Graduated ring
In commutative algebra and algebraic geometry, a graduated ring is a generalization of the polynomial ring in several variables. In algebraic geometry it is a means of describing projective varieties.
This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .
definition
A graduated ring A is a ring that has a representation as the direct sum of Abelian groups :
so that
Elements of are called homogeneous elements of degree . Each element of a graduated ring can be clearly written as the sum of homogeneous elements.
An ideal is said to be homogeneous if:
If there is an ideal of the ring , the ring associated with the ideal can be formed:
properties
- An ideal is homogeneous if and only if it can be generated by homogeneous elements.
- The sum, the product, the cut and the radical of homogeneous ideals are homogeneous again.
- A homogeneous ideal is prime if and only if it holds for all homogeneous :
- If it is noetherian and an ideal, then it is also noetherian.
Characterization of regular rings
If a local Noetherian ring, its maximal ideal, and a basis of the vector space , then the following statements are equivalent:
- (1) is regular.
- (2) The through
- defined homomorphism
- is an isomorphism of graduated algebras.
Examples
- If there is a body, then naturally is a graduated ring.
- This ring can also be provided with a different graduation:
- If , then the set of quasi-homogeneous polynomials is of degree :
See also
literature
- Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
- Atiyah , Macdonald : Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
- Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
- Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9