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Local Shtukas and Divisible Local Anderson Modules

Part of: Arithmetic algebraic geometry Algebraic groups General commutative ring theory

Published online by Cambridge University Press:  12 March 2019

Urs Hartl  and
Rajneesh Kumar Singh
Urs Hartl
Affiliation:
Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany URL: https://www.uni-muenster.de/Arithm/hartl
Rajneesh Kumar Singh
Affiliation:
Ramakrishna Vivekananda University, PO Belur Math, Dist Howrah, West Bengal 711202, India Email: rajneeshkumar.s@gmail.com
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Abstract

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We develop the analog of crystalline Dieudonné theory for $p$-divisible groups in the arithmetic of function fields. In our theory $p$-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson’s abelian $t$-modules and $t$-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings’s and Abrashkin’s theories of strict modules, which we review briefly.

Keywords

local shtukaformal Drinfeld moduleformal t-module

MSC classification

Primary: 11G09: Drinfel'd modules; higher-dimensional motives, etc.
Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 14L05: Formal groups, $p$-divisible groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1163 - 1207
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2019

Footnotes

Both authors were supported by the Deutsche Forschungsgemeinschaft (DFG) in form of the research grant HA3002/2-1 and the SFB’s 478 and 878.

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