;(T

?

Remark: The proofsofLemma3.9andLemma3.11 showthat W( 

P(A) h

) is isomorphi tothe ategory



WP(A) in[Ur ℄.

Proposition 3.12 a)Suppose wehaveaK su hthat(P;)(H(K); K

)  = (M; ). Then there isa morphismfrom (P;) to (M; ) in W(

 P(A)

h ). b) If 2 is invertible and there is a morphism in W(

 P(A)

h

) from (P;) to (M; ). Then there exists a K su h that (P;)(H(K);

K )

 =

(M; ) .

Proof: a)The lass of (;(P K;0);) is su h a morphism.

b) Let [(q;(L;);i)℄ be a morphism. Choose a se tion s of q. Then P is a dire t summand of M as a hermitian module (see [Kn, II.2.5.2℄ where it is shown that if we have a non-degenerate hermitian obje t with a subobje t on whi h the form is non-degenerate, then this is a dire t summand as a hermitiansubobje t. Here weuse the fa tthat 2isinvertible,whi himplies that any hermitian module an be onsidered as the equivalen e lass of a module equipped with a bilinear form). So the problemredu es to showing that the hermitian modules whi h are onne ted with 0 by a morphism are pre isely the metaboli modules (whi h are hyperboli if 2 is invertible). In this ase, (M; )  = (L ?  t L ? ;  a b  t b d  ). As a = 0 and b is an isomorphism(be auseL t L [0b℄

// //

t

Lis asplitepimorphismwithkernel L), onjugation with  b 1 0 0 1 

gives the desiredmodule.

We therefore see that W( 

P(A) h

) is isomorphi to the ategory 

^ W(A) in [CL2℄. Following[Ka2℄, we dene the higher Wittgroups:

Denition 3.13 W n (  P(A) h ):= oker(K n (P(A)) H !K n (  P(A) h )):

WewilloftenwriteW(  P(A) h )insteadofW 0 (  P(A) h

)forthe lassi alWitt group.

Let 

P(A) H

be the full sub ategory of obje ts in 

P(A) h

isomorphi to a hyperboli obje t. Be ause H



fa tors via the epimorphism K 0 (P(A)) ! K 0 (  P(A) H ),we have W(  P(A) h )  = oker(K 0 (  P(A) H )!K 0 (  P(A) h )) Lemma 3.14  0 (W(  P(A) h ))  =W(  P(A) h )

Proof: Any element of W( 

P(A) h

) an be represented by a dieren e (P;) (M;)where(M;)ismetaboli (Lemma3.5a)). One anshow[QSS, Lemma 5.4℄ that the metaboli modules are zero in W(

 P(A)

h

) even if 2 is not invertible. This implies the following:

() There existmetaboli modules (M;) and (N;) su hthat (P;)(M;)

 =

(Q; )(N;)(")" isa onsequen e of[QSS , 5.3℄ whi h says the following: If(L;) is a totallyisotropi submodule of (M; ), then (M; )(L

?

=L; j L

?)is metaboli .) () The lasses of (P;) and (Q; ) inW(

 P(A) h ) are equal. LetW(  P(A) H

)bethefullsub ategoryofW( 

P(A) h

)ofobje ts isomor-phi to a hyperboli obje t. We let S = Iso (P(A)); S

H = Iso(  P(A) H )  = Iso(W(  P(A) H ))and S h =Iso (  P(A) h )  = Iso(W(  P(A) h )). We have  = ÆS 1  : S h !W(  P(A) h

) where the a tion of S on S h

is given by thehyperboli fun tor. Here  isthe in lusionofS

h inW(  P(A) h ) and : S 1 W(  P(A) h ) ' ! W(  P(A) h

) exist be ause the a tion of S on W(

 P(A)

h

) is trivial (hen e is a homotopy equivalen e given by the pro-je tion on the level of obje ts). Then hermitian K-theory, the underlying lassi alK-theoryandtheW- onstru tionarerelatedby thefollowingresult of M. S hli hting.

Theorem 3.15 If 2 is invertible, then there is a homotopy bration

S 1 S S 1 H ! S 1 h S h  !W(  P(A) h ) where the a tion of S on S

h

is givenby the hyperboli fun tor.

Proof: Theproof an befound in[S h3 ℄; ituses the Waldhausen-likemodel of se tion 6and Karoubi's FundamentalTheorem (Theorem 3.7) and estab-lishes an additivity theorem for hermitianK-theory.

Remark: In [CL2, 3.1-3.4℄ R.Charney and R.Lee believed that they had a proof for Theorem 3.15. They wanted to show that all base hanges of the fun tor S

1

 are H 

( ;Z)-isomorphisms and then show by a rather expli it al ulation that we have a homotopy equivalen e S

1

S ! S 1

(0 # ). To provethatallbase hangesareisomorphismsinhomology(andhen e homo-topyequivalen esastheyare morphismsofH-groups),they nallyneededto show that the a tion indu ed by the swit h 



:(P P #)!(P P #) is theidentity inhomology. They laimedthat 



a ts asaninner automor-phism, whi his not true.

Remark: The interest of Charney and Lee in the ategory W( 

P(A) H

) omes from geometry: Let 

n

= fM(n;C )j t

 = ;Im() > 0g the Siegel spa e of dimension n. The symple ti group Sp

2n (Z)= 1 O n;n (R)= Aut(H(R n ); R n ) a ts on  n

transitively. The a tion of n = 1 O n;n (Z ) even extends tothe Satake ompa ti ation ofthis spa e. We an onstru t fullsub ategoriesW n ofW( 1 P(Z))withH i (W n ;Q)  = H i (W n+1 ;Q)forn> iandwe anestablishanisomorphismH

 (W n ;Q)  = !H  (  n = n ;Q)(see[CL1, CL2℄ formore details).

4 The hermitian K-theory of additive ate-gories

First, let us re all the denition of a symmetri monoidal ategory and its K-theory asdened by Quillen [Q2℄.

Denition 4.1 A symmetri monoidal ategory (C;;1 C

) is a ategory C together with a fun tor : C C ! C and an obje t 1

C

su h that we have natural isomorphisms 1 C A  = A  = A1 C , (AB)C  = A(BC) and AB 

=BA for any obje ts A;B;C of C su h that ertain diagrams [Q2, p.218℄ ommute.

Any additive ategory is a symmetri monoidal ategory, of ourse (take =),but there are other important examplesas we willsee below. Given two symmetri monoidal ategories C and D, we dene an a tion of C on D to be a fun tor F : C D ! D fullling some obvious onditions. We then dene the ategory <C;D>. Its obje ts are the obje ts of D. A morphism from D to D

0

is an equivalen e lass of a pair (C;f) with f : F(C;D)!D

0

. Forexample,wehavethe diagonala tionof C onCC, and we set C

1

C :=<C;CC>. Denition 4.2 Let (C;;1

C

) be a symmetri monoidal ategory and S = Iso(C) be the sub ategory of C with the same obje ts, and whose morphisms are the isomorphisms of C. Then we dene the K-theory of C by

K n (C)= n (S 1 S) 8n0:

Let C be a ategory equipped with a duality fun tor t : C ! C op and a natural isomorphism  : Id C ! t Æ t satisfying t  M Æt M = Id t M for all obje ts M of C. We \ hoose"and  as in the ase of modules; in general, the hoi e will often be  = Id and  = 1 (or  = -1 if C is additive). More pre isely, one should onsider the ategory of additive ategories with dualityandaskthatfun torsinthis ategory ommutewiththeidenti ation isomorphismsuptonaturalisomorphism;see[S h2 ℄forthedetails. Asbefore, we will suppress the isomorphism . This is justied by the fa t that any ategory withduality(C;

t

;)isequivalenttoa ategory withdualitywhere the isomorphismbetween the identityfun tor and the bidualisgiven by the identity, see [S h2 ℄for the details. We an then dene:

Denition 4.3 Let C h

be the ategory of hermitian obje ts relative to (C; t ;;). Anobje tisanisomorphism :M  = ! t M su hthat t Æ M =. A morphism :(M;)!(N; ) is a ommutative square

M 

//

 = 



N  =



t t t 

oo

Remark: As before (see Denition 3.4) we have the notion of a hyperboli obje t and we write C

H

for the full sub ategory of hyperboli obje ts in C h

. We an also drop the ondition that the morphisms  and are isomor-phisms. Then we obtainthe ategoryC

hd

of all hermitianobje ts, in luding the degenerateones. Of ourse, C

hd

ontains C h

asa full sub ategory.

If C is additive (resp. symmetri monoidal) and t

respe ts this stru -ture, we all C

h

the hermitian ategory asso iated to C. This ategory C h is still symmetri monoidal, so its K-theory is dened by the above S

1 S- onstru tion. A hermitianexa t ategory isno longerexa t.

Examples: 1)C = 

P(A) h

isanadditive ategorywithdualityHom A

( ;A). 2) The ategory of nite abelian groups together with

t

=Hom Z

( ;Q=Z ) gives a hermitian exa t ategory. This example will be extended to torsion modules over Dedeking rings inse tion 9.

3)Let (X;O X

)be as heme. Thenthe lo allyfreeO X

-sheavesof niterank Ve t(X) forman exa t ategory with Hom

O X ( ;O X )as dualityfun tor. 4)C C op

admitsadualityfun torsending(A;B) (f;g op ) ! (C;D)to(B;A) (g op ;f) ! (D;C) in (CC op ) op  = C op C. If C is exa t, C C op isalso exa t: (f;g op ) is an admissible monomorphismif f is an admissiblemonomorphismand g is anadmissibleepimorphism inC et .

For the rest of this hapter, let us x a hermitian additive ategory C h and its full sub ategory of hyperboli obje tsC

H .

Obviously,all of parts 3.4-3.14 (ex ept 3.7) remains true for hermi-tian additive ategories,repla ing \module"by\obje t", P(A)by C and 

P(A) h

by C h

. (In3.5b),\Ifthereexists  2 enter(A) su h that+  =1" has to be repla ed by \If for any obje t E, there exists  :

t M ! t M su h that t (Æ )+ Æ = 1 8 = t  : M  = ! t M"). We still write S H =Iso(C H )  =Iso (W(C H )) and S h =Iso (C h )  =Iso(W(C h )).

S hli hting's proves Theorem 3.15 for additive ategories as well using that we have the Fundamental Theorem 3.7 also for additive ategories. This is the ase, as there is a way to generalize ertain isomorphisms fromthe her-mitian K-theory of rings to additive pseudo-abelian ategories (also alled \karoubian",\idempotent omplete"or\saturated"). Re allthatafter[Ka5℄, a ategory is alled pseudo-abelian if for any obje t E and any proje tion p

2

=p2End(E); kerpexists. Moreover, for any additive ategorythere is a pseudo-abelian ategory

e

C whose obje tsare ouples(E;p) where E is an obje t of C and p

2

=p2End(E). Then we have a full in lusionC

//

//

e C indu ed by E 7! (E;1

E

). M. S hli hting drew my attention to the idea of the following proposition whi h also applies to more general situations su h as homotopy artesian squares. We write add at for the ategory of

free R -modules.

Proposition 4.4 Let D and E be two fun tors add at!
op

Set ommut-ing with ltered olimits and sending equivalen es of ategories to homotopy equivalen esof simpli ialsets. Supposethatthereisa naturaltransformation  : D ! E su h that DP(R )

 P(R)

! EP(R ) is a homotopy equivalen e for any ring R . Then DA

 A

! EA is a homotopy equivalen e for any pseudo-abelian ategory A.

Proof: Wesaythatanadditive ategoryisnitelygeneratedifthereisan obje tA



su hthat any obje tof is adire t summand of (A  ) n for some n. Dene R := End(A  ). We have in lusions F(R )

//

//



//

//

P(R ). (givenbytheisomorphismofthesub ategoryofP(R ) onsistingofthesingle obje t R and the sub ategory of P(A



) onsisting of the single obje t A 

.) The pseudo-abelianization ~ gives P(R ) '

g F(R )

//

//

~ 

//

//

g P(R ) ' P(R ) and we therefore get an equivalen e of ategories P(R ) '

~

. It fol-lows that  is a homotopy equivalen e for any nitely generated pseudo-abelian ategory. Let fgA be the ategory of nitely generated sub ate-gories of A. Then this implies ho olim

fgA D( ~ ) ' ho olim fgA E( ~ ) and onsequently olim fgA D( ~ ) ' olim fgA E( ~

) be ause our olimits are l-tered[BK,XII,3.5℄. Finally, olim and~ ommuteandany additive ategory isthe ltered(see the followingremark) olimitof itsnitelygenerated sub- ategories. Repla ingEnd(A  ) by End(A   t A 

) in the above argument, we an alsoprovethatanydualityfun toronanynitelygeneratedadditive pseudo-abelian ategory is isomorphi to Hom

R

( ;R ) onP(R ).

Remark: Wesay that a ategory(and inparti ulara partiallyordered set) is \ oltered" (or \right ltered") if i) it is non-empty, ii) for any two ob-je ts, there is a \bigger"obje t and iii) for any pair of morphisms between two obje ts there is a morphismwhi h oequalizes them (Quillen[Q1℄ alls su h a ategory \ltering"). We also have the dual notion of \ltered" (or \left ltered"). Nevertheless, we say \ltered olimits"as in[BK℄ insteadof \ oltered olimits".

Of ourse,wewanttohaveourresultsnotonlyforpseudo-abelian ategories, butforadditive ategoriesingeneral. Foranarbitraryadditive ategoryC we observethat thein lusionC

//

//

~

C is onal(be ause(A;p)(A;1 p)  = (A;1) ). This implies that the in lusion (S

1 S) 0 ! ( ~ S 1 ~ S) 0 indu es a ho-motopyequivalen ewhere D

0

always standsforthe onne ted omponentof 0 of a ategory D[Q2, p.221and 224℄.

In fa t, two years ago I was not aware of the mistake in the proof of [CL2℄. I generalized their result (whi h was onlyvalidfor rings) to additive ategories [Ho℄usingessentiallyProposition4.4, onalityand thefollowing:

Lemma 4.5 The in lusion W(C H )!W(C h ) 0 is a homotopy equivalen e. Proof: Let W N

be the full sub ategory of W(C h ) 0 of obje ts (P;) su h that (P;)(H(N); N

) is hyperboli . Then we have that 8M;N obje ts of C, the in lusion  : W

M

! W MN

is a homotopy equivalen e. To see this, let  : W MN ! W M be given by (P;) = (P H(N); N ) and (f)=fid H(N)

. We then have anatural transformation:Id W

M

)Æ denedby

(P;)

=[(p;(PN;0);i)℄andasimilarnaturaltransformation Id

W MN

)  Æ. Let symmon at be the ategory of (small) symmetri monoidal ategories. ForanyabelianmonoidA, wewrite

^

A forthe ategory whose obje ts are the elements of A and

^

A(a;b)=fCja+ =bg. Then this implies that the naturaltransformation

0

)W isahomotopy equivalen e for any obje t of

d  0 (S) where
0 : d  0

(S) ! symmon at is the onstant fun tor

0

(M) = W(C H

). The se ond fun tor is given by W(N) = W N on obje ts and on morphisms it is the in lusion of ategories. With [BK, XII,3.5℄ and Lemma 3.14 we an on lude that W(C

H ) 'ho olim d  0 (S)
0 ' ho olim d  0 (S) W ' olim d  0 (S) W 'W(C h ) 0 .

Remark: The ategory<S;S h

>(see[Q2℄)and thedoublemapping oneof Thomason[Th2 ℄givetwoothermodelsforthehomotopytypeofW(C

h );and the latter one ts the above homotopy bration even if 2 is not invertible. The advantage of our ategoryis thatit generalizeswell toexa t ategories, as we willsee in the next se tion.

AnothermodelforW(C h

)intheadditive ase anbe onstru tedinterms of \pseudo-simpli ial symmetri monoidal ategories" using [Ja2, Corollary 4.8℄. Inthe samearti le(p.192),Jardinesuggests adenitionofetale hermi-tian K-theorywith nite oeÆ ients fors hemes, usingaglobally brant re-pla ement(withrespe ttothe losedmodelstru tureof[Ja1℄)ofthepresheaf of hermitian K-theory asdened in this se tion. We ould repla e theetale topology by the Zariski or the Nisnevish topology and work with integral oeÆ ients as well, ompare with the lastparagraph of se tion10.

5 The hermitian K-theory of exa t ategories

Re all the denition of an exa t ategory due to [Q1℄ [Ke1℄:

Denition 5.1 Anexa t ategory C is an additive ategory, together with a lass of sequen es fA

//

//

B

// //

Cg, alled\exa t sequen es". Wesay that A

//

//

B isan \admissible monomorphism" and that

B

// //

C is an \admissible epimorphism". They fulll the following ax-ioms:

i) The lass of admissible monomorphisms is losed under omposition and under obase- hange.

ii) The lass of admissible epimorphisms is losed under omposition and under base- hange.

iii) Any sequen e isomorphi to an exa t sequen e is exa t. Any sequen e of the form A  1

//

//

AB  2

// //

B is exa t.

iv)Inany exa tsequen eA

//

//

B

// //

C, AisakernelforB

// //

C and C is a okernel for A

//

//

B.

Infa t,thedenitionwegivehereisduetoKellerwhosimpliedthe original denitionofQuillen,showingthatthefollowingaxiom(anditsdual)isa on-sequen e oftheotheraxioms: Ifanadmissiblemonomorphismi:C

//

//

D an be fa tored as i = v Æu su h that u has a okernel in C, then u is an admissiblemonomorphism.

Any additive ategory an be onsidered as an exa t ategory, taking the split monomorphisms and epimorphisms to be the admissible ones. Of ourse,ingeneralthereareother hoi esforfamiliesofadmissibleshortexa t sequen es. We will frequently use \additive" as a synonym for an exa t ategory in whi h every exa t sequen e splits. Another example for exa t ategories is to take a onvenient sub ategory of an abelian ategory and taking all monomorphisms and epimorphisms to be admissible. Following Quillen,we then asso iate toany exa t ategory C a ategoryQC whi h has the same obje ts as C. A morphism from C to D in QC is given by the equivalen e lass of a diagrammC

oooo

E

//

//

D.

Denition 5.2 Let C be an exa t ategory. Then its K-theory is dened by

K n

(C)= n

(QC) :

It isthis Q- onstru tion whi h allows Quillentoprove histheorems on reso-lution,lo alizationand devissage. Hisproof thatthe S

1

S- onstru tionand the Q- onstru tion oin idefor additive ategories and that they generalize

the Plus onstru tion for proje tive modules of nite type over a given ring an befound in[Q2℄.

Letusxanexa t ategoryC withduality. Obviously, all of parts 3.5-3.12a)(ex ept 3.7) remains truefor hermitian exa t ategories, re-pla ing\module"by \obje t", \split monomorphism"by \admissible mono-morphism", \split epimorphism" by \admissible epimorphism", P(A) by C and  P(A) h par C h .

The asso iated hermitian ategory to an exa t ategory with duality is no longer exa t orWaldhausen; itdoesnot even havea nal obje t. Therefore, its K-theory is not dened in general. If all short exa t sequen es split, we an denetheK-theoryas(thehomotopy groupsof)S

1 h

S h

. Butinallother ases, this denition would be as bad as taking S

1

S to dene lassi al K-theory of an exa t ategory inwhi h not allshort exa t sequen es split. The aim of this hapter is to dene for any exa t ategory C with duality a ategory L(C

h

) su h that there is a homotopy bration ontaining Q(C) and L(C

h

)andwhi h oin ides withthe one indu ed bythe hyperboli fun -tor as given in orollary 4.8 if the exa t stru ture is given by the additive stru ture (i.e. anyshort exa tsequen e splits). Inparti ular, wewouldhave ahomotopy equivalen eL(C h )'S 1 h S h

inthis ase. Furthermore,we would like to have (as it is known for modules, see [Ka7℄) L((C C

op ) h ) 'Q(C) . Ifsu h a ategory L(C h

) exists, itseems tobejustied to all it\the her-mitianK-theoryof the hermitianexa t ategoryC

h

". Bytheway, weshould mentionthatuptonow,everythingwehave alled\hermitianK-theory"was a spe ial ase of lassi al K-theoryfor symmetri monoidal ategories.

For any exa t ategory C with duality, we dene its Witt group by W(C h ) :=  0 (W(C h

)). As we now onsider short exa t sequen es whi h do not split in general, the fa t that (M;) ontains a \lagrangian" L = L

?

//

//

M nolongerimpliesanisomorphism(M;)  = (L ?  t L ? ;  0 1  d 

). Wethereforehavetodistinguishthesetwo lassesofobje tsand allthem metaboli andsplitmetaboli ,respe tively. Lemma5.3of[QSS ℄impliesthat givenamorphismfrom(P;)to(M;)inW(C

h

),theobje t(M;)(P; ) ismetaboli . ItfollowsthatW(C

h

)stillisthe monoidofisomorphism lasses of hermitian obje ts, divided by the equivalen e relationgenerated by iden-tifying the metaboli obje ts with 0.

Lemma 5.3 The fun tor H dened below is an equivalen e of ategories

H :Q(C)!W((C C op

) h

)

Proof: We set H (A) =((A;A);(1;1 op )) and H [A p

oooo

B i

//

//

C℄= the lass of

(A;A) (1;1 op )



(B;Ct B A) (p; ~ i op )

oooo



//

(i;~p op )

//

(C;C) (1;1 op )



(A;A)

//

( ~ i;p op )

//

(C t B A;B) (C;C) (~p;i op )

oooo

where ~

i and p~are dened by the bi artesian square B

//

i

//

p



C ~ p



A

//

~ i

//

Ct B A As the obje ts of (C C op ) h

are ne essarily of the form (E;F) (f;f op ) ! (F;E), the isomorphism[((1;1 op );((E;F);(f;f op ));(f;1 op )℄2 W((CC op ) h )(((E;F);(f;f op

));H (F))shows thatHisessentiallysurje tive. Itis learlyfaithful. Finally,Hisfullbe auseanymorphisminW((CC

op )

h ) is given by a artesian square.

Denition 5.4 Let F(C h

) be the homotopy ber of the forgetful fun tor W(C

h )

F

!Q(C).

This isonlyatopologi alspa e,nota ategory. Itenablesustogiveamodel for the K-theory of anexa t ategory:

Proposition 5.5 There is a homotopy equivalen e

Q(C)'F((CC op

) h

)

Proof: We havethe following diagram F((CC op ) h ) f

//

W((CC op ) h ) F

//

Q(CC op )  = 



Q(C)Q(C op )  = 



Q(C)

//

_

_

_

_

_

_

H '

OO

Q(C)Q(C) whi h is ommutative: ÆÆF ÆH (P q

oooo

L j

//

//

M) =  Æ((P;P) (q; ~ j op )

oooo

(L;Mt L P) (j;~q op )

//

//

(M;M))= (P;P) (q;q)

oooo

(L;L) (j;j)

//

//

(M;M)=
(P q

oooo

L j

//

//

M) where

is the diagonalmap,  is the anoni al isomorphism and  is the equiva-len e of ategories dened in [Q1℄. The existen e of the homotopy bration F((CC op ) h )!Q(C)
!Q(C)Q(C)implies F((CC op ) h )'hob(
) '

To nd a ategory whose lassifying spa e is homotopy equivalent to F(C

h

) (whi h will be nally given in Denition 5.10) we remark rst of all that a artesian square in the ategory of ategories is still artesian in the ategory of simpli ial sets be ause nerve : at !

op

Set has a left ad-joint[Th1 ℄ and therefore respe ts limits.

Re all [Q1,p.93℄that afun torg :B !C is alled\prebered"inthe sense of Quillen if for any obje t of C the in lusiong

1

( ) !( #g)has aright adjoint, g

1

( ) being the sub ategory of morphisms of B mappingto id C

.

Proposition 5.6 Given a diagram in the ategory of ategories F



A C B

//

g 



B g



A

//

C

where the square is artesian, (realizations of) A and C are onne ted or H-groups and F ! B

g

!C is a homotopy bration su h that g is bered in the sense of Quillen [Q1℄ and g fulllls the onditions of Quillen's theorem B (i.e., all base hanges are homotopy equivalen es). Theng

 isalso bered and F !A C B g  !A is a homotopy bration. Moreover, if B is ontra tible, then A

C B

g

! A ! C is also a homotopy bration.

Proof: Following [SGA1, VI,6.9℄, if g is bered, then g 

is also bered. A trivial al ulationshowsthatifgfullllsthe onditionsoftheoremB,thenso does g



. Itfollows that F 'hob (g)'g 1 ( )  =g  1 (a)'hob(g  ) 8a2A su h that f(a)= .

For the se ond assertion, repla eB g !C by a(Serre) brationp: B '

//

g

?

?

?

?

?

?

?

?

B C C I p

zzuu

uu

uu

uu

u

C

Then we have to he k that A  C B ! A  C C I  C B is a homotopy equivalen e. This is a onsequen e of the ve lemma applied to the long exa t homotopy sequen es of the homotopy brations F ! A

C B ! A and F !A C C I  C B !A

Of ourse,wewanttoapplythis propositionwhenA!C isthe forgetful fun torF :W(C

h

)!Q(C). LetE =E(C)bethe ategoryin[Q2℄. Itsobje ts aretheshortexa tsequen esinC,andamorphismfromA

//

//

B

// //

C toA 0

//

//

B 0

// //

C 0

diagram A

//

//

B

// //

C A 0

OO

OO

//

//

B





// //

C 0

OOOO





A 0

//

//

B 0

// //

C 0

where standsforabi artesiansquare. Observethatsayingthatthissquare is bi artesian is not an extra ondition but a onsequen e of the diagram. S = IsoC a ts on E by D+(A

//

//

B

// //

C) = DA

//

//

D B

// //

C. Following [Q2℄, we know that the proje tion on the third om-ponent g : S

1

E ! Q(C) fulllls all the onditions required in Proposition 5.5 ifallshortexa tsequen es inC split. Itfollowsthatwe haveahomotopy bration S 1 E 0  !S 1 E Q(C) W(C h ) g !W(C h ) where S 1 E 0 := g 1

(0). The somewhat surprising fa t now is that this homotopy bration indu ed by the forgetful fun tor F : W(C

h

) ! Q(C) oin ides with the homotopy bration of the previous hapter indu ed by the hyperboli fun tor. More pre isely, we have the following theorem:

Theorem 5.7 Let C h

be a hermitian exa t ategory in whi h all short exa t sequen es splitand 2isinvertible. Thenwehave adiagram of ategoriesand fun tors S 1 S S 1 H

//

j



S 1 S h 

//

J



W(C h ) S 1 E 0 

//

S 1 E Q(C) W(C h ) g

//

W(C h )

where the verti al morphisms are homotopy equivalen es, the rows are ho-motopy brations, the right-hand square ommutes and the left-hand square ommutes up to homotopy.

Proof: Observe rst of all that we have a homotopy equivalen e S 1 S h ' ! S 1 h S h

if 2 is invertible by onality. We let j be the trivial equivalen e of ategories given by (A;B) ! (A;B

1

//

//

B

// //

0). The

fun -tor J is dened on the obje ts by J(A;(M;)) := (A;0

//

//

M 1

// //

M;(M;)). A morphismfrom (A;(M;)) to (A 0 ;(M 0 ; 0 ) given by (C;C A  = ! A 0 ;(H(C); C )(M;)  = ! (M 0 ; 0 )) is mapped by J to (C;CA  = ! A; ;Æ) where is given by

C

//

//

CM

// //

M 0

OO

OO

//

//

CM

// //





CM





OOOO

0

//

//

M 0

// //

M 0 and Æ isgiven by M 



CM

oooo

0



//

//

M 0  0



t M

//

//

t C t M t M 0

oooo

It isobviousthat the right-handsquare ommutes. Fortheleft-hand side,it is straightforward to he k thatthere is anatural transformation :Æj ) J ÆS 1 H dened by  (A;B) =(0;A 1 !A; ;) where  is given by B

//

//

B

// //

0 0

OO

OO

//

//

B





// //

B

OOOO





0

//

//

H(B)

// //

H(B) and  is given by 0



B

oooo

0



//

//

H(B)  B



0

//

//

t B H(B)

oooo

TheupperrowisahomotopybrationbyTheorem3.15andtheloweroneby Proposition5.6. Finally, thevelemma appliedtothe long exa t homotopy sequen es implies that J is alsoahomotopy equivalen e.

Remark: In the appendix of [CL1℄, Charney and Lee give an alternative des ription of the ategoryE

Q(C) W(C

h

) whi h they alled E sp

. Moreover, they prove that 

 (S 1 E sp )Q  =   (S 1 S h )Q for C h = 1 P(Z) h . Karoubi [unpublished℄ introdu ed the following ategory: An obje t is a morphism M  ! t M with t  Æ  M =   su h that M 0

//

//

M is an admissible monomorphism. A morphism from from (M;) to (N; )) is a ommutativediagram

M ? N

||

||y

y

y

y

""

""D

D

D

D

D

D

D

D

M

//

i

//





N



t M t N t i

oooo

The interested reader an he k that E  Q(C)

W(C h

) is equivalent to this ategory where the equivalen e isindu ed by sending

(A

//

//

B p

// //

C;(C;)) to (B; t

pÆÆp). Hen e the \lo alization" by S of these ategoriesyieldsalternativemodelsforthe hermitianK-theory of an additive ategory (but not for anexa t ategory in general).

WenowgivethedenitionofthehermitianK-theoryofanexa t ategory whi h generalizesthe hermitianK-theoryof an additive ategory:

Denition 5.8 For anyhermitianexa t ategoryC h

inwhi h2isinvertible, we dene its hermitian K-theory by

K h n (C h ):= n (F(C h )) 8n 0

and its U-theory by

U(C h ):=W(C h ) U n (C h ):= n+1 (W(C h )) 8n0

Looking at the long exa t sequen e of homotopy groups for a bration, we immediatlyget along exa t sequen e

:::!U n (C h )!K n (C) H  !K h n (C h )!U n 1 (C h )!K n 1 (C) H  !K h n 1 (C h )!:::

where the morphismH is dened by topology, but we sti k tothis notation as it oin ides with the hyperboli fun tor inthe additive ase. Proposition 5.5 allows us to onstru t ategories whose lassifying spa es are homotopy equivalent to F(C

h

). For this, we need ontra tible ategories whi h are bered over Q(C) su h that all base hanges are homotopy equivalen es. Gien [Gif℄ onstru ts two su h ategories. As I amnot quite onvin ed by theproofofhis\bigK- onstru tion",Iwillrestri tmyattentiontohis\small K- onstru tion". Gien rst denes the ategory C :=(QC QC)

QC EC wherethe morphismstoQC aregivenby thedire tsumand theproje tion q to the third omponent,respe tively. He then denes EC by the artesian square of bered fun tors

EC

//

q 



EC q



C v

//

QC

where v is dened by the proje tion to the se ond fa tor QC QC ! QC. Moreexpli itly,anobje tofEC isgivenbyU

oooo

M

// //

V

oooo

N su h that the indu ed morphism M

// //

U V isalso an admissible epi-morphism. Using Quillen's Theorem A, Gien on ludes that EC is on-tra tible. Let u : C !QC be indu ed by the proje tion to the rst fa tor QC QC !QC. Letu  :=uÆq  and kC :=u 1  (0).

Proposition 5.9 [Gif, 3.4℄ For any exa t ategory C, there is a homotopy bration

kC !EC u !QC with ontra tible total spa e EC.

Proof: We want to apply Quillen's Theorem B. As u 

is bered, we have to show that all the base hanges are homotopy equivalen es whi h is done in [Gif,3.3℄.

Denition 5.10 Let L(C h

) be the ategory dened by

L(C h ):=EC QC W(C h )

Corollary 5.11 There is a homotopy equivalen e

L(C h )'F(C h ) Proof: As u 

is bered, we an apply proposition5.6.

Of ourse, the interested reader an trytogivea more beautiful des rip-tion of this ategory;one might alsolook out for ategories other than EC fullllingthe onditions of Proposition 5.6.

It should also be pointed out that there is a Waldhausen model in U -theory

 s

e

C (see[SY ℄)togetherwithaforgetfulfun tortothe lassi al Wald-hausen model s

e

C (see [Wald℄) whi h oin ides with our forgetful fun tor W(C

h

)!QC. This impliesa homotopy equivalen e L(C h )' n (hob(  s e C forget ! s e C)).

6 Lo alization I: Negative hermitian K-theory of additive ategories

The following se tion is joint work with Mar o S hli hting. Re all the fol-lowing lo alization theorem of Pedersen and Weibel for additive ategories ([PW℄, see also[Ka1℄):

Theorem 6.1 Let B be an additive ategory and A a pseudo-abelian full sub ategory su h that B is A-ltered in the sense of Karoubi. Then there is a homotopy bration

Iso(A) 1

Iso(A)!Iso(B) 1

Iso (B)!Iso (B=A) 1

Iso (B=A):

Here B=A is the ategory with the same obje ts asB. The morphisms from U to V inB=A are those in B modulo the ones fa toring through an obje t of A. As we want toprove a hermitian analogueof this theorem, we give a sket h of the proof:

Proof: One anshowthatB=Aisthelo alizationofBwithrespe ttothose morphisms whi h are a omposition of split monomorphisms with okernel in A and split epimorphismswith kernel in A. We even have left and right al ulusoffra tions(see[GZ℄forthedenitions). TheproofofPedersen and Weibeluses Thomason'sdoublemapping one (see[Th2, 5.1℄ andDenition 6.2 below) and the following two properties whi h are a onsequen e of the assumptions (see [S h1 ,Lemma 3.16,i)-iv)℄):

(P1)AnyisomorphismU 

!V inB=Aisrepresentedbyafra tionU 

oo

oo

W 

//

//

V where 

//

//

denotes a split monomorphism whose okernel

lies inA.

(P2) Given two morphisms f :U 

//

//

V and g :U 

//

//

V in B su h

that f =g in B=A, then there is a h:W 

//

//

U su h that f Æh=gÆh

in B.

Observethat the 

//

//

in(P2) are notmorphisms in<IsoA;IsoB>aswe

did not hoose a splitting. Wedenote by (i;p):U 



_//

V a\dire t mor-phism"fromU toV. Thismeansbydenitionthati:U



//

//

V asabove

and p is a retra tion of i. Hen e (i;p) is a morphismin <IsoA;IsoB> . As the dualstatement of (P2) for split epimorphisms isalsotrue, (P2) remains true when we repla e the



//

//

by 



_//

.

We will always note the obje ts of B by T;U;V::: and those of A by A;B;C:::. We will use the double mapping one of Thomason [Th2, 5.1℄ only in the spe ial ase of a symmetri monoidalfun tor A !B and allit the \mapping one"in this ase:

be a symmetri monoidal fun tor. Then the mapping one T =T(F) is the symmetri monoidal ategory with the same obje ts as AB. A morphism (A;U)!(B;V)isgivenbytheequivalen e lassof (C;D;:A!CB; : F(D)U !V) (see [Th2℄ for the equivalen e relation).

In fa t, this mapping one is a spe ial ase of Thomason's double mapping one. S hli hting [S h1 , Remarque 14.15℄ gave an easier proof of Theorem 6.1whi halsoneedsthemapping oneandthe abovetwoproperties. Instead of givinga hermitian version of the original proof of Pedersen and Weibel -whi his possible aswell- we willworkwith this simpliedversion. It is the following:

We rst observe that the mapping one T of IsoA ! IsoB is homotopy equivalent to <IsoA;IsoB> . In fa t, one an he k that the bers (U #p) of the evident proje tion p : T ! <IsoA;IsoB> are all ontra tible be- ause the fun tor <IsoA;IsoA>

op

! (U # p) given by A 7! ((A;U);1 U

) has a right adjoint and <IsoA;IsoA>

op

is ontra tible be ause it has a nal obje t. Thus p is a homotopy equivalen e by Quillen's theorem A [Q1℄. Applying [Th2, Lemma 2.3℄, it remains to show that the morphism  : <IsoA;IsoB> ! Iso (B=A) indu ed by the identity on the obje ts is a homotopy equivalen e. But applying (P1), (P2) and the statement dual to (P2) yields that the bers (U # ) are oltered (in the sense of [Q1℄) and hen e ontra tible.

PedersenandWeibelthendeneforeveryadditive ategoryAanadditive ategoryCAwhose lassifyingspa efor itsK-theoryis ontra tibleandsu h that CA is A-ltered. In fa t, they work with a slightly modied version of Karoubi's CA (whi h they all C

+

A). An obje t of CA is given by a sequen e A = (A

0 ;A

1

;:::) of obje ts of A, and a morphism f : A ! B is given by a matrix of morphisms f

ji : A

i ! B

j

in A su h that there exists an n with f

ji

= 0 whenever jj ij > n, and the ompositon is just dened by matrix multipli ation. Dene SA=CA=A to be the \suspension" of A, the quotient CA=A dened as in Theorem 6.1.. Then Iso(SA)

1

Iso(SA) is a delooping of Iso(A)

1

Iso (A) by their lo alizationtheorem if A is pseudo-abelian. Iterating this pro ess allows to dene the negative K-theory of an additive ategoryA, following Karoubi[Ka1℄.

Denition 6.3 The negative K-theory of an additive ategory A is dened by K n (A):=K 1 (S n+1 A) 8 n 0:

The hermitiananalogue of Theorem 6.1 will allowus to dene the negative K-theory of a hermitianadditive ategory.

to repla eB by B H

or by B h

(the ategory of hyperboli resp. allhermitian obje ts in B, see Denition 4.3). Roughly spoken, the ategory B

H

is too small, as the indu ed form on the okernel of a monomorphism between hyperboli obje ts isnot ne essarilyhyperboli . The ategoryB

h

istoo big, aswedonotknowhowtondaW asin(P1)and(P2)whi hisequippedwith a non-degenerate form. Wetherefore introdu e anintermediate ategory:

Denition 6.4 Let f B H A h

bethefullsub ategoryof B h

ofthose obje ts(U;) su h that there is an (A; ) in A

h

with (U;)(A; ) an obje t of B H

.

We willneed the following,whi his trivialif B=CA:

Lemma 6.5 For any dire t morphism Y 

//

//

H(X), there is a dire t

morphism H(Z) 

//

//

Y su h that the omposition H(Z) 

//

//

H(X)

is indu ed by the hyperboli fun tor.

Proof: The se tion Y  A  = ! X 

t

X and the ltration on X (as B is A-ltered) yieldsa diagram

A 

//

 0 b  



X t 



 =

vvnnn

nn

nn

nn

nn

nn

n

ZB ( 0 t b  )

//

t A where  
:A !Y A  = !X t

X. Then one he ks that the monomor-phism Z  t Z ! Z  t Z B  t B  = ! X t

X fa tors over Y su h that everything ommutes.

Theorem 6.6 Let B be anadditive ategory with duality fun tor in whi h 2 is invertible. Let A be a pseudo-abelian sub ategory losed under the duality fun tor su hthat B is A-ltered. Thenwe have a homotopy bration

Iso(A h ) 1 Iso (A h )!Iso( f B H A h ) 1 Iso( f B H A h )!Iso ((B=A) H ) 1 Iso((B=A) H )

Proof: As above in the additive ase, we will show that the bers of  : <Iso (A h );Iso( f B H A h )> ! Iso((B=A) H

) are oltered. Let H(X) be an obje tof (B=A) H . i) As H(X)is alsoanobje t of f B H A h ,( #H(X))is non-empty. ii)Giventwoobje ts(U;)and(V; )of

f B

H A

h

equippedwithanisomorphism in(B=A)

H

toH(X),we anapply(P1) and(P2)togeta ommonsubobje t (without form) W of U and V. Applying (P2) again, we an assume that j

W = j

W

. By Lemma 6.5, we an hoose W = H(Z). Applying (P2) and the lemmaagain, we nd a ommun hermitiansubobje t (H(T); ) of

(U;)and (V; ). Finally,observe thatthe indu ed formonthe omplement in A is alsonon-degenerate as2 is invertible [Kn, II,2.5.2℄.

iii) Given again two obje ts (U;) and (V; ) of f B H A h equipped with an isomorphism in (B=A) H

to H(X), together with two split monomorphisms g 1 ;g 2 :(U;) 

//

//

(V; ) respe ting the forms. Then we rst equalize g 1 and g

2

withoutformsand then pro eed asin ii),using both Lemma 6.5and the dual statement of Lemma 6.5.

Corollary 6.7 Under the hypotheses of Theorem 6.6, we have a homotopy bration Iso (A h ) 1 Iso(A h )!Iso(B h ) 1 Iso (B h )!Iso((B=A) h ) 1 Iso ((B=A) h )

Proof: We willwriteK (C) insteadof Iso (C) 1

Iso(C). Assume that wehave full in lusions of symmetri monoidal ategories C  D  E where C  E is onal (whi h implies that 

i

(K (C)) !  i

(K (E)) is an isomorphism for i1and amonomorphismfori=0). ThenDE isalso onaland hen e 

i

(K (C))! i

(K (D)) is anisomorphism for i1 and a monomorphismfor i = 0. This implies in parti ular that K (SA)

H

' K (A h

) when we apply Theorem 6.6 toACA, onsider the in lusionsCA

H  g CA H A h CA h and re all thatK (CA

H )'K (CA h )'fptg. Wewrite b B A

forthe fullsub ategory of those obje ts X of the pseudo-abelianization

b

B for whi h there exists an obje tA of

b

A with AX anobje t of B. As B isA-ltered, SB isSA- l-tered[Ka1,p.152℄and

SB

SA is

d

SA-ltered[S h1,Lemma3.8℄. OurTheorem

gives a homotopy bration K ( d SA h ) ! K ( g ( SB SA ) H d SA h ) ! K (S(B=A) H ). Here we use the equivalen es

SB

SA =

d

SA ' SB=SA ' S(B=A), the rst equivalen e is trivialand the se ond is shown in [Ka1,p.154℄. Applying the loop spa e fun tor to this homotopy bration, we are redu ed to estab-lish two homotopy equivalen es K (

d SA h ) ' K (SA H ) and K (SB H ) ' K ( g ( SB SA ) H d SA h

). The rst homotopy equivalen e follows from the onal in lusions SA H  d SA H  d SA h

. For the se ond, onsider the in lusions

SB H  ( SB SA ) H  SB H and ( SB SA ) H  g ( SB SA ) H d SA h  ( SB SA ) h and use the onality argumentsgiven atthe beginningof the proof.

Using the suspension fun tor S, we see that this is in fa t a homotopy bration of non- onne tive spe tra. The following denition is impli itlyin [KV℄, althoughthey only onsider the ase P(A).

Denition 6.8 Let A be a pseudo-abelian ategory with duality fun tor. Then we dene K n (A h ):=K 1 ((S n+1 A) h ) 8n 0

The resulting long exa t sequen e extends the 5-term exa t sequen e of Karoubi and Villamayor[KV, Theoreme 3.4℄.

Of ourse, we an nowalso denenegative U-theory.

Re ently, S hli hting[S h1 ℄gaveanon- onne tivedeloopingof theK-theory ofanexa t ategoryingeneral. Imitatinghis onstru tion,the orresponding theorems inthe hermitiansetting remainto be proved.