[Formes différentielles en caractéristique positive en évitant la résolution de singularités]
This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities.
A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.
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DOI : 10.24033/bsmf.2739
Keywords: Differential forms, singularities, cdh-topology
Huber, Annette 1 ; Kebekus, Stefan 2 ; Kelly, Shane 1
@article{BSMF_2017__145_2_305_0,
author = {Huber, Annette and Kebekus, Stefan and Kelly, Shane},
title = {Differential forms in positive characteristic avoiding resolution of singularities},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {305--343},
year = {2017},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {145},
number = {2},
doi = {10.24033/bsmf.2739},
mrnumber = {3749788},
zbl = {1401.14103},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2739/}
}
TY - JOUR AU - Huber, Annette AU - Kebekus, Stefan AU - Kelly, Shane TI - Differential forms in positive characteristic avoiding resolution of singularities JO - Bulletin de la Société Mathématique de France PY - 2017 SP - 305 EP - 343 VL - 145 IS - 2 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2739/ DO - 10.24033/bsmf.2739 LA - en ID - BSMF_2017__145_2_305_0 ER -
%0 Journal Article %A Huber, Annette %A Kebekus, Stefan %A Kelly, Shane %T Differential forms in positive characteristic avoiding resolution of singularities %J Bulletin de la Société Mathématique de France %D 2017 %P 305-343 %V 145 %N 2 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2739/ %R 10.24033/bsmf.2739 %G en %F BSMF_2017__145_2_305_0
Huber, Annette; Kebekus, Stefan; Kelly, Shane. Differential forms in positive characteristic avoiding resolution of singularities. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 2, pp. 305-343. doi: 10.24033/bsmf.2739
Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Math., 305, Springer, Berlin-New York, 1973, 640 pages (avec la collaboration de P. Deligne et B. Saint-Donat) | MR
Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles, No. 1308, Hermann, Paris, 1964, 207 pages | MR | Zbl
Smoothness, semi-stability and alterations, Publ. Math. IHÉS, Volume 83 (1996), pp. 51-93 | Numdam | MR | Zbl | DOI
Commutative algebra, Graduate Texts in Math., 150, Springer, New York, 1995, 785 pages | MR | Zbl | DOI
Cohomology and holomorphic differential forms on complex analytic spaces, Ann. Scuola Norm. Sup. Pisa, Volume 24 (1970), pp. 65-77 | MR | Zbl | Numdam
Die Sätze von Bertini für lokale Ringe, Math. Ann., Volume 229 (1977), pp. 97-111 | MR | Zbl | DOI
Eléments de géométrie algébrique. I, Grundl. math. Wiss., 166, Springer, Berlin, 1971, 466 pages | MR | Zbl
Arithmetic cohomology over finite fields and special values of -functions, Duke Math. J., Volume 133 (2006), pp. 27-57 | MR | Zbl | DOI
A cohomological bound for the -topology, Amer. J. Math., Volume 123 (2001), pp. 425-443 http://muse.jhu.edu/... | MR | Zbl | DOI
Almost ring theory, Lecture Notes in Math., 1800, Springer, Berlin, 2003, 307 pages | MR | Zbl | DOI
Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. IHÉS, Volume 28 (1966), p. 255 | MR | Zbl | Numdam
Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. IHÉS, Volume 32 (1967), p. 361 | Numdam | MR | Zbl
Algebraic geometry, Graduate Texts in Math., 52, Springer, New York-Heidelberg, 1977, 496 pages | MR | Zbl | DOI
Differential forms in the -topology, Algebraic Geometry, Volume 4 (2014), pp. 449-478 | MR | Zbl | DOI
Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Illusie, Luc; Laszlo, Yves; Orgogozo, Fabrice, eds.), Astérisque, 363–364, Soc. Math. France, Paris, 2014, p. i-xxiv and 1–625 | MR | Numdam | Zbl
Pull-back morphisms for reflexive differential forms, Adv. Math., Volume 245 (2013), pp. 78-112 | MR | Zbl | DOI
Triangulated categories of motives in positive characteristic, Ph. D. Thesis , Université Paris 13 & Australian National University (2012) ( arXiv:1305.5349 )
Reciprocity sheaves, Compos. Math., Volume 152 (2016), pp. 1851-1898 | MR | Zbl | DOI
Autour de la platitude, Bull. Soc. Math. France, Volume 97 (1969), pp. 81-128 | Numdam | MR | Zbl | DOI
Commutative algebra, W. A. Benjamin, Inc., New York, 1970, 262 pp. paperbound pages | MR | Zbl
-algebraic topology over a field, Lecture Notes in Math., 2052, Springer, Heidelberg, 2012, 259 pages | MR | Zbl | DOI
Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. math., Volume 13 (1971), pp. 1-89 | MR | Zbl | DOI
Purity of reciprocity sheaves (preprint arXiv:1704.02442 ) | MR
Stacks Project (2014) ( http://stacks.math.columbia.edu )
Bloch-Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) (NATO Sci. Ser. C Math. Phys. Sci.), Volume 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117-189 | MR | Zbl
Homology of schemes, Selecta Math. (N.S.), Volume 2 (1996), pp. 111-153 | MR | Zbl | DOI
Analytical irreducibility of normal varieties, Ann. of Math., Volume 49 (1948), pp. 352-361 | MR | Zbl | DOI
Commutative algebra. Vol. II, Springer, New York-Heidelberg, 1975, 414 pages (reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29) | MR | Zbl
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