Some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-separated, meaning that the diagonal map is quasi-compact.One can always assume that R and U are affine schemes.Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.…

A scheme is a locally ringed space X admitting a covering by open sets U i, such that the restriction of the structure sheaf O X to each U i is an affine scheme.Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes.The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology.…

Every locally ringed space isomorphic to one of this form is called an affine scheme.General schemes are obtained by "gluing together" several affine schemes.…

If S is the complement of a prime ideal I the localization is denoted by R I, and R f is used to denote the localization by the powers of an element f. The two latter cases are fundamental in algebraic theory and scheme theory.In particular the definition of an affine scheme is based on the properties of these two kinds of localizations.An important related process is completion: one often localizes a ring, then completes.…

This translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic variety.More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme.The characteristic of an integral domain is either 0 or a prime number.…

For the etale topology, the two cohomologies agree for any sheaf, provided that any finite set of points in the base scheme X are contained in some open affine subscheme.This is satisfied, for example, if X is quasi-projective over an affine scheme.…

Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne-Mumford stacks, both often called algebraic stacks.Sometimes other algebraic sites replace the category of affine schemes.For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry.…

For any reasonable property, it turns out that the property is true generically on the subvariety (in the sense of being true on an open dense subset) if and only if the property is true at the generic point.Such results are frequently proved using the methods of limits of affine schemes developed in EGA IV 8.A related concept in algebraic geometry is general position, whose precise meaning depends on the context.…

Algebraic geometry proceeds by endowing Spec R with a sheaf \mathcal O (an entity that collects functions defined locally, i.e. on varying open subsets).The datum of the space and the sheaf is called an affine scheme.…

Every locally ringed space isomorphic to one of this form is called an affine scheme.General schemes are obtained by "gluing together" several affine schemes.It is useful to use the language of category theory and observe that Spec is a functor.…

This can sometimes be done by ad hoc means: for example, the left derived functors of Tor can be defined using a flat resolution rather than a projective one, but it takes some work to show that this is independent of the resolution.Not all categories of sheaves run into this problem; for instance, the category of sheaves on an affine scheme contains enough projectives.An acyclic sheaf F over X is one such that all higher sheaf cohomology groups vanish.…

In analogy to the duality between affine schemes and commutative rings, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings.There are certain analogues of Zariski topology in that context so that one can glue such affine schemes to more general objects.There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a Serre's theorem on Proj.…

Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes.The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology.In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme.…

Again, the coproduct of two nonzero commutative rings can be zero.The opposite category of CRing is equivalent to the category of affine schemes.The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.…

For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme.The concern here is to list the fundamental technical definitions and properties of scheme theory.…

Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces.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.In complex geometry, an affine variety is an analog of a Stein manifold.…

Recent ideas about higher algebraic stacks and homotopical or derived algebraic geometry have regard to further expanding the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to homotopy theory.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.

Altogether the equivalence of the two said categories is very apt to reflect algebraic properties of rings in a geometrical manner.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.Therefore, many notions that apply to rings and homomorphisms stem from geometric intuition.…

A variation on the theorem is that if every direct factor of an object in C is again in C, then the condition that the fiber of G at x be one-dimensional can be replaced by the condition that the fiber is non-empty.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.…