Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Homogeneous Ideals and Regular Rings

Description

This lecture covers the concept of homogeneous ideals and regular rings, exploring properties such as additive and generator characteristics. It also delves into N-graded rings and the relevance of closed sets in projective geometry.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Projective space

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs.

Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra.

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third.

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.

Algebraic geometry

Algebraic geometry is a branch of mathematics which classically studies 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. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations.

Related lectures (39)

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Varieties with nef anti-canonical: Surjective Albanese

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

Cohomology Real Projective Space

Covers cohomology in real projective spaces, focusing on associative properties and algebraic structures.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Elliptic Curves: Singular Points and Group Law