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Comptes Rendus. Mathématique
Algebraic geometry
Motives and homotopy theory in logarithmic geometry
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 717-727.

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.

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DOI: 10.5802/crmath.340
Classification: 14XX,  19XX,  55XX
Keywords: Logarithmic geometry, motives, motivic homotopy theory
Federico Binda 1; Doosung Park 2; Paul Arne Østvær 1, 3

1 Department of Mathematics F. Enriques, University of Milan, Via Cesare Saldini 50, 20133 Milan, Italy
2 Department of Mathematics and informatics, University of Wuppertal, Gaussstr. 20, 42119 Wuppertal, Germany
3 Department of Mathematics, University of Oslo, Niels Henrik Abels hus, Moltke Moes vei 35, 0851 Oslo, Norway
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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] Yves André Une introduction aux motifs. Motifs purs, motifs mixtes, périodes, Panoramas et Synthèses, 17, Société Mathématique de France, 2004, xi+261 pages | Zbl

[2] Aravind Asok; Jean Fasel Splitting vector bundles outside the stable range and 𝔸 1 – homotopy sheaves of punctured affine spaces, J. Am. Math. Soc., Volume 28 (2015) no. 4, pp. 1031-1062 | DOI | MR | Zbl

[3] Joseph Ayoub; Luca Barbieri-Viale 1-motivic sheaves and the Albanese functor, J. Pure Appl. Algebra, Volume 213 (2009) no. 5, pp. 809-839 | DOI | MR | Zbl

[4] Luca Barbieri-Viale; Bruno Kahn On the derived category of 1-motives, Astérisque, 381, Société Mathématique de France, 2016, xi+254 pages | MR

[5] Federico Binda; Tommy Lundemo; Doosung Park; Paul A. Østvær A logarithmic Hochschild-Kostant-Rosenberg theorem (2022) (in preparation)

[6] Federico Binda; Alberto Merici Connectivity and Purity for logarithmic motives (2022) (to appear in J. Inst. Math. Jussieu) | arXiv

[7] Federico Binda; Alberto Merici; Shuji Saito Derived log Albanese sheaves (2022) | arXiv

[8] Federico Binda; Doosung Park; Paul A. Østvær Logarithmic motivic homotopy theory (2022) (in preparation)

[9] Federico Binda; Doosung Park; Paul A. Østvær Triangulated categories of logarithmic motives over a field, Astérisque, 433, Société Mathématique de France, 2022

[10] Federico Binda; Kay Rülling; Shuji Saito On the cohomology of reciprocity sheaves (2021) | arXiv

[11] Marcel Bökstedt; Wu-Chung Hsiang; Ib Madsen The cyclotomic trace and algebraic K-theory of spaces, Invent. Math., Volume 111 (1993) no. 3, pp. 465-539 | DOI | MR | Zbl

[12] Pierre Deligne Théorie de Hodge. II. (Hodge theory. II), Publ. Math., Inst. Hautes Étud. Sci., Volume 40 (1971), pp. 5-57 | DOI | Zbl

[13] William Fulton 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] Michel Gros 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] Lars Hesselholt; Peter Scholze Arbeitsgemeinschaft: Topological Cyclic Homology, Oberwolfach Rep., Volume 15 (2018) no. 2, pp. 805-940 | DOI | MR | Zbl

[16] Daniel Isaksen; Paul A. Østvær Motivic stable homotopy groups, Handbook of homotopy theory, CRC Press, 2020, 35 pages | DOI | Zbl

[17] Bruno Kahn; Shuji Saito; Takao Yamazaki Motives with modulus (2019) | arXiv

[18] Bruno Kahn; Shuji Saito; Takao Yamazaki; Kay Rülling Reciprocity Sheaves, Compos. Math., Volume 152 (2016) no. 9, pp. 1851-1898 | DOI | MR | Zbl

[19] Jacob Lurie Higher algebra (2017) (available at http://www.math.harvard.edu/~lurie/papers/HA.pdf.)

[20] Carlo Mazza; Vladimir Voevodsky; Charles Weibel Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2, American Mathematical Society; Clay Mathematics Institute, 2006, xiv+216 pages | MR

[21] John W. Milnor; James D. Stasheff Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press, 1974 | DOI | Zbl

[22] Fabien Morel 𝔸 1 -algebraic topology over a field, Lecture Notes in Mathematics, 2052, Springer, 2012, x+259 pages | DOI | MR | Zbl

[23] Fabien Morel; Vladimir Voevodsky A 1 -homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. (1999) no. 90, pp. 45-143 | DOI | MR | Zbl

[24] Wiesława Nizioł K-theory of log-schemes. I, Doc. Math., Volume 13 (2008), pp. 505-551 | MR | Zbl

[25] Wiesława Nizioł K-theory of log-schemes II: Log-syntomic K-theory, Adv. Math., Volume 230 (2012) no. 4-6, pp. 1646-1672 | DOI | MR | Zbl

[26] Arthur Ogus Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, 178, Cambridge University Press, 2018, xviii+539 pages | DOI | Zbl

[27] Marco Robalo K-theory and the bridge from motives to noncommutative motives, Adv. Math., Volume 269 (2015), pp. 399-550 | DOI | MR | Zbl

[28] Oliver Röndigs; Markus Spitzweck; Paul A. Østvær The first stable homotopy groups of motivic spheres, Ann. Math., Volume 189 (2019) no. 1, pp. 1-74 | MR | Zbl

[29] Shuji Saito Reciprocity Sheaves and Logarithmic motives (2021) | arXiv

[30] Robert W. Thomason; Thomas Trobaugh 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] Vladimir Voevodsky Homology of schemes, Sel. Math., New Ser., Volume 2 (1996) no. 1, pp. 111-153 | DOI | MR | Zbl

[32] Vladimir Voevodsky 𝔸 1 -homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Volume Extra Vol. I (1998), pp. 579-604 | MR | Zbl

[33] Vladimir Voevodsky Motivic cohomology with /2-coefficients, Publ. Math., Inst. Hautes Étud. Sci. (2003) no. 98, pp. 59-104 | DOI | MR | Zbl

[34] Vladimir Voevodsky 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] Vladimir Voevodsky On motivic cohomology with Z/l-coefficients, Ann. Math., Volume 174 (2011) no. 1, pp. 401-438 | DOI | MR | Zbl

[36] Vladimir Voevodsky; Andrei Suslin; Eric M. Friedlander Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, 143, Princeton University Press, 2000, vi+254 pages | MR

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