Motives and homotopy theory in logarithmic geometry

Binda, Federico; Park, Doosung; Østvær, Paul Arne

doi:10.5802/crmath.340

Géométrie algébrique

Motives and homotopy theory in logarithmic geometry

Binda, Federico ¹ ; Park, Doosung ² ; Østvær, Paul Arne ^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

Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 717-727.

Résumé
Abstract

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.

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.

Reçu le : 2021-06-17
Révisé le : 2021-12-05
Accepté le : 2022-01-26
Publié le : 2022-06-22

Zbl

DOI : 10.5802/crmath.340

Classification : 14XX, 19XX, 55XX
Mots clés : Logarithmic geometry, motives, motivic homotopy theory

Affiliations des auteurs :

Binda, Federico ¹ ; Park, Doosung ² ; Østvær, Paul Arne ^{1, 3}

@article{CRMATH_2022__360_G6_717_0,
     author = {Binda, Federico and Park, Doosung and {\O}stv{\ae}r, Paul Arne},
     title = {Motives and homotopy theory in logarithmic geometry},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--727},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G6},
     year = {2022},
     doi = {10.5802/crmath.340},
     zbl = {07547270},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.340/}
}

TY  - JOUR
AU  - Binda, Federico
AU  - Park, Doosung
AU  - Østvær, Paul Arne
TI  - Motives and homotopy theory in logarithmic geometry
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 717
EP  - 727
VL  - 360
IS  - G6
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.340/
DO  - 10.5802/crmath.340
LA  - en
ID  - CRMATH_2022__360_G6_717_0
ER  -

%0 Journal Article
%A Binda, Federico
%A Park, Doosung
%A Østvær, Paul Arne
%T Motives and homotopy theory in logarithmic geometry
%J Comptes Rendus. Mathématique
%D 2022
%P 717-727
%V 360
%N G6
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.340/
%R 10.5802/crmath.340
%G en
%F CRMATH_2022__360_G6_717_0

Binda, Federico; Park, Doosung; Østvær, Paul Arne. Motives and homotopy theory in logarithmic geometry. Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 717-727. doi : 10.5802/crmath.340. http://www.numdam.org/articles/10.5802/crmath.340/

Bibliographie
Cité par

[1] André, Yves Une introduction aux motifs. Motifs purs, motifs mixtes, périodes, Panoramas et Synthèses, 17, Société Mathématique de France, 2004, xi+261 pages | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :