Authors:	Schedlmeier, Tobias Daniel
Title:	Cartier crystals and perverse constructible étale p-torsion sheaves
Online publication date:	28-Apr-2017
Year of first publication:	2017
Language:	english
Abstract:	In 2004, Emerton and Kisin established an analogue of the Riemann-Hilbert correspondence for smooth varieties over a field of positive characteristic p. It is a natural equivalence between the bounded derived categories of so-called locally finitely generated unit modules (lfgu for short) and constructible étale p-torsion sheaves. Here lfgu modules can be seen as an analogue of D-modules on complex varieties. By Emerton and Kisin's equivalence, the abelian category of lfgu modules corresponds to an abelian category, which was defined by Gabber intrinsically and is called perverse constructible étale p-torsion sheaves. In this thesis we generalize this Riemann-Hilbert type correspondence to F-finite embeddable varieties, i.e. varieties X which are not necessarily smooth but admit an embedding into a smooth variety Y of positive characteristic p, for which the absolute Frobenius is a finite morphism. This is done by defining lfgu modules on X as lfgu modules on Y supported in X. In order to prove the independence of the choice of an embedding, we generalize the adjunction between the derived push-forward and the twisted inverse image functor to morphisms which are only proper over the support of the considered objects. This adjunction is shown in the general context of quasi-coherent sheaves on Noetherian schemes and, in particular, independent of the characteristic of the involved schemes. Then we verify that this general adjunction induces an adjunction between the push-forward and pull-back on derived categories of lfgu modules. As an alternative analogue of D-modules for a Riemann-Hilbert correspondence for varieties of positive characteristic, we consider Cartier crystals, i.e. certain coherent sheaves with a right action of the Frobenius up to nilpotence of this action. In contrast to lfgu modules, Cartier crystals are defined on singular varieties as well. On smooth, F-finite varieties, the abelian categories of Cartier crystals and lfgu modules are equivalent by a result of Blickle and Böckle. Therefore, for an F-finite embeddable variety, our generalization of Emerton and Kisin’s result yields a natural equivalence between the bounded derived categories of Cartier crystals and constructible étale p-torsion sheaves, inducing an equivalence between the abelian categories of Cartier crystals and perverse constructible étale p-torsion sheaves. Finally, we show that this equivalence is compatible not only with the pull-back and push-forward for open and closed immersions but also with the intermediate extension functor for open immersions.
DDC:	510 Mathematik 510 Mathematics
Institution:	Johannes Gutenberg-Universität Mainz
Department:	FB 08 Physik, Mathematik u. Informatik
Place:	Mainz
ROR:	https://ror.org/023b0x485
DOI:	http://doi.org/10.25358/openscience-3883
URN:	urn:nbn:de:hebis:77-diss-1000012746
Version:	Original work
Publication type:	Dissertation
License:	In Copyright
Information on rights of use:	https://rightsstatements.org/vocab/InC/1.0/
Extent:	98 Seiten
Appears in collections:	JGU-Publikationen