 First edition | |
| Author | Robin Hartshorne |
|---|---|
| Language | English |
| Subject | Algebraic geometry |
| Genre | Textbook |
| Published | 1977 |
Algebraic Geometry is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977. [1]
| First edition | |
| Author | Robin Hartshorne |
|---|---|
| Language | English |
| Subject | Algebraic geometry |
| Genre | Textbook |
| Published | 1977 |
Algebraic Geometry is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977. [1]
It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students.
The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
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 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.
David Bryant Mumford is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology.
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group.
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups, and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields.
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism.
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005). Given a function defined on an open subset of an algebraic variety, local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone ; this obstruction corresponds precisely to a nonzero class in the local cohomology module .
Robin Cope Hartshorne is an American mathematician who is known for his work in algebraic geometry.
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal.
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, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.
In algebraic geometry, Chow's moving lemma, proved by Wei-Liang Chow (1956), states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z' intersect properly. The lemma is one of key ingredients in developing the intersection theory, as it is used to show the uniqueness of the theory.
In the mathematical discipline of algebraic geometry, Serre's theorem on affineness is a theorem due to Jean-Pierre Serre which gives sufficient conditions for a scheme to be affine. The theorem was first published by Serre in 1957.
In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yum-Tong Siu and Shing-Tung Yau.