Tannakian duality over Dedekind rings and applications (Q1745322)

Tannakian duality describes the problem of recovering a group or a group scheme from its category of representations. In this context, the notion of a Tannakian category was introduced by \textit{N. Saavedra Rivano} in [Categories tannakiennes. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0241.14008)], and later by \textit{P. Deligne} in [Prog. Math. 87, 111--195 (1990; Zbl 0727.14010)] who also corrected some inaccuracies in Saavedra's work. A Tannakian category over a field \(K\) consists of a \(K\)-linear rigid symmetric monoidal tensor category \(\mathcal{T}\) together with a faithful exact additive tensor functor \(\omega:\mathcal{T}\to \mathrm{Mod}_X\) (called \textit{fibre functor}) to the category \(\mathrm{Mod}(X)\) of \(\mathcal{O}_X\)-modules for some \(K\)-scheme \(X\) such that the endomorphism ring of the unit object \(1_{\mathcal{T}}\in \mathcal{T}\) is \(\mathrm{End}(1_{\mathcal{T}})\cong K\). In the neutral case, i.e. when \(X=\mathrm{Spec}(K)\), one obtains an affine group scheme \(G\) over \(K\) from \((\mathcal{T},\omega)\) as the group of tensor automorphisms of \(\omega\), and the Tannakian category is equivalent to the category \(\mathrm{Rep}_K(G)\) of finite dimensional \(K\)-representations of this group \(G\) via the functor \(\omega\). On the other hand, for every affine group scheme \(G\) over \(K\), the category \(\mathrm{Rep}_K(G)\) is a neutral Tannakian category with fibre functor being the forgetful functor, and the Tannakian group scheme obtained from this category is the group scheme \(G\) again. In this article, the authors generalise this Tannakian duality to a duality for flat affine group schemes \(G\) over a Dedekind ring \(R\). One such generalisation has already been considered by Saavedra in [loc. cit.] which is recalled by the authors in the first section. This approach defines the notion of a (neutral) Tannakian category \((\mathcal{C},\omega)\) over a Dedekind ring \(R\) which as above turns out to be equivalent to the category \(\mathrm{Rep}_R(G)\) of representations of \(G=\mathrm{Aut}^\otimes(\omega)\) on finitely generated \(R\)-modules, and \(G\) is a flat affine group scheme. In the core of this approach is a ``subcategory of definition'' \(\mathcal{C}^0\) of \(\mathcal{C}\) which corresponds to the full subcategory \(\mathrm{Rep}_R^0(G)\) of representations on finitely generated projective \(R\)-modules. The author's approach focuses on the subcategory \(\mathrm{Rep}_R^0(G)\). They introduce the notion of a ``neutral Tannakian lattice'' over \(R\) (a certain \(R\)-linear category \(\mathcal{T}\) with fibre functor \(\omega:\mathcal{T}\to \mathrm{Mod}^0(R)\); cf. Definition 2.2.2 ibid.). The main theorem in this context is Theorem 2.3.2: For a neutral Tannakian lattice \((\mathcal{T},\omega)\) over a Dedekind ring \(R\), the group scheme \(G=\mathrm{Aut}^\otimes(\omega)\) of tensor automorphisms of \(\omega\) is faithfully flat over \(R\) and \(\omega\) induces an equivalence between \(\mathcal{T}\) and the category \(\mathrm{Rep}_R^0(G)\). One should be aware that the definition they use is different from the definition of a Tannakian lattice over valuation rings introduced by \textit{T. Wedhorn} in [J. Algebra 282, No. 2, 575--609 (2004; Zbl 1088.18008)]. Since affine group schemes are in one-to-one correspondence with commutative Hopf algebras, and a representation of an affine group scheme is just a comodule of the corresponding Hopf algebra, large parts of the paper are formulated in terms of the latter. In the later parts of the article, the authors also discuss homomorphisms of flat coalgebras via the Tannakian duality as well as torsors of an affine flat group scheme corresponding to different fibre functors \(\eta:\mathcal{T}\to \mathrm{Mod}(S)\) (\(S\) being an \(R\)-algebra). This is a direct generalisation of Theorem 3.2 in [\textit{P. Deligne} and \textit{J. S. Milne}, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)].