Associated graded ring

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In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

\operatorname{gr}_I R = \oplus_{n=0}^\infty I^n/I^{n+1}.

Similarly, if M is a left R-module, then the associated graded module is the graded module over \operatorname{gr}_I R:

\operatorname{gr}_I M = \oplus_0^\infty I^n M/ I^{n+1} M.

Basic definitions and properties[edit]

For a ring R and ideal I, multiplication in \operatorname{gr}_IR is defined as follows: First, consider homogeneous elements a \in I^i/I^{i + 1} and b \in I^j/I^{j + 1} and suppose a' \in I^i is a representative of a and b' \in I^j is a representative of b. Then define ab to be the equivalence class of a'b' in I^{i + j}/I^{i + j + 1}. Note that this is well-defined modulo I^{i + j + 1}. 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 f \in M, the initial form of f in \operatorname{gr}_I M, written \mathrm{in}(f), is the equivalence class of f in I^mM/I^{m+1}M where m is the maximum integer such that f\in I^mM. If f \in I^mM for every m, then set \mathrm{in}(f) = 0. The initial form map is only a map of sets and generally not a homomorphism. For a submodule N \subset M, \mathrm{in}(N) is defined to be the submodule of \operatorname{gr}_I M generated by \{\mathrm{in}(f) | f \in N\}. This may not be the same as the submodule of \operatorname{gr}_IM 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 \operatorname{gr}_I R is an integral domain, then R is itself an integral domain.[1]

Examples[edit]

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

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[edit]

The associated graded can also be defined more generally for multiplicative descending filtrations of R, Let F be a descending chain of ideals of the form

R = I_0 \supset I_1 \supset I_2 \supset \dotsb

such that I_jI_k \subset I_{j + k}. The graded ring associated with this filtration is \operatorname{gr}_F R = \oplus_{n=0}^\infty I_n/ I_{n+1}. Multiplication and the initial form map are defined as above.

See also[edit]

References[edit]

  1. ^ Eisenbud, Corollary 5.5
  • 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.