## affine schemes

23 examples (0.02 sec)
• 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.