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 :

{\ displaystyle A = \ bigoplus _ {n \ in \ mathbb {N}} A_ {n} = A_ {0} \ oplus A_ {1} \ oplus A_ {2} \ oplus \ cdots}

so that ${\ displaystyle A_ {i} A_ {j} \ subseteq A_ {i + j}.}$

Elements of are called homogeneous elements of degree . Each element of a graduated ring can be clearly written as the sum of homogeneous elements. ${\ displaystyle A_ {j}}$ ${\ displaystyle j}$

An ideal is said to be homogeneous if: ${\ displaystyle I}$

{\ displaystyle I = \ bigoplus _ {n \ in \ mathbb {N}} (I \ cap A_ {n})}

If there is an ideal of the ring , the ring associated with the ideal can be formed: ${\ displaystyle I}$ ${\ displaystyle R}$ ${\ displaystyle I}$ ${\ displaystyle \ mathrm {gr} _ {I} (R)}$

{\ displaystyle \ mathrm {gr} _ {I} (R) = \ bigoplus _ {n \ in \ mathbb {N}} I ^ {n} / I ^ {n + 1}}

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 : ${\ displaystyle I}$ ${\ displaystyle f, g \ in R}$

{\ displaystyle fg \ in I \ \ Leftrightarrow (f \ in I \ lor g \ in I)}

If it is noetherian and an ideal, then it is also noetherian. ${\ displaystyle R}$ ${\ displaystyle I \ subset R}$ ${\ displaystyle \ mathrm {gr} _ {I} (R)}$

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: ${\ displaystyle R}$ ${\ displaystyle m}$ ${\ displaystyle k = R / m}$ ${\ displaystyle x_ {1}, \ dots x_ {n}}$ ${\ displaystyle k}$ ${\ displaystyle m / m ^ {2}}$

(1) is regular.

{\ displaystyle R}

(2) The through

{\ displaystyle f \ colon X_ {i} \ mapsto x_ {i}}

defined homomorphism

{\ displaystyle f \ colon k [X_ {1}, \ dots, X_ {n}] \ to \ mathrm {gr} _ {m} (R)}

is an isomorphism of graduated algebras.

{\ displaystyle k}

Examples

If there is a body, then naturally is a graduated ring. ${\ displaystyle K}$ ${\ displaystyle K [X_ {1}, \ dots X_ {n}]}$
This ring can also be provided with a different graduation:

If , then the set of quasi-homogeneous polynomials is of degree :

{\ displaystyle (i_ {1}, \ dots, i_ {n}) \ in \ mathbb {N} ^ {n}}

{\ displaystyle A_ {k}}

{\ displaystyle k}

{\ displaystyle A_ {k}: = \ sum _ {\ alpha _ {1} i_ {1} + \ dots + \ alpha _ {n} i_ {n} = k} s _ {\ alpha _ {1}, \ dots, \ alpha _ {n}} X ^ {i_ {1}} \ dots X ^ {i_ {n}}}

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