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:

gr I ⁡ R = ⨁ n = 0 ∞ I n / I n + 1 {\displaystyle \operatorname {gr} _{I}R=\bigoplus _{n=0}^{\infty }I^{n}/I^{n+1}} {\displaystyle \operatorname {gr} _{I}R=\bigoplus _{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 gr I ⁡ R {\displaystyle \operatorname {gr} _{I}R} {\displaystyle \operatorname {gr} _{I}R}:

gr I ⁡ M = ⨁ n = 0 ∞ I n M / I n + 1 M {\displaystyle \operatorname {gr} _{I}M=\bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M} {\displaystyle \operatorname {gr} _{I}M=\bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M}.

Basic definitions and properties

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

gr of a quotient module

Let N ⊂ M {\displaystyle N\subset M} {\displaystyle N\subset M} be left modules over a ring R and I an ideal of R. Since

I n ( M / N ) I n + 1 ( M / N ) ≃ I n M + N I n + 1 M + N ≃ I n M I n M ∩ ( I n + 1 M + N ) = I n M I n M ∩ N + I n + 1 M {\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}} {\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}}

(the last equality is by modular law), there is a canonical identification:[2]

gr I ⁡ ( M / N ) = gr I ⁡ M / in ⁡ ( N ) {\displaystyle \operatorname {gr} _{I}(M/N)=\operatorname {gr} _{I}M/\operatorname {in} (N)} {\displaystyle \operatorname {gr} _{I}(M/N)=\operatorname {gr} _{I}M/\operatorname {in} (N)}

where

in ⁡ ( N ) = ⨁ n = 0 ∞ I n M ∩ N + I n + 1 M I n + 1 M , {\displaystyle \operatorname {in} (N)=\bigoplus _{n=0}^{\infty }{I^{n}M\cap N+I^{n+1}M \over I^{n+1}M},} {\displaystyle \operatorname {in} (N)=\bigoplus _{n=0}^{\infty }{I^{n}M\cap N+I^{n+1}M \over I^{n+1}M},}

called the submodule generated by the initial forms of the elements of N {\displaystyle N} {\displaystyle N}.

Examples

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

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

R = I 0 ⊃ I 1 ⊃ I 2 ⊃ ⋯ {\displaystyle R=I_{0}\supset I_{1}\supset I_{2}\supset \dotsb } {\displaystyle R=I_{0}\supset I_{1}\supset I_{2}\supset \dotsb }

such that I j I k ⊂ I j + k {\displaystyle I_{j}I_{k}\subset I_{j+k}} {\displaystyle I_{j}I_{k}\subset I_{j+k}}. The graded ring associated with this filtration is gr F ⁡ R = ⨁ n = 0 ∞ I n / I n + 1 {\displaystyle \operatorname {gr} _{F}R=\bigoplus _{n=0}^{\infty }I_{n}/I_{n+1}} {\displaystyle \operatorname {gr} _{F}R=\bigoplus _{n=0}^{\infty }I_{n}/I_{n+1}}. Multiplication and the initial form map are defined as above.

See also

References

  1. Eisenbud 1995, Corollary 5.5
  2. Zariski & Samuel 1975, Ch. VIII, a paragraph after Theorem 1.