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Canonical q-deformations in arithmetic geometry
Peter Scholze
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, p. 1163-1192

In recent work with Bhatt and Morrow, we defined a new integral p-adic cohomology theory interpolating between étale and de Rham cohomology. An unexpected feature of this cohomology is that in coordinates, it can be computed by a q-deformation of the de Rham complex, which is thus canonical, at least in the derived category. In this short survey, we try to explain what we know about this phenomenon, and what can be conjectured to hold.

Dans un travail récent avec Bhatt et Morrow, nous avons défini une nouvelle théorie de cohomologie p-adique qui interpole entre cohomologie étale et de de Rham. Un aspect surprenant est que, en coordonnées, on peut calculer cette cohomologie par une q-déformation de la cohomologie de de Rham. Dans cet article, nous allons essayer d’expliquer ce que nous connaissons de ce phénomène, et ce qu’on peut conjecturer.

Received : 2016-06-08
Accepted : 2016-07-25
Published online : 2017-12-15
DOI : https://doi.org/10.5802/afst.1563
@article{AFST_2017_6_26_5_1163_0,
     author = {Peter Scholze},
     title = {Canonical $q$-deformations in arithmetic geometry},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {5},
     year = {2017},
     pages = {1163-1192},
     doi = {10.5802/afst.1563},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_5_1163_0}
}
Scholze, Peter. Canonical $q$-deformations in arithmetic geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, pp. 1163-1192. doi : 10.5802/afst.1563. afst.centre-mersenne.org/item/AFST_2017_6_26_5_1163_0/

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