Algebra schemes and their representations

Abstract

The equivalence (Cartier duality) between the category of topologically flat formal k-groups and the category of flat bialgebras has been treated as a duality of continuous vector spaces (of functions) [G, Expose VIIB by P. Gabriel, 2.2.1]. This is owing to the fact that the reflexivity of vector spaces of infinite dimension does not hold if one does not provide them with a certain topology and does not consider the continuous dual. In this paper we obtain this duality without providing the vector spaces of functions with a topology. Let R be a commutative ring with unit. It is natural to consider R-modules as R-module functors in the following way: if M is an R-module, let M be the Rmodule functor defined by M(S) := M ⊗R S for every R-algebra S which belongs to the category CR of R-algebras. Now, if M is a functor of R-modules, its dual M∗ can be defined in a natural way as the functor of R-modules defined M∗(S) := HomS(M|S ,S). In this work we will prove that the functor defined by an R-module is reflexive: M ∼ →M∗∗, even in the case of R being a ring. We call the functors M∗ R-module schemes and if they are R-algebra functors too, we will say they are R-algebra schemes. In section 2 we study and characterize the vector space schemes (2.3, 2.17) and we characterize when the module scheme closure of an R-module functor M is equal to M∗∗ (2.8, 2.9). P. Gabriel [G, Expose VIIB , 1.3.5] proved that the category of topologically flat formal R-varieties is equivalent to to the category of flat cocommutative Rcoalgebras, where R is a pseudocompact ring. We prove (4.2) that the category of R-algebra schemes is equivalent to the category of R-coalgebras, where R is a ring. From this perspective, on the theory of algebraic groups and their representations R-module schemes appear in a necessary way, as also doR-algebra schemes as linear envelopes of groups. Let G = Spec A be an R-group and let G· be the functor of points of G, i.e., G·(S) = HomR−sch(SpecS,G) for all S ∈ CR, and let RG· be the “linear envelope of G·” (see section 3). We prove that the R-algebra scheme closure ofRG· is theR-algebra scheme A∗ (3.3, 5.4) and the category of G-modules is equal to the category of A∗-modules (5.5). So, the theory of linear representations of a group G = Spec A is a particular case of the theory of A∗-modules (5.7, 5.8, 6.4, etc). Moreover, there is a bijective correspondence between the R-rational points of A∗ and the multiplicative characters of G (5.6). When R is an algebraically closed field and G is smooth we prove that the completion of RG· by its ideal functors of finite codimension is also A∗ (3.5, 5.9).