In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin[1] for use in deformation theory. Intuitively,
schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).
In mathematics, algebraicspaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory...
vector space (called vectors). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous...
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent...
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants...
an algebraic stack is a vast generalization of algebraicspaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are...
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in...
In mathematics, a piecewise algebraicspace is a generalization of a semialgebraic set, introduced by Maxim Kontsevich and Yan Soibelman. The motivation...
physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model...
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory...
empirical sciences. Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty...
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...
the basis of algebraic geometry, because they are rings of functions of algebraic geometric objects. Another crucial example are Lie algebras, which are...
are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are...
mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus...
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket...
the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The...
algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are...
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic...
noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring. A homomorphism between two R-algebras is an...
A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space. Measure algebra: A Banach algebra consisting...
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