This document is a short user’s guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.
Ce document est un petit guide d’utilisation de la théorie des motifs et de la théorie de l’homotopie dans le cadre de la géométrie logarithmique. Nous passons en revue certaines des idées de base et des résultats en relation avec d’autres travaux sur les motifs avec module, théorie de l’homotopie motivique, et les faisceaux de réciprocité.
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.340
Keywords: Logarithmic geometry, motives, motivic homotopy theory
Federico Binda 1; Doosung Park 2; Paul Arne Østvær 1, 3
@article{CRMATH_2022__360_G6_717_0, author = {Federico Binda and Doosung Park and Paul Arne {\O}stv{\ae}r}, title = {Motives and homotopy theory in logarithmic geometry}, journal = {Comptes Rendus. Math\'ematique}, pages = {717--727}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.340}, zbl = {07547270}, language = {en}, }
Federico Binda; Doosung Park; Paul Arne Østvær. Motives and homotopy theory in logarithmic geometry. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 717-727. doi : 10.5802/crmath.340. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.340/
[1] Une introduction aux motifs. Motifs purs, motifs mixtes, périodes, Panoramas et Synthèses, 17, Société Mathématique de France, 2004, xi+261 pages | Zbl
[2] Splitting vector bundles outside the stable range and – homotopy sheaves of punctured affine spaces, J. Am. Math. Soc., Volume 28 (2015) no. 4, pp. 1031-1062 | DOI | MR | Zbl
[3] 1-motivic sheaves and the Albanese functor, J. Pure Appl. Algebra, Volume 213 (2009) no. 5, pp. 809-839 | DOI | MR | Zbl
[4] On the derived category of 1-motives, Astérisque, 381, Société Mathématique de France, 2016, xi+254 pages | MR
[5] A logarithmic Hochschild-Kostant-Rosenberg theorem (2022) (in preparation)
[6] Connectivity and Purity for logarithmic motives (2022) (to appear in J. Inst. Math. Jussieu) | arXiv
[7] Derived log Albanese sheaves (2022) | arXiv
[8] Logarithmic motivic homotopy theory (2022) (in preparation)
[9] Triangulated categories of logarithmic motives over a field, Astérisque, 433, Société Mathématique de France, 2022
[10] On the cohomology of reciprocity sheaves (2021) | arXiv
[11] The cyclotomic trace and algebraic K-theory of spaces, Invent. Math., Volume 111 (1993) no. 3, pp. 465-539 | DOI | MR | Zbl
[12] Théorie de Hodge. II. (Hodge theory. II), Publ. Math., Inst. Hautes Étud. Sci., Volume 40 (1971), pp. 5-57 | DOI | Zbl
[13] Introduction to toric varieties. The 1989 William H. Roever lectures in geometry, Annals of Mathematics Studies, 131, Princeton University Press, 1993, xi+157 pages | Zbl
[14] Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. Fr., Nouv. Sér. (1985) no. 21, p. 87 | MR | Zbl
[15] Arbeitsgemeinschaft: Topological Cyclic Homology, Oberwolfach Rep., Volume 15 (2018) no. 2, pp. 805-940 | DOI | MR | Zbl
[16] Motivic stable homotopy groups, Handbook of homotopy theory, CRC Press, 2020, 35 pages | DOI | Zbl
[17] Motives with modulus (2019) | arXiv
[18] Reciprocity Sheaves, Compos. Math., Volume 152 (2016) no. 9, pp. 1851-1898 | DOI | MR | Zbl
[19] Higher algebra (2017) (available at http://www.math.harvard.edu/~lurie/papers/HA.pdf.)
[20] Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2, American Mathematical Society; Clay Mathematics Institute, 2006, xiv+216 pages | MR
[21] Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press, 1974 | DOI | Zbl
[22] -algebraic topology over a field, Lecture Notes in Mathematics, 2052, Springer, 2012, x+259 pages | DOI | MR | Zbl
[23] -homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. (1999) no. 90, pp. 45-143 | DOI | MR | Zbl
[24] -theory of log-schemes. I, Doc. Math., Volume 13 (2008), pp. 505-551 | MR | Zbl
[25] -theory of log-schemes II: Log-syntomic -theory, Adv. Math., Volume 230 (2012) no. 4-6, pp. 1646-1672 | DOI | MR | Zbl
[26] Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, 178, Cambridge University Press, 2018, xviii+539 pages | DOI | Zbl
[27] -theory and the bridge from motives to noncommutative motives, Adv. Math., Volume 269 (2015), pp. 399-550 | DOI | MR | Zbl
[28] The first stable homotopy groups of motivic spheres, Ann. Math., Volume 189 (2019) no. 1, pp. 1-74 | MR | Zbl
[29] Reciprocity Sheaves and Logarithmic motives (2021) | arXiv
[30] Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift Vol. III (Progress in Mathematics), Volume 88, Birkhäuser, 1990, pp. 347-435 | MR | Zbl
[31] Homology of schemes, Sel. Math., New Ser., Volume 2 (1996) no. 1, pp. 111-153 | DOI | MR | Zbl
[32] -homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Volume Extra Vol. I (1998), pp. 579-604 | MR | Zbl
[33] Motivic cohomology with -coefficients, Publ. Math., Inst. Hautes Étud. Sci. (2003) no. 98, pp. 59-104 | DOI | MR | Zbl
[34] Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra, Volume 214 (2010) no. 8, pp. 1384-1398 | DOI | MR | Zbl
[35] On motivic cohomology with -coefficients, Ann. Math., Volume 174 (2011) no. 1, pp. 401-438 | DOI | MR | Zbl
[36] Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, 143, Princeton University Press, 2000, vi+254 pages | MR
Cited by Sources:
Comments - Policy