affine scheme

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  • One can always assume that R and U are affine schemes.
  • Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes.
  • Every locally ringed space isomorphic to one of this form is called an affine scheme.
  • In particular the definition of an affine scheme is based on the properties of these two kinds of localizations.
  • More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.
  • This is satisfied, for example, if X is quasi-projective over an affine scheme.
  • Sometimes other algebraic sites replace the category of affine schemes.
  • Such results are frequently proved using the methods of limits of affine schemes developed in EGA IV 8.
  • The datum of the space and the sheaf is called an affine scheme.
  • General schemes are obtained by "gluing together" several affine schemes.
  • Not all categories of sheaves run into this problem; for instance, the category of sheaves on an affine scheme contains enough projectives.
  • There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.
  • The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology.
  • The opposite category of CRing is equivalent to the category of affine schemes.
  • For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme.
  • An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring.
  • An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring.
  • This gives then a presheaf of cohomology theories over the site of affine schemes flat over the moduli stack of elliptic curves.
  • Affine schemes are-much the same way as manifolds are locally given by open subsets of R n -local models for schemes, which are the object of study in algebraic geometry.
  • Suppose that is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type O X -module.

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