43 articles
Rational mapping
A rational mapping, or rational map, in algebraic geometry is a map between algebraic varieties given by rational functions, defined away from a closed subset.
Rigid analytic space
A rigid analytic space, the subject of rigid analytic geometry, is an analytic space over a non-Archimedean field, introduced by John Tate in the early 1960s.
Scheme (mathematics)
A scheme is the basic geometric object of modern algebraic geometry, a locally ringed space built from the spectra of commutative rings, introduced by Alexander Grothendieck in the late 1950s.
Scheme-theoretic image
The scheme-theoretic image of a morphism f: X → Y is the smallest closed subscheme of Y through which f factors, cut out by the kernel of O_Y → f*O_X.
Serre duality
Serre duality, or Serre's duality theorem, is a duality theorem in algebraic geometry, proved by Jean-Pierre Serre in 1955, that identifies top cohomology with duals of Hom groups.
Sheaf cohomology
Sheaf cohomology is the cohomology theory attached to a sheaf on a topological space or scheme, measuring the failure of local sections to glue into global ones.
Sheafification
Sheafification is the construction that turns a presheaf into a sheaf, its best approximation, and the standard way to build quotient sheaves, image sheaves, and inverse image sheaves.
Spectrum of a ring
The spectrum of a ring, written Spec(R), is the set of all prime ideals of a commutative ring R with the Zariski topology, forming an affine scheme.
Stack (mathematics)
In algebraic geometry, a stack is a category fibred in groupoids over schemes that behaves like a sheaf of geometric objects, introduced by Deligne and Mumford in 1969 to solve moduli problems with automorphisms.
Topos
A topos (plural: topoi or toposes) is a category behaving like a category of sheaves. Grothendieck toposes are sheaves on a site; elementary toposes, defined by Lawvere and Tierney, are axiomatized categories.
Torsor (algebraic geometry)
A torsor in algebraic geometry, also called a principal homogeneous space, is a scheme with a simply transitive group-scheme action, locally trivial over a covering of the base.
Valuative criterion
The valuative criterion is a test for separatedness and properness of a morphism of schemes, asking when maps from the generic point of a valuation ring extend.
Vector bundle
A vector bundle is a family of vector spaces attached to a base space, glued locally like a product; the tangent bundle of a manifold is the standard example.