EXTENDING SCHEMES TO A CARTESIAN CLOSED CATEGORY

Abstract

Our objective was to embed schemes of finite type over a field k in a suitably small Cartesian closed category. Two types of globalized versions of an ind-affine scheme were proposed: locally ind-affine ringed spaces and ind-schemes obtained by taking the inductive limit of closed subschemes of a locally ind-affine ringed space in ringed spaces. First in ε case some reasonably general conditions implying that translations, basic open subsets and closed subsets of an ind-affine scheme are again ind-affine schemes were obtained. Certain immersive properties of locally ind-affine ringed spaces are shown. As an adjunct we then determine a class of locally ind-affine ringed spaces which since they patch appropriately are ind-schemes. A restriction of locally ind-affine ringed spa1 leads to the category of locally ind-affine schemes (containing the category of schemes of finite type over k) which is see1 to be Cartesian closed with respect to the contravariant variable. Possible extensions to the covariant variable are studied.