Abstract
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 e 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 covar...
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