Annala, Toni Chern classes in precobordism theories. (English) Zbl 1523.14008 J. Eur. Math. Soc. (JEMS) 25, No. 4, 1379-1422 (2023). Summary: We construct Chern classes of vector bundles in the universal precobordism theory of Annala-Yokura over an arbitrary Noetherian base ring of finite Krull dimension. As an immediate corollary, we show that the Grothendieck ring of vector bundles can be recovered from the universal precobordism ring, and that we can construct candidates for Chow rings satisfying an analogue of the classical Grothendieck-Riemann-Roch theorem. We also strengthen the weak projective bundle formula of Annala-Yokura to the case of arbitrary projective bundles. Cited in 3 Documents MSC: 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C40 Riemann-Roch theorems 19E08 \(K\)-theory of schemes Keywords:algebraic cobordism; derived algebraic geometry; Chern classes; projective bundle formula × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Annala, T.: Bivariant derived algebraic cobordism. J. Algebraic Geom. 30, 205-252 (2021) Zbl 1470.14006 MR 4233182 · Zbl 1470.14006 [2] Annala, T.: Precobordism and cobordism. Algebra Number Theory 15, 2571-2646 (2021) MR 4377859 · Zbl 1493.14036 [3] Annala, T., Yokura, S.: Bivariant algebraic cobordism with bundles. arXiv:1911.12484 (2019) [4] Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V., Yakerson, M.: Modules over algebraic cobordism. Forum Math. Pi 8, art. e14, 44 pp. (2020) Zbl 1458.14027 MR 4190058 · Zbl 1458.14027 [5] Khan, A. A., Rydh, D.: Virtual Cartier divisors and blow-ups. arXiv:1802.05702 (2018) [6] Levine, M.: Comparison of cobordism theories. J. Algebra 322, 3291-3317 (2009) Zbl 1191.14023 MR 2567421 · Zbl 1191.14023 [7] Levine, M., Morel, F.: Algebraic Cobordism. Springer Monogr. Math., Springer, Berlin (2007) Zbl 1188.14015 MR 2286826 · Zbl 1188.14015 [8] Levine, M., Pandharipande, R.: Algebraic cobordism revisited. Invent. Math. 176, 63-130 (2009) Zbl 1210.14025 MR 2485880 · Zbl 1210.14025 [9] Lowrey, P. E., Schürg, T.: Derived algebraic cobordism. J. Inst. Math. Jussieu 15, 407-443 (2016) Zbl 1375.14087 MR 3466543 · Zbl 1375.14087 [10] Lurie, J.: Higher Topos Theory. Ann. of Math. Stud. 170, Princeton Univ. Press, Princeton, NJ (2009) Zbl 1175.18001 MR 2522659 · Zbl 1175.18001 [11] Lurie, J.: Higher algebra. http://www.math.harvard.edu/ lurie/papers/HA.pdf (2017) [12] Lurie, J.: Spectral algebraic geometry. http://www.math.harvard.edu/ lurie/papers/SAG-rootfile.pdf (2018) [13] Quillen, D.: Elementary proofs of some results of cobordism theory using Steenrod operations. Adv. Math. 7, 29-56 (1971) Zbl 0214.50502 MR 290382 · Zbl 0214.50502 [14] Toën, B., Vezzosi, G.: Homotopical algebraic geometry. I. Topos theory. Adv. Math. 193, 257-372 (2005) Zbl 1120.14012 MR 2137288 · Zbl 1120.14012 [15] Toën, B., Vezzosi, G.: Homotopical algebraic geometry. II. Geometric stacks and applications. Mem. Amer. Math. Soc. 193, no. 902, x+224 pp. (2008) Zbl 1145.14003 MR 2394633 · Zbl 1145.14003 [16] Yokura, S.: Oriented bivariant theories. I. Int. J. Math. 20, 1305-1334 (2009) Zbl 1195.55006 MR 2574317 · Zbl 1195.55006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.