Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-29T18:08:17.121Z Has data issue: false hasContentIssue false
  • English
  • Français

DERIVED LOGARITHMIC GEOMETRY I

Part of: Commutative algebra: Homological methods Families, fibrations Algebraic geometry: Foundations

Published online by Cambridge University Press:  28 October 2014

Steffen Sagave ,
Timo Schürg  and
Gabriele Vezzosi
Steffen Sagave
Affiliation:
Department of Mathematics and Informatics, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany (sagave@math.uni-wuppertal.de)
Timo Schürg
Affiliation:
University of Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany (timo.schuerg@math.uni-augsburg.de)
Gabriele Vezzosi
Affiliation:
Institut de Mathématiques de Jussieu - UMR7586 Batiment Sophie Germain, Case 7012, 75205 PARIS Cedex 13, France (gabriele.vezzosi@math.imj-prg.fr) DiMaI, Università di Firenze, Firenze, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

In order to develop the foundations of derived logarithmic geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log-étale maps, and use them to define derived log stacks.

Keywords

log structuresderived algebraic geometrycotangent complexsimplicial commutative rings

MSC classification

Primary: 14A20: Generalizations (algebraic spaces, stacks) 14D23: Stacks and moduli problems 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 2 , April 2016 , pp. 367 - 405
Copyright
© Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M. and Sun, S., Logarithmic geometry and moduli, in Handbook of Moduli, Volume II, pp. 1–62. (2013).Google Scholar
Artin, M., Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189, MR 0399094 (53 #2945).CrossRefGoogle Scholar
Beilinson, A., p-adic periods and derived de Rham cohomology, J. Amer. Math. Soc. 25(3) (2012), 715–738, MR 2904571.Google Scholar
Bhatt, B., $p$-adic derived de Rham cohomology (2012), arXiv:1204.6560.Google Scholar
Bousfield, A. K. and Friedlander, E. M., Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, in Geometric Applications of Homotopy Theory (Proc. Conf., Evanston, IL, 1977), II, pp. 80–130. (1978), MR 513569 (80e:55021).CrossRefGoogle Scholar
Friedlander, E. M. and Mazur, B., Filtrations on the homology of algebraic varieties, Mem. Amer. Math. Soc. 110(529) (1994),. x+110. With an appendix by Daniel Quillen, MR 1211371 (95a:14023).Google Scholar
Gabber, O. and Ramero, L., Foundations for Almost Ring Theory (2014).Google Scholar
Goerss, P. G. and Jardine, J. F., Simplicial Homotopy Theory, Progress in Mathematics, Volume 174 (Birkhäuser Verlag, Basel, 1999), MR 1711612 (2001d:55012).Google Scholar
Gross, M. and Siebert, B., Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26(2) (2013), 451–510, MR 3011419.Google Scholar
Hirschhorn, P. S., Model Categories and Their Localizations, Mathematical Surveys and Monographs, Volume 99 (American Mathematical Society, Providence, RI, 2003), MR 1944041 (2003j:18018).Google Scholar
Illusie, L., Complexe Cotangent et Déformations. I, Lecture Notes in Mathematics, Volume 239 (Springer-Verlag, Berlin, 1971), MR 0491680 (58 #10886a).Google Scholar
Kato, K., Logarithmic structures of Fontaine–Illusie, in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), pp. 191–224 (Johns Hopkins University Press, Baltimore, MD, 1989), MR 1463703 (99b:14020).Google Scholar
Kato, F., Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11(2) (2000), 215–232, MR 1754621 (2001d:14016).Google Scholar
Li, J., Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57(3) (2001), 509–578, MR 1882667 (2003d:14066).Google Scholar
Li, J., A degeneration formula of GW-invariants, J. Differential Geom. 60(2) (2002), 199–293, MR 1938113 (2004k:14096).Google Scholar
Lurie, J., Representability theorems (2012), Preprint, available at http://www.math.harvard.edu/∼lurie/.Google Scholar
Marty, F., Smoothness in relative geometry, J. K-Theory 12(3) (2013), 461–491, MR 3165184.Google Scholar
Olsson, M. C., The logarithmic cotangent complex, Math. Ann. 333(4) (2005), 859–931, MR 2195148 (2006j:14017).Google Scholar
Quillen, D. G., Homotopical Algebra, Lecture Notes in Mathematics, Volume 43 (Springer-Verlag, Berlin, 1967), MR 0223432 (36 #6480).Google Scholar
Rezk, C., Every homotopy theory of simplicial algebras admits a proper model, Topology Appl. 119(1) (2002), 65–94, MR 1881711 (2003g:55033).Google Scholar
Rognes, J., Topological logarithmic structures, in New Topological Contexts for Galois Theory and Algebraic Geometry (BIRS 2008), Geometry & Topology Monographs, Volume 16, pp. 401–544 (Geometry & Topology Publications, Coventry, 2009), MR 2544395 (2010h:14029).Google Scholar
Rognes, J., Sagave, S. and Schlichtkrull, C., Localization sequences for logarithmic topological Hochschild homology (2014), arXiv:1402.1317.Google Scholar
Sagave, S. and Schlichtkrull, C., Group completion and units in I-spaces, Algebr. Geom. Topol. 13(2) (2013), 625–686, MR 3044590.CrossRefGoogle Scholar
Schwede, S., Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120(1) (1997), 77–104, MR 1466099 (98h:55027).Google Scholar
Simpson, C., Homotopy Theory of Higher Categories, New Mathematical Monographs, Volume 19 (Cambridge University Press, Cambridge, 2012), MR 2883823 (2012m:18019).Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193(2) (2005), 257–372, MR 2137288 (2007b:14038).Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193(902) (2008), x+224, MR 2394633 (2009h:14004).Google Scholar
Toën, B. and Vaquié, M., Au-dessous de Spec ℤ, J. K-Theory 3(3) (2009), 437–500, MR 2507727 (2010j:14006).Google Scholar