Marco Schlichting

Comptes Rendus Mathématique

Comptes RendusMathématiqueMarco SchlichtingCorrigendum to “Symplectic and orthogonal K -groups of the integers”Volume 362 (2024), p. 307-308Online since: 2 May 2024https://doi.org/10.5802/crmath.606This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2024, Vol. 362, p. 307-308https://doi.org/10.5802/crmath.606Erratum / ErratumNumber Theory / Théorie des nombresCorrigendum to “Symplectic and orthogonalK -groups of the integers”Corrigendum à « K -groupes symplectiques etorthogonaux de l’anneau des entiers »Marco Schlichting aa Marco Schlichting, Mathematics Institute, Zeeman Building, University of Warwick,Coventry CV4 7AL, UKE-mail: m.schlichting@warwick.ac.ukAbstract. The action of the duality functor on the odd torsion of Kn (Z) was stated incorrectly in [3], in half ofthe cases, and lead to incorrect formulas for the odd primary torsion of πn BSp(Z)+ and πn BO∞,∞(Z)+.Résumé. L’action du foncteur de dualité sur la torsion impaire de Kn (Z) était énoncé incorrectement dans [3],dans la moitié des cas, et a conduit à des formules incorrectes pour la torsion primaire impaire deπn BSp(Z)+et πn BO∞,∞(Z)+.Manuscript received 6 November 2023, accepted 7 January 2024.Let R be a ring of integers in a number field and ℓ an odd prime. The formula [3, (2.3)] for theℓ-primary torsion subgroups of Kn(R) is false. The correct formulas extracted from [4, Proof ofTheorem 70] are for i > 1K2i (R){ℓ} = K2i (R ′){ℓ} = H 2et (R ′, (K2i+2)ℓ)where the right hand term is the inverse limit limν H 2et (R′, (K2i+2)/ℓν). In particular, the dualityacts by (−1)i+1 on this group. In odd degree for i > 1 we haveK2i−1(R){ℓ} = K2i−1(R ′){ℓ} = H 0et (R ′, (K2i )/ℓ∞)where the right hand term is the direct limit colimν H 0et (R′, (K2i )/ℓν). In particular, the dualityacts by (−1)i on this group. These formulas are not new; see [4, Theorem 70 and proof thereof];see also [1, Lemma 3.4.4]. Together with [3, Lemma 2.1] this yields now a correction of [3,Theorem 2.2] in the cases n ≡ 0,2 mod 4.Theorem 1. Let R be a ring of integers in a number field, and ℓ ∈ Z an odd prime. Then for alln ≥ 1 we have isomorphismsGWn(R){ℓ} ∼= K Spn(R){ℓ} ∼= KQn(R){ℓ} ∼={Kn(R){ℓ} n ≡ 2,3 mod 40 n ≡ 0,1 mod 4.[3, Section 3] is unaffected by this error and thus, we obtain the following table of homotopygroups correcting [3, Théorème 0.1 and Theorem 1.1] where the 2-primary part comes from [2,4.7.2] and [3, Theorem 3.3] and the odd primary part from Theorem 1. See also [1, Theorem 3.2.1].ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/308 Marco SchlichtingTheorem 2. The homotopy groups of the spaces B Sp(Z)+ and BO∞,∞(Z)+ for n ≥ 1 are given inthe following tablen mod 8 0 1 2 3 4 5 6 7πnB Sp(Z)+ 0 0 Z⊕Kn(Z)odd Kn(Z) Z/2 Z/2 Z⊕Kn(Z)odd Kn(Z)πnBO∞,∞(Z)+ Z⊕Z/2 (Z/2)3(Z/2)2⊕Kn(Z)oddZ/8⊕Kn(Z)oddZ 0 Kn(Z)odd Kn(Z)Finally, this leads to a correction of [3, Remark 1.3] and the following table for n > 0. Denote byBk the k-th Bernoulli number [4, Example 24] and let dn denote the denominator of1n+1 B(n+1)/4for n = 3 mod 4. By [4, Introduction, Lemma 27] we have Kn(Z) = Z/2dn for n = 3 mod 8 andKn(Z) =Z/dn for n = 7 mod 8. Similarly, denote by ck the numerator of Bk /4k. Then K4k−2(Z) is afinite group of order ck when k is even and of order 2ck when k is odd. This group is conjecturedto be cyclic.n mod 8 0 1 2 3 4 5 6 7πnB Sp(Z)+ 0 0 Z⊕|Z/ck | Z/2dn Z/2 Z/2 Z⊕|Z/ck | Z/dnπnBO∞,∞(Z)+ Z⊕Z/2 (Z/2)3 (Z/2)2 ⊕|Z/ck | Z/dn Z 0 |Z/ck | Z/dnwhere |Z/m| denotes a finite group of order n conjectured to be cyclic and n = 4k −2.Full proof of the claims in [3] are now available in [1].References[1] B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus, W. Steimle, “Hermitian K -theoryfor stable ∞-categories, Part III: Grothendieck-Witt groups of rings”, https://arxiv.org/abs/2009.07225, 2020.[2] M. Karoubi, “Bott periodicity in topological, algebraic and Hermitian K -theory”, in Handbook of K -theory. Vol. 1, 2,Springer, 2005, p. 111-137.[3] M. Schlichting, “Symplectic and orthogonal K -groups of the integers”, C. R. Math. Acad. Sci. Paris 357 (2019), no. 8,p. 686-690.[4] C. Weibel, “Algebraic K -theory of rings of integers in local and global fields”, in Handbook of K -theory. Vol. 1, 2,Springer, 2005, p. 139-190.

Corrigendum to “Symplectic and orthogonal $K$-groups of the integers”

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Corrigendum to “Symplectic and orthogonal KKK-groups of the integers”

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Corrigendum to “Symplectic and orthogonal $K$ -groups of the integers”