Editing Scheme (mathematics) (section)

==Generalizations==
Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways.  One is to use the [[étale topology]]. [[Michael Artin]] defined an '''[[algebraic space]]''' as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the [[Artin representability theorem]], gives simple conditions for a functor to be represented by an algebraic space.<ref name=St07Y1>{{Citation | title=Stacks Project, Tag 07Y1 | url=http://stacks.math.columbia.edu/tag/07Y1}}.</ref>

A further generalization is the idea of a [[stack (mathematics)|stack]]. Crudely speaking, '''[[algebraic stack]]s''' generalize algebraic spaces by having an [[algebraic group]] attached to each point, which is viewed as the automorphism group of that point. For example, any [[Group action (mathematics)|action]] of an algebraic group ''G'' on an algebraic variety ''X'' determines a '''[[quotient stack]]''' [''X''/''G''], which remembers the [[stabilizer subgroup]]s for the action of ''G''. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.

Grothendieck originally introduced stacks as a tool for the theory of [[descent (mathematics)|descent]]. In that formulation, stacks are (informally speaking) sheaves of categories.{{sfn|Vistoli|2005|loc=Definition 4.6}} From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include [[Deligne–Mumford stack]]s (similar to [[orbifold]]s in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The [[Keel–Mori theorem]] says that an algebraic stack with finite stabilizer groups has a [[coarse moduli space]] that is an algebraic space.

Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to [[homotopy theory]]. In this setting, known as [[derived algebraic geometry]] or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of [[highly structured ring spectrum|E-infinity ring spectra]]). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that [[derived functor]]s in homological algebra yield higher information about operations such as [[tensor product]] and the [[Hom functor]] on modules.