Algebra schemes and their representations (Q2368797)

Let \(R\) be a commutative ring with identity. A functor \(F\) from the category of \(R\)-algebras to abelian groups is an \(R\)-module functor if there is a morphism of functors which gives an \(R\)-module structure \(A\times F\left( A\right) \rightarrow F\left( A\right) \) for all \(R\)-modules \(A\). One class of examples of such a functor is as follows: for \(E\) and \(R\)-module the functor \(\mathbb{E}\) given by \(\mathbb{E}\left( A\right) =E\otimes _{R}A\) is an \(R\)-module functor. This functor \(\mathbb{E}\) is said to be quasi-coherent, coherent if \(E\) is finitely generated. The dual to \(\mathbb{ E,}\) denoted \(\mathbb{E}^{\ast }\), is an \(R\)-module functor which is called an \(R\)-module scheme. If furthermore \(\mathbb{E}^{\ast }\) is an \(R\)-algebra functor then it is an \(R\)-algebra scheme. This work is a study of \(R\)-module schemes and \(R\)-algebra schemes. One result of this paper is that quasi-coherent \(R\)-modules are reflexive, i.e. \(\mathbb{E\cong E}^{\ast \ast }.\) The significance here is that this theorem is proved without using topology. Now suppose that \(R=k\) a field. Then it is natural to refer to \(k\)-module schemes as vector space schemes. The authors investigate\ when a reflexive \( k \)-vector space functor \(F\) is a vector space scheme. The functor \(\text{Hom}_{k}\left( F,-\right) \) commutes with direct limits if and only if \(F\) is a vector space scheme. It is also shown that \(F\) is a vector space scheme if and only if \(F\) is complete and separate, i.e. \(\widehat{F} =F.\) Given an \(R\)-module functor \(F\) its closure \(\bar{F}\) is defined to be the \( R \)-module scheme such that \(\text{Hom}_{R}\left( F,\mathbb{V}^{\ast }\right) \cong \text{Hom}_{R}\left( \bar{F},\mathbb{V}^{\ast }\right) \) for \(V\) an \(R\)-module. Let \(G=\text{Spec}\) \(A\) be an \(R\)-group and define \(G^{\cdot }\left( B\right) =\text{Hom}\left( \text{Spec}B,G\right) \) for \(B\) a commutative \(R\)-algebra. Define \(R\left[ G^{\cdot }\right] \) to be the functor such that \(R\left[ G^{\cdot }\right] \left( B\right) \) is the set of formal finite \(B\)-linear combinations of points of \(G\) in \(B\), the so-called linear envelope of \(G\)''. Then the \(R\)-algebra scheme closure of \(R \left[ G^{\cdot }\right] \) is in fact \(\mathbb{A}\) and hence the category of \(G\)-modules is equivalent to the category of \(\mathbb{A}^{\ast }\)-modules. From this we get that the representation theory for \(G\) can be found in the representation theory of \(\mathbb{A}^{\ast }\)-modules. Every \(R\)-algebra scheme \(\mathbb{A}^{\ast }\) is shown to be the inverse limit of certain finite \(R\)-algebra schemes. Furthermore, for \(k\) a field conditions are given for when the \(k\)-algebra scheme \(\mathbb{A}^{\ast }\) is separable, e.g. \(\mathbb{A}^{\ast }\) is separable if and only if \(\mathbb{A} ^{\ast }\bar{\otimes}_{k}\left( \mathbb{A}^{\ast }\right) ^{op}\) is semisimple. Finally, the principal theorem of Wedderburn-Malcev is proved in the context of algebra schemes (over a field).