In algebraic geometry, given algebraic stacks p : X → C , q : Y → C {\displaystyle p:X\to C,\,q:Y\to C} over a base category C, a morphism f : X → Y {\displaystyle...

Deligne–Mumford stacks, which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced...

specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves M g ,...

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by...

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety...

function A morphism from an algebraic variety to the affine line. representable morphism A morphism F → G {\displaystyle F\to G} of stacks such that,...

In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion...

specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities...

quasi-separated. Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not...

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

géométrie algébrique Fiber product of schemes Flat morphism Smooth scheme Finite morphism Quasi-finite morphism Proper morphism Semistable elliptic curve Grothendieck's...

{\displaystyle |X|} is called the moduli space of X. If f : X → Y {\displaystyle f:X\to Y} is a morphism of algebraic stacks that induces a homeomorphism f : | X...

In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully flat morphisms. Sometimes an fpqc...

In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski...

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

sending an affine morphism f : Y → X {\displaystyle f:Y\to X} to f ∗ O Y . {\displaystyle f_{*}{\mathcal {O}}_{Y}.} A morphism of schemes f : X → Y {\displaystyle...

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules Ω...

Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of...

classes each determined by a morphism f: X → Ω, the characteristic morphism of that class, which we take to be the subobject of X characterized or named by...

In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory...

In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following...

a morphism between smooth varieties, then f factors as X → X × Y → Y {\displaystyle X\to X\times Y\to Y} where the first map is the graph morphism and...

Quasi-coherent sheaf on an algebraic stack Mumford 1999, Ch. III, § 1, Theorem-Definition 3. Stacks Project, Tag 01LA. Stacks Project, Tag 01BU. Serre 1955...

algebraic geometry: Introduction to intersection theory in algebraic geometry https://mathoverflow.net/questions/25218/why-is-riemann-roch-for-stacks-so-hard...

In algebraic geometry, a Deligne–Mumford stack is a stack F such that the diagonal morphism F → F × F {\displaystyle F\to F\times F} is representable,...

restriction morphism res U , U : F ( U ) → F ( U ) {\displaystyle \operatorname {res} _{U,U}\colon F(U)\rightarrow F(U)} is the identity morphism on F ( U...

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common...

object or a morphism in F {\displaystyle F} is called its projection (by ϕ {\displaystyle \phi } ). If f {\displaystyle f} is a morphism of E {\displaystyle...

algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some...