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| Affine Structures on a Ringed Space and Schemes |
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Citation： |
Fengwen AN.Affine Structures on a Ringed Space and Schemes[J].Chinese Annals of Mathematics B,2011,32(1):139~160 |
| Page view： 1791
Net amount： 1319 |
Authors： |
Fengwen AN; |
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| 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 |
Classification： |
14A15, 14A25, 57R55 |
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