en>TakuyaMurata at 03:35, 24 February 2013

2013-02-24T03:35:23Z

New page

In [[mathematics]], the '''associated graded ring''' of a [[commutative ring]] ''R'' with respect to a proper ideal ''I'' is the [[graded ring]] <math>\operatorname{gr}_I R = \oplus_{n=0}^\infty I^n/I^{n+1}</math>. Similarly, if ''M'' is a ''R''-module, then the '''associated graded module''' is the [[graded ring|graded module]] <math>\operatorname{gr}_I M = \oplus_0^\infty I^n M/ I^{n+1} M</math>.

== Basic definitions and properties ==
For a ring ''R'' and ideal ''I'', multiplication in <math>\operatorname{gr}_IR</math> is defined as follows: First, consider [[homogeneous element|homogeneous elements]] <math>a \in I^i/I^{i + 1}</math> and <math>b \in I^j/I^{j + 1}</math> and suppose <math>a' \in I^i</math> is a representative of ''a'' and <math>b' \in I^j</math> is a representative of ''b''. Then define <math>ab</math> to be the equivalence class of <math>a'b'</math> in <math>I^{i + j}/I^{i + j + 1}</math>. Note that this is [[well-defined]] modulo <math>I^{i + j + 1}</math>. Multiplication of inhomogeneous elements is defined by using the distributive property.

A ring or module may be related to its associated graded through the '''initial form map'''. Let ''M'' be an ''R''-module and ''I'' an ideal of ''R''. Given <math>f \in M</math>, the '''initial form''' of ''f'' in <math>\operatorname{gr}_I M</math>, written <math>\mathrm{in}(f)</math>, is the equivalence class of ''f'' in <math>I^mM</math> where ''m'' is the maximum integer such that <math>f\in I^mM</math>. If <math>f \in I^mM</math> for every ''m'', then set <math>\mathrm{in}(f) = 0</math>. The initial form map is only a map of sets and generally not a [[module homomorphism|homomorphism]]. For a [[submodule]] <math>N \subset M</math>, <math>\mathrm{in}(N)</math> is defined to be the submodule of <math>\operatorname{gr}_I M</math> generated by <math>\{\mathrm{in}(f) | f \in N\}</math>. This may not be the same as the submodule of <math>\operatorname{gr}_IM</math> generated by the only initial forms of the generators of ''N''.

A ring inherits some "good" properties from its associated graded ring. For example, if ''R'' is a [[Noetherian ring|noetherian]] [[local ring]], and <math>\operatorname{gr}_I R</math> is an [[integral domain]], then ''R'' is itself an integral domain.<ref>{{harvnb|Eisenbud|loc=Corollary 5.5}}</ref>

== Examples ==

Let ''U'' be the enveloping algebra of a Lie algebra <math>\mathfrak{g}</math> over a field ''k''; it is filtered by degree. The [[Poincaré–Birkhoff–Witt theorem]] implies that <math>\operatorname{gr} U</math> is a polynomial ring; in fact, it is the [[coordinate ring]] <math>k[\mathfrak{g}^*]</math>.

== Generalization to multiplicative filtrations ==
The associated graded can also be defined more generally for multiplicative [[Filtration (mathematics)|descending filtrations]] of ''R'', Let ''F'' be a descending chain of ideals of the form
:<math>R = I_0 \supset I_1 \supset I_2 \supset \dotsb</math>
such that <math>I_jI_k \subset I_{j + k}</math>. The graded ring associated with this filtration is <math>\operatorname{gr}_F R = \oplus_{n=0}^\infty I^n/ I^{n+1}</math>. Multiplication and the initial form map are defined as above.

== See also ==
* [[Graded (mathematics)]]
* [[Rees algebra]]

==References==
{{reflist}}
* [[David Eisenbud|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.

{{abstract-algebra-stub}}
[[Category:Ring theory]]

Projection formula - Revision history

en>TakuyaMurata at 03:35, 24 February 2013