The stable Adams operations on Hermitian K-theory

Fasel, Jean¹ ; Haution, Olivier²

¹ Institut Fourier - UMR 5582, Université Grenoble Alpes, CNRS, Grenoble, France
² Mathematisches Institut, Ludwig-Maximilians-Universität München, München, Germany, Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Milano, Italy

Geometry & topology, Tome 29 (2025) no. 1, pp. 127-169.

Résumé

We prove that exterior powers of (skew-)symmetric bundles induce a $\lambda$-ring structure on the ring ${\mathrm{GW}}^0(X)\oplus {\mathrm{GW}}^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory.

Détail

DOI : 10.2140/gt.2025.29.127

Keywords: Hermitian K-theory, Grothendieck–Witt groups, Adams operations, $lambda$-rings

Bibliographie

[1] J F Adams, Vector fields on spheres, Ann. of Math. 75 (1962) 603 | DOI

[2] A Ananyevskiy, Stable operations and cooperations in derived Witt theory with rational coefficients, Ann. -Theory 2 (2017) 517 | DOI

[3] A Asok, B Doran, J Fasel, Smooth models of motivic spheres and the clutching construction, Int. Math. Res. Not. 2017 (2017) 1890 | DOI

[4] M F Atiyah, D O Tall, Group representations, λ-rings and the J-homomorphism, Topology 8 (1969) 253 | DOI

[5] T Bachmann, Motivic and real étale stable homotopy theory, Compos. Math. 154 (2018) 883 | DOI

[6] T Bachmann, M J Hopkins, η-periodic motivic stable homotopy theory over fields, preprint (2020)

[7] P Balmer, Witt groups, from: "Handbook of -theory, II" (editors E M Friedlander, D R Grayson), Springer (2005) 539 | DOI

[8] P Balmer, C Walter, A Gersten–Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. 35 (2002) 127 | DOI

[9] P Berthelot, Généralités sur les λ-anneaux, from: "Théorie des intersections et théorème de Riemann–Roch (SGA 6)", Lecture Notes in Math. 225, Springer (1971) 297

[10] P Berthelot, Le K∙ d’un fibre projectif : calculs et conséquences, from: "Théorie des intersections et théorème de Riemann–Roch (SGA 6)", Lecture Notes in Math. 225, Springer (1971) 365

[11] N Bourbaki, Algèbre, Chapitres I–III, Hermann (1970)

[12] B Calmès, E Dotto, Y Harpaz, F Hebestreit, M Land, K Moi, D Nardin, T Nikolaus, W Steimle, Hermitian K-theory for stable ∞-categories, II : Cobordism categories and additivity, preprint (2020)

[13] B Calmès, E Dotto, Y Harpaz, F Hebestreit, M Land, K Moi, D Nardin, T Nikolaus, W Steimle, Hermitian K-theory for stable ∞-categories, III : Grothendieck–Witt groups of rings, preprint (2020)

[14] B Calmès, E Dotto, Y Harpaz, F Hebestreit, M Land, K Moi, D Nardin, T Nikolaus, W Steimle, Hermitian K-theory for stable ∞-categories, I : Foundations, Selecta Math. 29 (2023) 10 | DOI

[15] F Déglise, J Fasel, The Borel character, J. Inst. Math. Jussieu 22 (2023) 747 | DOI

[16] J Fasel, V Srinivas, A vanishing theorem for oriented intersection multiplicities, Math. Res. Lett. 15 (2008) 447 | DOI

[17] H Gillet, C Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987) 243 | DOI

[18] J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445 | DOI

[19] B Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1 | DOI

[20] J Milnor, D Husemoller, Symmetric bilinear forms, 73, Springer (1973) | DOI

[21] I Panin, C Walter, On the motivic commutative ring spectrum BO, Algebra i Analiz 30 (2018) 43 | DOI

[22] I Panin, C Walter, Quaternionic Grassmannians and Borel classes in algebraic geometry, Algebra i Analiz 33 (2021) 136 | DOI

[23] J Riou, Catégorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math. France 135 (2007) 495 | DOI

[24] J Riou, Algebraic K-theory, A1-homotopy and Riemann–Roch theorems, J. Topol. 3 (2010) 229 | DOI

[25] M Schlichting, Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra 221 (2017) 1729 | DOI

[26] M Schlichting, G S Tripathi, Geometric models for higher Grothendieck–Witt groups in A1-homotopy theory, Math. Ann. 362 (2015) 1143 | DOI

[27] J P Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Inst. Hautes Études Sci. Publ. Math. 34 (1968) 37 | DOI

[28] A Grothendieck, P Berthelot, L Illusie, Théorie des intersections et théorème de Riemann–Roch (SGA 6), 225, Springer (1971)

[29] P Belmans, A J De Jong, Others, The Stacks project, electronic reference (2005–)

[30] C Walter, Grothendieck–Witt groups of triangulated categories, preprint (2003)

[31] M Zibrowius, Symmetric representation rings are λ-rings, New York J. Math. 21 (2015) 1055

[32] M Zibrowius, The γ-filtration on the Witt ring of a scheme, Q. J. Math. 69 (2018) 549 | DOI