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Affine structures on a ringed space and schemes
- Source :
- Chinese Annals of Mathematics, Series B. 32:139-160
- Publication Year :
- 2010
- Publisher :
- Springer Science and Business Media LLC, 2010.
-
Abstract
- In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number fields, behave like differential structures on a smooth manifold. As one does for differential manifolds, we will use pseudogroups of affine transformations 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 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 are in action in some special cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, we will obtain respectively necessary and sufficient conditions that two spaces are homeomorphic and that two schemes are isomorphic, which are the two main theorems of the paper. It follows 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.<br />Final version. 22 pages. to appear in Chinese Ann of Math, Series B
- Subjects :
- Discrete mathematics
Pure mathematics
Applied Mathematics
General Mathematics
14A15, 14A25, 57R55
Affine plane
Affine coordinate system
Affine geometry
Mathematics - Algebraic Geometry
Complex space
Ringed space
Affine hull
Affine group
FOS: Mathematics
Affine space
Algebraic Geometry (math.AG)
Mathematics
Subjects
Details
- ISSN :
- 18606261 and 02529599
- Volume :
- 32
- Database :
- OpenAIRE
- Journal :
- Chinese Annals of Mathematics, Series B
- Accession number :
- edsair.doi.dedup.....423ebdec82303587cf3e833a5efd1d11