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Hermitian K-theory via oriented Gorenstein algebras

  • Marc Hoyois EMAIL logo , Joachim Jelisiejew , Denis Nardin and Maria Yakerson

Abstract

We show that the hermitian K-theory space of a commutative ring R can be identified, up to 𝐀 1 -homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor.

Funding statement: Marc Hoyois, Denis Nardin, and Maria Yakerson were partially supported by SFB 1085 “Higher invariants”. Joachim Jelisiejew was supported by NCN grant 2017/26/D/ST1/00913 and by the START fellowship of the Foundation for Polish Science.

Acknowledgements

We are thankful to Tom Bachmann, Joseph M. Landsberg, Rahul Pandharipande and Burt Totaro for helpful discussions. We would like to thank SFB 1085 “Higher invariants” and Regensburg University for its hospitality. Yakerson was supported by a Hermann-Weyl-Instructorship and is grateful to the Institute of Mathematical Research (FIM) and to ETH Zürich for providing perfect working conditions.

References

[1] H. Ananthnarayan, L. L. Avramov and W. F. Moore, Connected sums of Gorenstein local rings, J. reine angew. Math. 667 (2012), 149–176. 10.1515/CRELLE.2011.132Search in Google Scholar

[2] A. Ananyevskiy, O. Röndigs and P. A. Ø stvær, On very effective hermitian K-theory, Math. Z. 294 (2020), no. 3–4, 1021–1034. 10.1007/s00209-019-02302-zSearch in Google Scholar

[3] T. Bachmann, The generalized slices of Hermitian K-theory, J. Topol. 10 (2017), no. 4, 1124–1144. 10.1112/topo.12032Search in Google Scholar

[4] T. Bachmann, Cancellation theorem for motivic spaces with finite flat transfers, Doc. Math. 26 (2021), 1121–1144. 10.4171/dm/837Search in Google Scholar

[5] T. Bachmann, η-periodic motivic stable homotopy theory over Dedekind domains, J. Topol. 15 (2022), no. 2, 950–971. 10.1112/topo.12234Search in Google Scholar

[6] T. Bachmann, The very effective covers of KO and KGL over Dedekind schemes, preprint (2022), http://arxiv.org/abs/2201.02786. Search in Google Scholar

[7] T. Bachmann, E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo and M. Yakerson, On the infinite loop spaces of algebraic cobordism and the motivic sphere, Épijournal Géom. Algébrique 5 (2021), Article ID 9. 10.46298/epiga.2021.volume5.6581Search in Google Scholar

[8] T. Bachmann and J. Fasel, On the effectivity of spectra representing motivic cohomology theories, preprint (2018), http://arxiv.org/abs/1710.00594v3. Search in Google Scholar

[9] T. Bachmann and M. Hoyois, Norms in motivic homotopy theory, Astérisque 425, Société Mathématique de France, Paris 2021. 10.24033/ast.1147Search in Google Scholar

[10] T. Bachmann and K. Wickelgren, Euler classes: Six functors formalisms, dualities, integrality and linear subspaces of complete intersections, preprint (2021), http://arxiv.org/abs/2002.01848; to appear in J. Inst. Math. Jussieu. 10.1017/S147474802100027XSearch in Google Scholar

[11] T. Bachmann and M. Yakerson, Towards conservativity of 𝔾 m -stabilization, Geom. Topol. 24 (2020), no. 4, 1969–2034. 10.2140/gt.2020.24.1969Search in Google Scholar

[12] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin 1990. 10.1007/978-3-642-51438-8Search in Google Scholar

[13] P. Bürgisser, M. Clausen and M. A. Shokrollahi, Algebraic complexity theory, Grundlehren Math. Wiss. 315, Springer, Berlin 1997. 10.1007/978-3-662-03338-8Search in Google Scholar

[14] B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus and W. Steimle, Hermitian K-theory for stable -categories I: Foundations, preprint (2020), http://arxiv.org/abs/2009.07223. 10.1007/s00029-022-00758-2Search in Google Scholar

[15] B. Calmès, E. Dotto, Y. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus and W. Steimle, Hermitian K-theory for stable -categories II: Cobordism categories and additivity, preprint (2020), http://arxiv.org/abs/2009.07224. Search in Google Scholar

[16] B. Calmès, Y. Harpaz and D. Nardin, A motivic spectrum representing hermitian K-theory, in preparation, 2022. Search in Google Scholar

[17] G. Casnati and R. Notari, On some Gorenstein loci in i l b 6 ( k 4 ) , J. Algebra 308 (2007), no. 2, 493–523. 10.1016/j.jalgebra.2006.09.023Search in Google Scholar

[18] D. Clausen and A. Mathew, Hyperdescent and étale K-theory, Invent. Math. 225 (2021), no. 3, 981–1076. 10.1007/s00222-021-01043-3Search in Google Scholar

[19] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math. 150, Springer, New York 2013. Search in Google Scholar

[20] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo and M. Yakerson, Framed transfers and motivic fundamental classes, J. Topol. 13 (2020), no. 2, 460–500. 10.1112/topo.12134Search in Google Scholar

[21] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo and M. Yakerson, Modules over algebraic cobordism, Forum Math. Pi 8 (2020), Paper No. e14. 10.1017/fmp.2020.13Search in Google Scholar

[22] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo and M. Yakerson, Motivic infinite loop spaces, Camb. J. Math. 9 (2021), no. 2, 431–549. 10.4310/CJM.2021.v9.n2.a3Search in Google Scholar

[23] E. Elmanto and A. A. Khan, Perfection in motivic homotopy theory, Proc. Lond. Math. Soc. (3) 120 (2020), no. 1, 28–38. 10.1112/plms.12280Search in Google Scholar

[24] G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), J. Amer. Math. Soc. 34 (2021), no. 1, 261–313. 10.1090/jams/958Search in Google Scholar

[25] S. Gille, A transfer morphism for Witt groups, J. reine angew. Math. 564 (2003), 215–233. 10.1515/crll.2003.092Search in Google Scholar

[26] S. Gille, Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), no. 2, 211–233. 10.1007/s00209-003-0489-zSearch in Google Scholar

[27] R. Hartshorne, Deformation theory, Grad. Texts in Math. 257, Springer, New York 2010. 10.1007/978-1-4419-1596-2Search in Google Scholar

[28] F. Hebestreit and W. Steimle, Stable moduli spaces of hermitian forms, preprint (2021), http://arxiv.org/abs/2103.13911. Search in Google Scholar

[29] J. Hornbostel, A 1 -representability of Hermitian K-theory and Witt groups, Topology 44 (2005), no. 3, 661–687. 10.1016/j.top.2004.10.004Search in Google Scholar

[30] M. Hoyois, Cdh descent in equivariant homotopy K-theory, Doc. Math. 25 (2020), 457–482. 10.4171/dm/754Search in Google Scholar

[31] M. Hoyois, The localization theorem for framed motivic spaces, Compos. Math. 157 (2021), no. 1, 1–11. 10.1112/S0010437X20007575Search in Google Scholar

[32] M. Hoyois, J. Jelisiejew, D. Nardin, B. Totaro and M. Yakerson, The Hilbert scheme of infinite affine space and algebraic K-theory, preprint (2021), http://arxiv.org/abs/2002.11439; to appear in J. Eur. Math. Soc. Search in Google Scholar

[33] J. A. Jacobson, Cohomological invariants for quadratic forms over local rings, Math. Ann. 370 (2018), no. 1–2, 309–329. 10.1007/s00208-017-1561-zSearch in Google Scholar

[34] J. Jelisiejew and K. Šivic, Components and singularities of Quot schemes and varieties of commuting matrices, J. reine angew. Math. 788 (2022), 129–187.10.1515/crelle-2022-0018Search in Google Scholar

[35] J. M. Landsberg, Geometry and complexity theory, Cambridge Stud. Adv. Math. 169, Cambridge University, Cambridge 2017. 10.1017/9781108183192Search in Google Scholar

[36] J. M. Landsberg and M. Michał ek, Abelian tensors, J. Math. Pures Appl. (9) 108 (2017), no. 3, 333–371. 10.1016/j.matpur.2016.11.004Search in Google Scholar

[37] M. Levine, The homotopy coniveau tower, J. Topol. 1 (2008), no. 1, 217–267. 10.1112/jtopol/jtm004Search in Google Scholar

[38] A. López-Ávila, E , -ring structures in Motivic Hermitian K-theory Ph.D. thesis, Universität Osnabrück, 2017, https://osnadocs.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2018030216685/7/thesis_lopez-avila.pdf. Search in Google Scholar

[39] J. Lurie, On the classification of topological field theories, Current developments in mathematics 2008, International Press, Somerville (2009), 129–280. 10.4310/CDM.2008.v2008.n1.a3Search in Google Scholar

[40] J. Lurie, Higher algebra, Harvard University, Harvard 2017. Search in Google Scholar

[41] J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergeb. Math. Grenzgeb. 73, Springer, New York 1973. 10.1007/978-3-642-88330-9Search in Google Scholar

[42] F. Morel, 𝔸 1 -algebraic topology over a field, Lecture Notes in Math. 2052, Springer, Heidelberg 2012. 10.1007/978-3-642-29514-0Search in Google Scholar

[43] F. Morel and V. Voevodsky, 𝐀 1 -homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143. 10.1007/BF02698831Search in Google Scholar

[44] I. Panin, K. Pimenov and O. Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology Homotopy Appl. 10 (2008), no. 2, 211–226. 10.4310/HHA.2008.v10.n2.a11Search in Google Scholar

[45] I. Panin and C. Walter, On the algebraic cobordism spectra MSL and MSp, Algebra i Analiz 34 (2022), no. 1, 144–187. Search in Google Scholar

[46] M. Schlichting, Hermitian K-theory of exact categories, J. K-Theory 5 (2010), no. 1, 105–165. 10.1017/is009010017jkt075Search in Google Scholar

[47] M. Schlichting, Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra 221 (2017), no. 7, 1729–1844. 10.1016/j.jpaa.2016.12.026Search in Google Scholar

[48] M. Schlichting and G. S. Tripathi, Geometric models for higher Grothendieck–Witt groups in 𝔸 1 -homotopy theory, Math. Ann. 362 (2015), no. 3–4, 1143–1167. 10.1007/s00208-014-1154-zSearch in Google Scholar

[49] M. Spitzweck, A commutative 1 -spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. Fr. (N. S.) 157 (2018), 1–110. Search in Google Scholar

[50] The Stacks Project Authors, The Stacks Project, http://stacks.math.columbia.edu. Search in Google Scholar

[51] V. Voevodsky, Notes on framed correspondences, preprint 2001, https://www.math.ias.edu/vladimir/node/98. Search in Google Scholar

[52] M. Yakerson, The unit map of the algebraic special linear cobordism spectrum, J. Inst. Math. Jussieu 20 (2021), no. 6, 1905–1930. 10.1017/S1474748019000720Search in Google Scholar

Received: 2021-07-22
Revised: 2022-09-06
Published Online: 2022-10-27
Published in Print: 2022-12-01

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