Homological characterization of regularity in logarithmic algebraic geometry
Authors:
Jesús Conde-Lago and Javier Majadas
Journal:
J. Algebraic Geom. 31 (2022), 205-260
DOI:
https://doi.org/10.1090/jag/787
Published electronically:
December 7, 2021
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Abstract |
References |
Additional Information
Abstract: We characterize K. Kato’s log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.
References
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- Michel André, Localisation de la lissité formelle, Manuscripta Math. 13 (1974), 297–307 (French, with English summary). MR 357403, DOI 10.1007/BF01168230
- B. Bhatt, p-Adic derived de Rham cohomology, arXiv:1204.6560v1 [math.AG] (2012).
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- Robert Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR 741678
- A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680, DOI 10.1007/BFb0059052
- Luc Illusie, Chikara Nakayama, and Takeshi Tsuji, On log flat descent, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 1, 1–5. MR 3017719, DOI 10.3792/pjaa.89.1
- L. Illusie and A. Ogus, Géométrie Logarithmique, Exposé I (L. Illusie): le langage des log schémas. Exposé II (L. Illusie): différentielles et log lissité. Mimeographed notes of a course held at IHP, Paris, Spring 1997.
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI 10.2307/2374941
- Javier Majadas and Antonio G. Rodicio, Smoothness, regularity and complete intersection, London Mathematical Society Lecture Note Series, vol. 373, Cambridge University Press, Cambridge, 2010. MR 2640631, DOI 10.1017/CBO9781139107181
- J. Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original. MR 1206474
- Arthur Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, vol. 178, Cambridge University Press, Cambridge, 2018. MR 3838359, DOI 10.1017/9781316941614
- Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. MR 2195148, DOI 10.1007/s00208-005-0707-6
- D. Quillen, Homology of commutative rings, Mimeographed notes, MIT, Cambridge, MA, 1967.
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
- Steffen Sagave, Timo Schürg, and Gabriele Vezzosi, Derived logarithmic geometry I, J. Inst. Math. Jussieu 15 (2016), no. 2, 367–405. MR 3480969, DOI 10.1017/S1474748014000322
References
- Michel André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes in Mathematics, Vol. 32, Springer-Verlag, Berlin-New York, 1967 (French). MR 0214644
- Michel André, Homologie des algèbres commutatives, Die Grundlehren der mathematischen Wissenschaften, Band 206, Springer-Verlag, Berlin-New York, 1974 (French). MR 0352220
- Michel André, Localisation de la lissité formelle, Manuscripta Math. 13 (1974), 297–307 (French, with English summary). MR 357403, DOI 10.1007/BF01168230
- B. Bhatt, p-Adic derived de Rham cohomology, arXiv:1204.6560v1 [math.AG] (2012).
- O. Gabber and L. Ramero, Foundations for almost ring theory—Release 7.5, arXiv:arXiv:math/0409584v13 (2004).
- Robert Gilmer, Commutative semigroup rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR 741678
- A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
- Luc Illusie, Chikara Nakayama, and Takeshi Tsuji, On log flat descent, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 1, 1–5. MR 3017719, DOI 10.3792/pjaa.89.1
- L. Illusie and A. Ogus, Géométrie Logarithmique, Exposé I (L. Illusie): le langage des log schémas. Exposé II (L. Illusie): différentielles et log lissité. Mimeographed notes of a course held at IHP, Paris, Spring 1997.
- Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
- Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI 10.2307/2374941
- Javier Majadas and Antonio G. Rodicio, Smoothness, regularity and complete intersection, London Mathematical Society Lecture Note Series, vol. 373, Cambridge University Press, Cambridge, 2010. MR 2640631, DOI 10.1017/CBO9781139107181
- J. Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original. MR 1206474
- Arthur Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, vol. 178, Cambridge University Press, Cambridge, 2018. MR 3838359, DOI 10.1017/9781316941614
- Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. MR 2195148, DOI 10.1007/s00208-005-0707-6
- D. Quillen, Homology of commutative rings, Mimeographed notes, MIT, Cambridge, MA, 1967.
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
- Steffen Sagave, Timo Schürg, and Gabriele Vezzosi, Derived logarithmic geometry I, J. Inst. Math. Jussieu 15 (2016), no. 2, 367–405. MR 3480969, DOI 10.1017/S1474748014000322
Additional Information
Jesús Conde-Lago
Affiliation:
Departamento de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
Email:
jesus.conde@usc.es
Javier Majadas
Affiliation:
Departamento de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
MR Author ID:
269001
Email:
j.majadas@usc.es
Received by editor(s):
March 8, 2019
Received by editor(s) in revised form:
April 23, 2021
Published electronically:
December 7, 2021
Additional Notes:
This work was partially supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE), and by Xunta de Galicia through the Competitive Reference Groups (GRC) ED431C 2019/10. The first author was also supported by a scholarship, Xunta de Galicia (European Social Fund support included, UE)
Article copyright:
© Copyright 2021
University Press, Inc.
