Marco Schlichting

Schlichting, Marco

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 357 (2019) 686–690Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comTopologySymplectic and orthogonal K -groups of the integers ✩K -groupes symplectiques et orthogonaux de l’anneau des entiersMarco SchlichtingMathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UKa r t i c l e i n f o a b s t r a c tArticle history:Received 8 June 2019Accepted after revision 5 August 2019Available online 20 September 2019Presented by Claire VoisinWe explicitly compute the homotopy groups of the topological spaces B Sp(Z)+,B O ∞,∞(Z)+, and B O ∞(Z)+.Crown Copyright © 2019 Published by Elsevier Masson SAS on behalf of Académie des sciences. All rights reserved.r é s u m éNous calculons explicitement les groupes d’homotopie des espaces topologiques B Sp(Z)+, B O ∞,∞(Z)+ et B O ∞(Z)+.Crown Copyright © 2019 Published by Elsevier Masson SAS on behalf of Académie des sciences. All rights reserved.Version française abrégéeSoient Sp(Z), O ∞,∞(Z) et O ∞(Z) le groupe symplectique infini, le 〈1, −1〉-groupe orthogonal infini et le groupe or-thogonal hyperbolique sur l’anneau des entiers Z. Ils sont obtenus comme réunion des sous-groupes Sp2n(Z), O n,n(Z) et O 2n(Z) de GL2n(Z) laissant invariantes les formes bilinéaires des matrices de Gram :⎛⎜⎜⎝0 1−1 0. . .0 1−1 0⎞⎟⎟⎠ ,⎛⎜⎜⎝1 00 −1. . .1 00 −1⎞⎟⎟⎠ and⎛⎜⎜⎝0 11 0. . .0 11 0⎞⎟⎟⎠ .Les groupes Sp(Z), O ∞,∞(Z) et O ∞(Z) ont des sous-groupes de commutateurs parfaits. Rappelons que, pour un tel groupe G , la construction plus de Quillen BG+ appliquée à l’espace classifiant BG de G est munie d’une application continue BG → BG+ qui induit un isomorphisme sur les groupes d’homologie intégrale et vaut G → G/[G, G] sur π1.Le but de cet article est de calculer explicitement les groupes d’homotopie des espaces topologiques B Sp(Z)+ , B O ∞,∞(Z)+ et B O ∞(Z)+ . Ces espaces sont des espaces de lacets infinis, puisqu’ils sont les composants connexes des ✩ This work was partially funded by the Leverhulme Trust.E-mail address: m.schlichting@warwick.ac.uk.https://doi.org/10.1016/j.crma.2019.08.0011631-073X/Crown Copyright © 2019 Published by Elsevier Masson SAS on behalf of Académie des sciences. All rights reserved.M. Schlichting / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 686–690 687espaces de la K -théorie K Sp(Z), GW (Z) et K Q (Z) des formes non dégénérées symplectiques, bilinéaires symétriques et quadratiques sur Z. On sait que les groupes d’homotopie de ces espaces sont des groupes abéliens de génération finie.Pour un groupe abélien A, on note Aodd le sous-groupe des éléments d’ordre impair fini.Théorème 0.1. Les groupes d’homotopie des espaces B Sp(Z)+ et B O ∞,∞(Z)+ pour n ≥ 1 sont donnés dans le tableau du Théo-rème 1.1.Théorème 0.2. L’application qui envoie une forme quadratique sur sa forme bilinéaire symétrique associée induit un morphisme d’es-paces de K -théorie K Q (Z) → GW (Z) qui est un isomorphismeπn B O ∞(Z)+∼=−→ πn B O ∞,∞(Z)+ en degré n ≥ 2et le monomorphisme (Z/2)2 ⊂ (Z/2)3 en degré n = 1.Remarque 0.1. Notons par Bk le k-ième nombre de Bernoulli [12, Exemple 24] et par dn le dénominateur de 1n+1 B(n+1)/4pour n = 3 mod 4. Selon [12, Introduction, Lemme 27], on a Kn(Z) = Z/2dn pour n = 3 mod 8 et Kn(Z) = Z/dn pour n = 7 mod 8. En outre, pour k > 0, les groupes K4k(Z) sont finis d’ordre impair et conjecturés zéro [12, Introduction]. Par exemple, K4(Z) = 0 [5]. Donc, on a pour n ≥ 1 le tableau de groupes d’homotopie présenté dans la Remarque 1.3.1. Statement of resultsLet Sp(Z), O ∞,∞(Z) and O ∞(Z) be the infinite symplectic, infinite 〈1, −1〉-orthogonal and infinite hyperbolic orthogo-nal groups over the integers. They are obtained as the union of subgroups Sp2n(Z), O n,n(Z) and O 2n(Z) of GL2n(Z) fixing the bilinear forms with Gram matrix⎛⎜⎜⎝0 1−1 0. . .0 1−1 0⎞⎟⎟⎠ ,⎛⎜⎜⎝1 00 −1. . .1 00 −1⎞⎟⎟⎠ and⎛⎜⎜⎝0 11 0. . .0 11 0⎞⎟⎟⎠ .The groups Sp(Z), O ∞,∞(Z) and O ∞(Z) have perfect commutator subgroups. Recall that, for such groups G , Quillen’s plus construction BG+ applied to the classifying space BG of G comes with a continuous map BG → BG+ , which induces an isomorphism on integral homology groups and is G → G/[G, G] on π1.The purpose of this article is to compute explicitly the homotopy groups of the topological spaces B Sp(Z)+ , B O ∞,∞(Z)+ and B O ∞(Z)+ . These spaces are infinite loop spaces since they are the connected components of the spaces K Sp(Z), GW (Z), and K Q (Z), which are the K -theory spaces of non-degenerate symplectic, symmetric bilinear and quadratic forms over Z. It is known that the homotopy groups of these spaces are finitely generated abelian groups.For an abelian group A, denote by Aodd the subgroup of elements of finite odd order.Theorem 1.1. The homotopy groups of the spaces B Sp(Z)+ and B O ∞,∞(Z)+ for n ≥ 1 are given in the following table:n mod 8 0 1 2 3 4 5 6 7πn B Sp(Z)+ Kn(Z) 0 Z Kn(Z) Z/2 ⊕ Kn(Z) Z/2 Z Kn(Z)πn B O ∞,∞(Z)+Z⊕Z/2⊕Kn(Z)(Z/2)3 (Z/2)2Z/8⊕Kn(Z)oddZ⊕ Kn(Z) 0 0 Kn(Z)Theorem 1.2. The map that sends a quadratic form to its associated symmetric bilinear form induces a map of K -theory spaces K Q (Z) → GW (Z) which is an isomorphismπn B O ∞(Z)+∼=−→ πn B O ∞,∞(Z)+ in degree n ≥ 2and the monomorphism (Z/2)2 ⊂ (Z/2)3 in degree n = 1.Remark 1.3. Denote by Bk the k-th Bernoulli number [12, Example 24] and let dn denote the denominator of 1n+1 B(n+1)/4for n = 3 mod 4. By [12, Introduction, Lemma 27], we have Kn(Z) = Z/2dn for n = 3 mod 8 and Kn(Z) = Z/dn for n = 7mod 8. Moreover, for k > 0, the groups K4k(Z) are finite of odd order and are conjectured to be zero [12, Introduction]. For example, K4(Z) = 0 [5]. In particular, for n ≥ 1, we have the following table of homotopy groups688 M. Schlichting / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 686–690n mod 8 0 1 2 3 4 5 6 7πn B Sp(Z)+ (0?) 0 Z Z/2dn Z/2 ⊕ (0?) Z/2 Z Z/dnπn B O ∞,∞(Z)+ Z⊕Z/2 ⊕ (0?) (Z/2)3 (Z/2)2 Z/dn Z⊕ (0?) 0 0 Z/dnwhere (0?) denotes a finite group of odd order conjectured to be zero.2. Proof part 1: odd torsionLemma 2.1. Let R be the ring of integers in a number field F . Then, for all n ≥ 0, there are isomorphismsK Q n(R)odd ∼= GWn(R)odd ∼= K Spn(R)odd ∼= (Kn(R)odd)C2where the action of C2 on K -theory is induced by GL(R) → GL(R) : M → t M−1 .Proof. The natural map K Q n(R)odd → GWn(R)odd is an isomorphism with inverse the cup product with the quadratic space associated with the Leech lattice 8 [4, Ch. 2, §6]. Write GW [0](R) and GW [2](R) for GW (R) and K Sp(R); see Section 3below for general GW [n] . The hyperbolic and forgetful maps factor as K [r](R)hC2 → GW [r](R) → K [r](R)hC2 ; see [7, (7.3) and Lemma 7.4], which does not use 1/2 ∈ R . Here K [n] denotes the K -theory spectrum K with C2-action induced by the n-th shifted duality Hom( , R[n]). On the spectrum level, this action depends on n = 0, 2. However, on homotopy groups, the actions agree for n = 0, 2. Denote by L[r] the homotopy cofibre of the map of spectra1 K [r](R)hC2 → GW [r](R), then L[r]i = L[r−1]i−1 only depends on the difference n − i, i ≥ 1 [9] andGW [r]n (R)[1/2] ∼= K [r]n (R)[1/2]C2 ⊕ L[r]n (R)[1/2]since the composition K [r](R)[1/2]hC2 → GW [r](R)[1/2] → K [r](R)[1/2]hC2 is an equivalence [7, Lemma B.14]. Strictly speak-ing, we define a non-connective version of L[r] as the homotopy colimit of the sequenceGW [r] → S1 ∧ GW [r−1] → S2 ∧ GW [r−2] → · · · (2.1)with appropriate delooping of GW [n] as in [7] using the definition of E [n] as below. The maps in (2.1) are the connecting maps of the homotopy fibration (3.1). Then we have by definition L[n]i = L[n−i]0 and, as in [7], we formally obtain the homotopy fibration whose connected cover we used above:(K [n])hC2 → GW [n] → L[n].By Lemma 3.4 below, the canonical map L[r]i (R)[1/2] → L[r]i (F )[1/2] is an isomorphism for i ≥ r. By [7, Proposition 7.2] and [1, Theorem 5.6], we haveL[r]i (F )[1/2] ={W (F )[1/2] r ≡ i mod 40 elsewhere W (F ) is the usual Witt group of F . But it is well known that W (F )[1/2] is a free Z[1/2]-module of rank the number of orderings of F . This proves the lemma for Kn Q , GWn for n ≥ 0 and Kn Sp for n ≥ 2. From the Zariski local-to-global spectral sequence, we see L[2]1 (R)[1/2] = L[1]0 (R)[1/2] = H1(R, L00[1/2]) = 0 since L[0]0 [1/2] is constant (flasque) on a ring of integers R and L[1]0 is Zariski-locally trivial. So, K1 Sp(R)odd = (K1(R)odd)C2 . Finally, L[2]0 (R) = 0 for a ring of integers since K0 Sp(R) = H0(R, Z), by the Zariski spectral sequence, hence H : K0(R) → K0 Sp(R) is surjective and L[2]0 = 0. Continue to assume that R is a ring of integers in a number field. Let  ∈Z be an odd prime and set R ′ = R[1/]. Then the inclusion R ⊂ R ′ induces an isomorphism: Kn(R){} ∼= Kn(R ′){} on -primary torsion subgroups for n ≥ 1. For i ≥ 1, the abelian group K2i(R ′) is finite and the group K2i−1(R ′) is finitely generated. For all i ≥ 1 and large ν , we therefore have an exact sequence0 → K2i(R ′){} → K2i(R ′,Z/ν) → K2i−1(R ′){} → 0 (2.2)[12, Lemma 68]. Since  is invertible in R ′ , which has cd(R ′) ≤ 2, the proved Quillen–Lichtenbaum conjecture says that the following change of topology map is an isomorphism K2i(R ′, Z/ν) ∼= K ét2i (R ′, Z/ν) for i ≥ 1. The change of topology map is C2-equivariant. From the etale local-to-global spectral sequence for K ét, we obtain the C2-equivariant isomorphism1 All spectra in this paper are −1-connected, and all homotopy fibrations are in the category of −1-connected spectra unless otherwise stated. In particular, the second map of a homotopy fibration need not be surjective on π0.M. Schlichting / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 686–690 689K2i(R′,Z/ν) ∼= K ét2i (R ′,Z/ν) ∼= H0ét(R′, K2i/ν) (2.3)[12, Proof of Theorem 70], on which the action on the left is GL(R) → GL(R) : M → t M−1 and on the right-hand side it is multiplication with (−1)i . Combining (2.2) and (2.3), Lemma 2.1 yields the following.Theorem 2.2. Let R be a ring of integers in a number field, and  ∈Z an odd prime. Then for all n ≥ 1 we have isomorphismsGWn(R){} ∼= K Spn(R){} ∼= K Q n(R){} ∼={Kn(R){} n ≡ 0,3 mod 40 n ≡ 1,2 mod 4.3. Proof part 2: 2-adic computationsFor an exact category with weak equivalences and duality (E , w, , can), denote by GW (E , w, , can) the associated Grothendieck–Witt space of symmetric bilinear forms [6, Definition 3]. If E has a strong symmetric cone [6, Definition 4], [9] I denote by E [1] = (MorE , wcone, , can) the exact category with weak equivalences and duality of morphisms in E with duality and double dual identification induced by functoriality of  and can and weak equivalences those maps f → g of arrows in E such that cone( f ) → cone(g) is a weak equivalence in E . By functoriality, E [1] also has a strong symmetric cone. Set GW [0](E ) = GW (E ) and define inductivily for r ≥ 1GW [r+1](E ) = GW [r](E [1]).By [6, Theorem 6], the sequenceEE →1E−→ Mor E 1−→ E [1]induces a homotopy fibration GW (E ) → K (E ) → GW [1](E ) of −1-connected spectra and by iteration the homotopy fibra-tionGW [r](E ) → K (E ) → GW [r+1](E ); (3.1)compare [7, Proof of Proposition 4.9]. For details and a generalisation, see [9]. For r < 0, we define GW [r](E ) such that (3.1)holds for all r ∈ Z. For a commutative ring R , we denote by GW [r](R) the space GW [r](Chb P(R), quis, Hom( , R), can)where P(R) is the category of finitely generated projective R-modules and quis is the set of quasi-isomorphisms.Theorem 3.1 ([10]). Let R be a commutative ring, then(1) GW [0](R) is the K -theory space GW (R) of the category of non-degenerate symmetric bilinear forms over R,(2) GW [2](R) is the K -theory space K Sp(R) of the category of non-degenerate symplectic forms over R, and(3) GW [4](R) is the K -theory space K Q (R) of the category of non-degenerate quadratic forms over R.In particular, by [8, Theorem 6.6, Example 3.11 and Remark 2.19], we haveGW [0](Z) = GW (Z)  Z×Z× B O ∞,∞(Z)+,GW [2](Z) = K Sp(Z)  Z× B Sp(Z)+,GW [4](Z) = K Q (Z)  Z×Z× B O ∞(Z)+.Theorem 3.2 ([11]). Let R be a Dedekind domain and S ⊂ R a multiplicative set of non-zero divisors. Then there is a natural homotopy fibration⊕℘∩S =∅GW [−1](R/℘) → GW [0](R) → GW [0](S−1 R).Recall that Friedlander [2] shows that Kn Sp(F2) is a finite group of odd order for n ≥ 1. In particular, its 2-adic comple-tion Kn Sp(F2)∧2 = 0 for n ≥ 1. Since the same is true for K (F2), we obtain GWn(F2)∧2 = 0 for n ≥ 1, GW [±1]n (F2)∧2 = 0 for n ≥ 0 and the following from Theorems 3.1, 3.2 and the homotopy fibration (3.1).Theorem 3.3. Let Z′ =Z[1/2] then the ring homomorphism Z →Z′ induces isomorphisms after 2-adic completionKn Sp(Z)∧2 ∼= Kn Sp(Z′)∧2 , n ≥ 0,GWn(Z)∧2 ∼= GWn(Z′)∧2 , n ≥ 1,K Q n(Z)∧2 ∼= K Q n(Z′)∧2 , n ≥ 2.690 M. Schlichting / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 686–690Finally, the 2-adic homotopy groups of K Sp(Z′) and GW (Z′) = K Q (Z′) can be found in [3, 4.7.2]. This proves the theorems in Section 1 apart from the following, which was needed in the proof of Lemma 2.1.Lemma 3.4. Let R be the ring of integers in a number field F . Then the inclusion R ⊂ F induces an isomorphismL[r]i (R)[1/2]  L[r]i (F )[1/2], i ≥ r.Proof. It suffices to prove the case r = 0 since L[r]i = L[0]i−r . From Theorem 3.2, we deduce the homotopy fibration of −1-connected spectra⊕℘ =(0)GW [−1](R/℘)[1/2] → GW [0](R)[1/2] → GW [0](F )[1/2]in which the right horizontal map is also surjective on π0, by the computations in [4]. Using the analogous statement for K -theory, we obtain the homotopy fibration of spectra⊕℘ =(0)L[−1](R/℘)[1/2] → L[0](R)[1/2] → L[0](F )[1/2].The left term in that fibration is trivial since for a finite field Fq , we haveL[−1](Fq)[1/2]  0.This is well known for q odd, and for q even, L[−1](Fq) is a module spectrum over L[0](F2) whose homotopy groups are 2-primary torsion, since on π0 it isL[0]0 (F2) = W (F2) = Z/2. References[1] P. Balmer, Triangular Witt groups. II. From usual to derived, Math. Z. 236 (2) (2001) 351–382.[2] E.M. Friedlander, Computations of K -theories of finite fields, Topology 15 (1) (1976) 87–109.[3] M. Karoubi, Bott periodicity in topological, algebraic and Hermitian K -theory, in: Handbook of K -Theory, Vol. 1, 2, Springer, Berlin, 2005, pp. 111–137.[4] J. Milnor, D. Husemoller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, New York, Heidelberg, 1973.[5] J. Rognes, K4(Z) is the trivial group, Topology 39 (2) (2000) 267–281.[6] M. Schlichting, The Mayer–Vietoris principle for Grothendieck–Witt groups of schemes, Invent. Math. 179 (2) (2010) 349–433.[7] M. Schlichting, Hermitian K -theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra 221 (7) (2017) 1729–1844.[8] M. Schlichting, Higher K -theory of forms I. From rings to exact categories, J. Inst. Math. Jussieu (2019), https://doi .org /10 .1017 /S1474748019000410, in prees.[9] M. Schlichting, Higher K -theory of forms II. From exact categories to chain complexes, in preparation.[10] M. Schlichting, Higher K -theory of forms III. From chain complexes to derived categories, in preparation.[11] M. Schlichting, Higher K -theory of forms for Dedekind domains, in preparation.[12] C. Weibel, Algebraic K -theory of rings of integers in local and global fields, in: E.M. Friedlander, D.R. Grayson (Eds.), Handbook of K -Theory, Vol. 1, Springer, Berlin, 2005, pp. 139–190.

Symplectic and orthogonal <i>K</i>-groups of the integers

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Symplectic and orthogonal <i>K</i>-groups of the integers

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