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 (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding line bundles. On singular varieties, this property can also fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same. The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain. An algebraic cycle is a higher codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1. (Wikipedia).
Definition of a Zero Divisor with Examples of Zero Divisors
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Zero Divisor with Examples of Zero Divisors - Examples of zero divisors in Z_m the ring with addition modulo m and multiplication modulo m. Examples are done with Z_8 and Z_4. - Example of a zero divisor with the D
From playlist Abstract Algebra
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From playlist Algebraic geometry II: Schemes
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From playlist Algebraic geometry II: Schemes
Schemes 36: Weil and Cartier divisors
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. In this lecture we define Weil and Cartier divisors and divisor classes, and give some simple examples of the groups of divisor classes.
From playlist Algebraic geometry II: Schemes
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From playlist Divisibility and the Euclidean Algorithm
Schemes 37: Comparison of Weil and Cartier divisors
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. In this lecture we compare Cartier and Weil divisors, showing that for Noethernian integral schems the map from Cartier to Weil divisors is injective if the sc
From playlist Algebraic geometry II: Schemes
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From playlist Abstract Algebra
Schemes 39: Divisors and Dedekind domains
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From playlist Algebraic geometry II: Schemes
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From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)
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From playlist Mathematics
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From playlist Algebraic geometry: extra topics
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From playlist Special / Prizes Lectures
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From playlist Divide Polynomials using Long Division with monomial divisor