Associated graded ring
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:
- [math]\displaystyle{ \operatorname{gr}_I R = \oplus_{n=0}^\infty I^n/I^{n+1} }[/math].
Similarly, if M is a left R-module, then the associated graded module is the graded module over [math]\displaystyle{ \operatorname{gr}_I R }[/math]:
- [math]\displaystyle{ \operatorname{gr}_I M = \oplus_{n=0}^\infty I^n M/ I^{n+1} M }[/math].
Basic definitions and properties
For a ring R and ideal I, multiplication in [math]\displaystyle{ \operatorname{gr}_IR }[/math] is defined as follows: First, consider homogeneous elements [math]\displaystyle{ a \in I^i/I^{i + 1} }[/math] and [math]\displaystyle{ b \in I^j/I^{j + 1} }[/math] and suppose [math]\displaystyle{ a' \in I^i }[/math] is a representative of a and [math]\displaystyle{ b' \in I^j }[/math] is a representative of b. Then define [math]\displaystyle{ ab }[/math] to be the equivalence class of [math]\displaystyle{ a'b' }[/math] in [math]\displaystyle{ I^{i + j}/I^{i + j + 1} }[/math]. Note that this is well-defined modulo [math]\displaystyle{ 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 ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given [math]\displaystyle{ f \in M }[/math], the initial form of f in [math]\displaystyle{ \operatorname{gr}_I M }[/math], written [math]\displaystyle{ \mathrm{in}(f) }[/math], is the equivalence class of f in [math]\displaystyle{ I^mM/I^{m+1}M }[/math] where m is the maximum integer such that [math]\displaystyle{ f\in I^mM }[/math]. If [math]\displaystyle{ f \in I^mM }[/math] for every m, then set [math]\displaystyle{ \mathrm{in}(f) = 0 }[/math]. The initial form map is only a map of sets and generally not a homomorphism. For a submodule [math]\displaystyle{ N \subset M }[/math], [math]\displaystyle{ \mathrm{in}(N) }[/math] is defined to be the submodule of [math]\displaystyle{ \operatorname{gr}_I M }[/math] generated by [math]\displaystyle{ \{\mathrm{in}(f) | f \in N\} }[/math]. This may not be the same as the submodule of [math]\displaystyle{ \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 local ring, and [math]\displaystyle{ \operatorname{gr}_I R }[/math] is an integral domain, then R is itself an integral domain.[1]
gr of a quotient module
Let [math]\displaystyle{ N \subset M }[/math] be left modules over a ring R and I an ideal of R. Since
- [math]\displaystyle{ {I^n(M/N) \over I^{n+1}(M/N)} \simeq {I^n M + N \over I^{n+1}M + N} \simeq {I^n M \over I^n M \cap (I^{n+1} M + N)} = {I^n M \over I^n M \cap N + I^{n+1} M} }[/math]
(the last equality is by modular law), there is a canonical identification:[2]
- [math]\displaystyle{ \operatorname{gr}_I (M/N) = \operatorname{gr}_I M / \operatorname{in}(N) }[/math]
where
- [math]\displaystyle{ \operatorname{in}(N) = \bigoplus_{n=0}^{\infty} {I^n M \cap N + I^{n+1} M \over I^{n+1} M}, }[/math]
called the submodule generated by the initial forms of the elements of [math]\displaystyle{ N }[/math].
Examples
Let U be the universal enveloping algebra of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that [math]\displaystyle{ \operatorname{gr} U }[/math] is a polynomial ring; in fact, it is the coordinate ring [math]\displaystyle{ k[\mathfrak{g}^*] }[/math].
The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.
Generalization to multiplicative filtrations
The associated graded can also be defined more generally for multiplicative descending filtrations of R (see also filtered ring.) Let F be a descending chain of ideals of the form
- [math]\displaystyle{ R = I_0 \supset I_1 \supset I_2 \supset \dotsb }[/math]
such that [math]\displaystyle{ I_jI_k \subset I_{j + k} }[/math]. The graded ring associated with this filtration is [math]\displaystyle{ \operatorname{gr}_F R = \bigoplus_{n=0}^\infty I_n/ I_{n+1} }[/math]. Multiplication and the initial form map are defined as above.
See also
References
- ↑ Eisenbud 1995, Corollary 5.5
- ↑ Zariski & Samuel 1975, Ch. VIII, a paragraph after Theorem 1.
- Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. 150. New York: Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.
- Matsumura, Hideyuki (1989). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Translated from the Japanese by M. Reid (Second ed.). Cambridge: Cambridge University Press. ISBN 0-521-36764-6.
- Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8
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