Affine structures on a ringed space and schemes

Abstract

The author first introduces the notion of affine structures on a ringed space and then obtains several related properties. Affine structures on a ringed space, arising from complex analytical spaces of algebraic schemes, behave like differential structures on a smooth manifold.

As one does for differential manifolds, pseudogroups of affine transformations are used to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is said to be admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures make a contribution to the cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, two necessary and sufficient conditions, that two spaces are homeomorphic and that two schemes are isomorphic, coming from the main theorems of the paper, are obtained respectively. A conclusion is drawn that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively.

Keywords

Affine structure / Pseudogroup of affine transformations / Ringed space / Scheme

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