Using an idea due to R.Thomason, we define a "homology theory" on the
category of rings which satisfies excision, exactness, homotopy (in the
algebraic sense) and periodicity of order 4. For regular noetherian rings, we
find P. Balmer's higher Witt groups. For more general rings, this homology
isomorphic to the KT-theory of J. Hornbostel, inspired by the work of B.
Williams. For real or complex C*-algebras, we recover - up to 2 torsion -
topological K-theory.Comment: 6 pages ; see also http://www.math.jussieu.fr/~karoubi

Karoubi, Max

English

arXiv

Max Karoubi

Comptes Rendus Mathématique

1/anceexcision,gher Witts.orie degébrique)e Balmer.l dehelC. R. Acad. Sci. Paris, Ser. I 342 (2006) 165–168http://france.elsevier.com/direct/CRASSAlgebra/TopologyStabilization of the Witt groupMax KaroubiDépartement de mathématiques, UMR 7586 du CNRS, case 7012, université Paris 7, 2, place Jussieu, 75251 Paris cedex 05, FrReceived 26 September 2005; accepted after revision 16 November 2005Presented by Alain ConnesAbstractIn this Note, using an idea due to Thomason, we define a “homology theory” on the category of rings which satisfiesexactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmer’s higroups. For more general rings, this homology is isomorphic to theKT-theory of Hornbostel, inspired by the work of WilliamFor real or complexC∗-algebras, we recover – up to 2 torsion – topologicalK-theory.To cite this article: M. Karoubi, C. R. Acad.Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméStabilisation du groupe de Witt. En utilisant une idée due à Thomason, nous définissons dans cette Note une « thél’homologie » sur la catégorie des anneaux qui satisfait aux propriétés d’excision, d’exactitude, d’homotopie (au sens alet de périodicité d’ordre 4. Pour les anneaux noethériens réguliers, nous retrouvons les groupes de Witt supérieurs dPour des anneaux plus généraux, cette homologie est isomorphe à laKT-théorie définie par Hornbostel et inspirée par le travaiWilliams. Pour les algèbres stellaires, réelles ou complexes, nous retrouvons – à la 2 torsion près – laK-théorie topologique.Pourciter cet article : M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.1.Let A be a ring with an anti-involutiona → ā and letε be an element of the center ofA such thatεε̄ = 1. Weassume also that 2 is invertible in the ring. There are now well known definitions of the higher hermitianK-group(denoted byεLn(A), as in [5]) and the higher Witt groupεWn(A): this is the cokernel of the map induced by thyperbolic functorKn(A) −→ εLn(A)where theKn(A) denote the QuillenK-groups (which are defined for all values ofn ∈ Z). One of the fundamentaresults of higher Witt theory is the periodicity isomorphism (whereZ′ = Z[1/2], cf. [4])εWn(A) ⊗ Z′ ∼= −εWn−2(A) ⊗ Z′.E-mail address:karoubi@math.jussieu.fr (M. Karoubi).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.12.014166 M. Karoubi / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 165–168all2srtesianeheofIt is induced by the cup-product with a genuine elementu2 ∈ −1L−2(Z′). By analogy with algebraic topology, we shcall u2 the Bott element in Witt theory. This element is explicitly described in the following way. We consider the×2matrix (with the involution defined bȳz = z−1 andt̄ = t−1 and where we putλ = λ̄ = 1/2), defined in [4] p. 243. Thi2× 2 matrix represents an element of−1L0(Z′[t, t−1, z, z−1]) whose image in−1L−2(Z′) ∼= −1W−2(Z′) ∼= Z ⊕ Z/2is a free generator (cf. [5] for the details).2.The higher Witt groupsεWn(A) have not all the nice formal properties one should expect. For instance, a casquare of rings with anti-involutions (where the vertical maps are surjective)AA1A2 A′does not induce, in general, a long Mayer–Vietoris exact sequence of Witt groups−→ εWn+1(A′) −→ εWn(A) −→ εWn(A1) ⊕ εWn(A2) −→ εWn(A′) −→ .Following an idea due to Thomason [8], one may overcome this difficulty bystabilizingthe higher Witt groups. Morprecisely, we define a new theoryεWn(A) as the limit of the inductive systemεWn(A) −→ −εWn−2(A) −→ εWn−4(A) −→ · · ·where the arrows are induced by the cup-product with the Bott elementu2 mentioned above. As a matter of fact, tperiodicity mapεWn(A) −→ −εWn−2(A) can be factored asεWn(A) −→ −εLn−2(A) −→ −εWn−2(A).ThereforeεWn(A) is also the limit of the inductive systemεLn(A) −→ −εLn−2(A) −→ εLn−4(A) −→ · · · .Theorem 2.1. This new theoryεWn(A) satisfies the following properties:(a) Homotopy invariance.The polynomial extensionA → A[t], wheret̄ = t , induces an isomorphismεWn(A) ∼= εWn(A[t]).(b) Exactness and excision.From a cartesian square as above(with ψ surjective), one deduces an isomorphismthe associated relative groupsεWn(φ) ∼= εWn(ψ)and therefore aMayer–Vietorisexact sequence for alln ∈ Z−→ εWn+1(A′) −→ εWn(A) −→ εWn(A1) ⊕ εWn(A2) −→ εWn(A′) −→ .(c) Periodicity.The cup-product with theBott element induces the isomorphismsεWn(A) ∼= −εWn−2(A) ∼= εWn−4(A).(d) Normalization.Let us assume now thatA is a regular noetherian ring. Then the natural mapεW0(A) −→ εW0(A)is an isomorphism and the groupεW1(A) is isomorphic to the cokernel of the map defined in[5].k0(A) −→ εW1(A).Moreover, the groups1Wn(A) coincide with the higherWitt groups of Balmer[1].M. Karoubi / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 165–168 167ce isessreedermitianeheingasd in [4],entc 0 [7].ase, theons afterSketch of the proof. Periodicity is imposed by the definition (as in Thomason’s theory). Homotopy invariana consequence of the same property for the Witt groups. Since theLn-groups satisfy the excision and exactnproperties forn < 0 (cf. [6] for instance), this is also true of the theoryεWn: as we have noticed before,εWn(A) isalso the limit of the inductive systemεLn(A) −→ −εLn−2(A) −→ εLn−4(A) −→ .The rest of the theorem follows immediately from [5], Theorem 4.3 and from next theorem.Theorem 2.2. The homologyεW∗(A) is isomorphic to the KT* -theory of Hornbostel(cf. [3], Section5).Proof. ThisKT-theory is the direct limit of the systemεLn(A) −→ εLn(UA) −→ · · · −→ εLn(UrA) −→whereUA is the ring defined in [5], p. 263 andUrA the r-iteration of the “U -construction”. All the arrows above aL∗-module maps as defined in [4] p. 233 and [5] p. 276. This implies that the homomorphismεLn(A) −→ εLn(UrA)is the cup-product with a well defined elementwr in 1L0(UrZ′) (this is probably related to the question 6.6 raisby Hornbostel in his paper [3]). On the other hand, as a consequence of the fundamental theorem of hK-theory (cf. [5] p. 264), we have an isomorphism ofL∗-modules betweenεLn(UrA) and εLn(Ur+4A ) (as noticedalso by Williams [9]). Therefore, the previous direct limit is simply the limit of the systemεLn(A) −→ εLn−4(A) −→ · · ·where the arrows are defined by the cup-product with a specific elementw i 1L−4(Z′). On the other hand, wknow that if we apply this construction to the ringA = Z′ andε = 1, we find an isomorphism between1W0(Z′) and1L−4(Z′) = 1W−4(Z′) (because the ringZ′ is regular). As a matter of fact, we find a chain of isomorphisms1W0(Z′) ∼= 1L0(U2Z′) ∼= 1L0(U3Z′) ∼= 1L0(U3Z′) ∼= 1L0(U4Z′) ∼= 1L−4(Z′).I claim thatw, the image of 1 by this chain of isomorphisms, is(u2)2 up to a unipotent element. This is exactly twell known computation of the classical Witt ring ofZ′ which isZ ⊕ Z/2 generated by the classes of the followelements in the Grothendieck Witt group:〈x2〉 and〈x2〉 − 〈2x2〉. If A is regular noetherian andε = 1, Hornbostel has proved moreover in [3] thatKTn(A) is isomorphic to then-Witt group defined by Balmer, which proves the last part of Theorem 2.1.Remark 1. For simplicity’s sake, we have just considered hermitianK-theory groups. One could have takenwell homotopy colimits of the corresponding classifying spaces, using for instance the machinery developeSection 1, greatly generalized by Schlichting.Remark 2. Using [2] and most of the above properties of the theoryεWn, Schlichting was able to prove cdh descfor the theoryεW∗ extended to the category of (commutative) schemes of finite type over a field of characteristi3.WhenA is a Banach algebra, we may consider the topological analogs of the previous definitions. In that cgroupεWtopn (A) is simply isomorphic toεWtopn (A) ⊗ Z′. One should also notice thatεW topn (A) ⊗ Z′ is isomorphic toεWn(A) ⊗ Z′ as a consequence of the periodicity theorem and the well-known computation ofεW0 andεW1. Finally,if A is aC∗algebra, it is also well known that1W topn (A) is isomorphic to the topologicalK-theory ofA.AcknowledgementsIt is a pleasure to thank Jens Hornbostel and Marco Schlichting for their interesting remarks and suggestia first draft of this Note.168 M. Karoubi / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 165–168.References[1] P. Balmer, Witt groups, in: Handbook ofK-Theory, Springer-Verlag, 2005.[2] G. Cortinas, C. Haesemeyer, M. Schlichting, C. Weibel, Cyclic homology, cdh-cohomology and negativeK-theory, http://www.math.uiucedu/K-theory/0722/.[3] J. Hornbostel,A1 representability of HermitianK-theory and Witt groups, Topology 44 (2005) 661–687.[4] M. Karoubi, Théorie de Quillen et homologie du groupe orthogonal, Ann. Math. 112 (1980) 207–257.[5] M. Karoubi, Le théorème fondamental de laK-théorie hermitienne, Ann. Math. 112 (1980) 259–282.[6] M. Karoubi, O. Villamayor,K-théorie algébrique etK-théorie topologique II, Math. Scand. 32 (1973) 57–86.[7] M. Schlichting, in press.[8] R. Thomason, AlgebraicK-theory and etale cohomology, Ann. Sci. École Norm. Sup. 13 (1985) 437–552.[9] B. Williams, QuadraticK-theory and geometric topology, in: Handbook ofK-Theory, Springer-Verlag, 2005.

Stabilization of the Witt group

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