On Tannakian duality over valuation rings
type: Item
instance of: scholarly article
title: On Tannakian duality over valuation rings
zbMATH Open document ID: 1088.18008
author: Torsten Wedhorn
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: The Tannakian duality may be divided in two parts: (1) The reconstruction problem: Given a ``group-like'' object \(G\) (e.g., a group scheme, a quantum group\dots), is it possible to recover \(G\) from the category of its representations, using the forgetful functor? (2) The description problem: Give a purely categorical description which ensures that a given category is equivalent to a category of representations of some ``group-like'' object. The paper under review proposes a solution to the reconstruction problem in the non neutral case (affine group scheme over field). This reconstruction is given in thm 2.14 and proposition 2.18 after some recalls on monoidal categories and on the comonoid of coendomorphism of a functor. This abstract reconstruction comes from \textit{P. Deligne} [``Catégories tannakiennes'', Prog. Math. 87, 111--195 (1990; Zbl 0727.14010)]. Then the author adopts this formal reconstruction to the case of Prüfer rings. A Prüfer ring is a commutative integral domain such that every finitely generated submodule of a flat module is projective (a noetherian Prüfer ring is a Dedekind ring). Some technical results on comodules over Prüfer rings are given. Thm 5.13 and 5.20 give the reconstruction theorem for an \(R\)-coalgebroid (with \(R\) a Prüfer ring). In the last section, the author gives a partial answer to the description problem for a groupoid over a valuation ring of height at most one. In order to do that, he defines the notion of Tannakian lattice over a valuation ring of height at most one (definition 6.9).
reviewed by: Jérôme Petit
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 2153376
zbMATH Keywords: Tannakian duality
zbMATH Keywords: group like object
zbMATH Keywords: Prüfer ring
zbMATH Keywords: groupoid
zbMATH Keywords: gerbes
zbMATH Keywords: Tannakian lattice
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.024
OpenAlex ID: W2032145676
Wikidata QID: Q59650023
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DOI: 10.1016/J.JALGEBRA.2004.07.024
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description: scientific article; zbMATH DE number 2153376