On the Riemann-Roch formula without projective hypotheses
Authors:
A. Navarro and J. Navarro
Journal:
Trans. Amer. Math. Soc. 374 (2021), 755-772
MSC (2010):
Primary 14C40, 14F42, 19E15, 19E20, 19L10
DOI:
https://doi.org/10.1090/tran/8107
Published electronically:
November 3, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a finite dimensional noetherian scheme. For any proper morphism between smooth
-schemes, we prove a Riemann-Roch formula relating higher algebraic
-theory and motivic cohomology, thus with no projective hypotheses either on the schemes or on the morphism. We also prove, without projective assumptions, an arithmetic Riemann-Roch theorem involving Arakelov's higher
-theory and motivic cohomology as well as an analogous result for the relative cohomology of a morphism.
These results are obtained as corollaries of a motivic statement that is valid for morphisms between oriented absolute spectra in the stable homotopy category of .
- [Ayo07] Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque 314 (2007), x+466 pp. (2008) (French, with English and French summaries). MR 2423375
- [BFM75] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101–145. MR 412190
- [Blo86] Spencer Bloch, Algebraic cycles and higher 𝐾-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, https://doi.org/10.1016/0001-8708(86)90081-2
- [BS58] Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136 (French). MR 116022
- [Cis13] Denis-Charles Cisinski, Descente par éclatements en 𝐾-théorie invariante par homotopie, Ann. of Math. (2) 177 (2013), no. 2, 425–448 (French, with English and French summaries). MR 3010804, https://doi.org/10.4007/annals.2013.177.2.2
- [CD19] Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, Springer Monographs in Mathematics, Springer, Cham, [2019] ©2019. MR 3971240
- [CD12] Denis-Charles Cisinski and Frédéric Déglise, Mixed Weil cohomologies, Adv. Math. 230 (2012), no. 1, 55–130. MR 2900540, https://doi.org/10.1016/j.aim.2011.10.021
- [Dég08] Frédéric Déglise, Around the Gysin triangle. II, Doc. Math. 13 (2008), 613–675. MR 2466188
- [Dég18] Frédéric Déglise, Orientation theory in arithmetic geometry, 𝐾-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, 2018, pp. 239–347. MR 3930052
- [Dég18b] Frédéric Déglise, Bivariant theories in motivic stable homotopy, Doc. Math. 23 (2018), 997–1076. MR 3874952
- [Del87] P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177 (French). MR 902592, https://doi.org/10.1090/conm/067/902592
- [Dye62]
E. Dyer,
Relations between cohomology theories,
Colloquium on Algebraic Topology, Aarthus Universitet, 1962, pp. 89-93. - [FG83] William Fulton and Henri Gillet, Riemann-Roch for general algebraic varieties, Bull. Soc. Math. France 111 (1983), no. 3, 287–300 (English, with French summary). MR 735307
- [Ful77] William Fulton, A Hirzebruch-Riemann-Roch formula for analytic spaces and non-projective algebraic varieties, Compositio Math. 34 (1977), no. 3, 279–283. MR 460323
- [Ful98] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- [FM81] William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, https://doi.org/10.1090/memo/0243
- [Gil81] Henri Gillet, Comparison of 𝐾-theory spectral sequences, with applications, Algebraic 𝐾-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 141–167. MR 618303, https://doi.org/10.1007/BFb0089520
- [Gil81b] Henri Gillet, Riemann-Roch theorems for higher algebraic 𝐾-theory, Adv. in Math. 40 (1981), no. 3, 203–289. MR 624666, https://doi.org/10.1016/S0001-8708(81)80006-0
- [HS15] Andreas Holmstrom and Jakob Scholbach, Arakelov motivic cohomology I, J. Algebraic Geom. 24 (2015), no. 4, 719–754. MR 3383602, https://doi.org/10.1090/S1056-3911-2015-00648-8
- [Jin16] Jin Fangzhou, Borel-Moore motivic homology and weight structure on mixed motives, Math. Z. 283 (2016), no. 3-4, 1149–1183. MR 3519998, https://doi.org/10.1007/s00209-016-1636-7
- [Jin16b]
F. Jin,
Quelques aspects sur l'homologie de Borel-Moore dans le cadre de l'homotopie motivique : poids et-théorie de Quillen,
PhD Dissertation, E.N.S. de Lyon, 2016. - [Jin17]
F. Jin,
Algebraic-theory in motivic homotopy categories,
1806.03927, 2018. - [Lev]
M. Levine,
-theory and motivic cohomology of schemes, I,
https://www.uni-due.de/~bm0032/publ/KthyMotI12.01.pdf. - [Lev01] Marc Levine, Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom. 10 (2001), no. 2, 299–363. MR 1811558
- [LM07] M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
- [MV99] Fabien Morel and Vladimir Voevodsky, 𝐴¹-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- [Nav81]
J. A. Navarro González,
Cálculo de las clases de Chern de los esquemas lisos y singulares,
PhD Dissertation, Universidad de Salamanca, 1981. - [Nav16] Alberto Navarro, On Grothendieck’s Riemann-Roch theorem, Expo. Math. 35 (2017), no. 3, 326–342. MR 3689905, https://doi.org/10.1016/j.exmath.2016.09.005
- [Nav16b] A. Navarro, Riemann-Roch for homotopy invariant 𝐾-theory and Gysin morphisms, Adv. Math. 328 (2018), 501–554. MR 3771136, https://doi.org/10.1016/j.aim.2018.01.001
- [Pan03] I. Panin, Oriented cohomology theories of algebraic varieties, 𝐾-Theory 30 (2003), no. 3, 265–314. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. MR 2064242, https://doi.org/10.1023/B:KTHE.0000019788.33790.cb
- [Pan04] I. Panin, Riemann-Roch theorems for oriented cohomology, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261–333. MR 2061857, https://doi.org/10.1007/978-94-007-0948-5_8
- [Pan09] Ivan Panin, Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov), Homology Homotopy Appl. 11 (2009), no. 1, 349–405. MR 2529164
- [PPR08] Ivan Panin, Konstantin Pimenov, and Oliver Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology Homotopy Appl. 10 (2008), no. 2, 211–226. MR 2475610
- [PPR09] Ivan Panin, Konstantin Pimenov, and Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s 𝐾-theory, Invent. Math. 175 (2009), no. 2, 435–451. MR 2470112, https://doi.org/10.1007/s00222-008-0155-5
- [Rio10] Joël Riou, Algebraic 𝐾-theory, 𝐴¹-homotopy and Riemann-Roch theorems, J. Topol. 3 (2010), no. 2, 229–264. MR 2651359, https://doi.org/10.1112/jtopol/jtq005
- [SGA4] M. Artin, A. Grothendieck, J-L Verdier, eds: Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3 (PDF). Lecture Notes in Mathematics (in French). 305. Berlin; New York: Springer-Verlag. pp. vi+640 (1972).
- [SGA6] P. Berthelot, A. Grothendieck, L. Illusie: Séminaire de Géométrie Algébrique du Bois Marie -- 1966-67 -- Théorie des intersections et théorème de Riemann-Roch -- (SGA 6). Lecture notes in mathematics, vol. 225 (1971)
- [Spi12] Markus Spitzweck, A commutative ℙ¹-spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. Fr. (N.S.) 157 (2018), 110. MR 3865569, https://doi.org/10.24033/msmf.465
- [TT90] R. W. Thomason and Thomas Trobaugh, Higher algebraic 𝐾-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, https://doi.org/10.1007/978-0-8176-4576-2_10
- [Voe98] Vladimir Voevodsky, 𝐀¹-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604. MR 1648048
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14C40, 14F42, 19E15, 19E20, 19L10
Retrieve articles in all journals with MSC (2010): 14C40, 14F42, 19E15, 19E20, 19L10
Additional Information
A. Navarro
Affiliation:
Departamento de Matemática Aplicada, E.T.S. de Arquitectura, Universidad Politéc- nica de Madrid, 28040 Madrid, Spain
J. Navarro
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, 06006 Badajoz, Spain
DOI:
https://doi.org/10.1090/tran/8107
Received by editor(s):
March 1, 2018
Received by editor(s) in revised form:
September 9, 2019
Published electronically:
November 3, 2020
Additional Notes:
The first author was supported by MTM2016-79400-P (MINECO)
The second author was supported by grants GR18001 and IB18087
Dedicated:
To our father
Article copyright:
© Copyright 2020
by the authors