Related Research Articles
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.
In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over . This definition can be extended to the quasi-projective varieties, such that a regular map between quasiprojective varieties is finite if any point like has an affine neighbourhood V such that is affine and is a finite map.
In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets , noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after Emmy Noether.
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that every subset is compact.
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.
In algebraic geometry, Zariski's main theorem, proved by Oscar Zariski (1943), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational.
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
This is a glossary of algebraic geometry.
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents S-derivations in the sense: for any -modules F, there is an isomorphism
In algebraic geometry, a closed immersion of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.
In algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its properties are much the same as in the special case of smooth schemes.
References
- ↑ Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer, p. 238, ISBN 9780387902449 . Note that the cited definition that Hartshorne gives is slightly misleading. A locally Noetherian scheme is regular if all its local rings are regular, but it is not the case for schemes which are not locally Noetherian. See the cited Stacks Project page for more details.
- ↑ "Section 28.9 (02IR): Regular schemes—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-02-18.
- ↑ Demazure, Michel (1980), Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies, vol. 39, North-Holland, Proposition 3.2, p. 168, ISBN 9780080871509 .
 | This algebraic geometry related article is a stub. You can help Wikipedia by expanding it. |