On Tannakian duality over valuation rings
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).
scientific article; zbMATH DE number 2153376
scientific article