Associated graded ring

For a ring R and ideal I, multiplication in $\backslash operatorname\{gr\}\_IR$ is defined as follows: First, consider homogeneous elements $a\; \backslash in\; I^i/I^\{i\; +\; 1\}$ and $b\; \backslash in\; I^j/I^\{j\; +\; 1\}$ and suppose $a\text{'}\; \backslash in\; I^i$ is a representative of a and $b\text{'}\; \backslash in\; I^j$ is a representative of b. Then define $ab$ to be the equivalence class of $a\text{'}b\text{'}$ 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 ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given $f\; \backslash in\; M$, the initial form of f in $\backslash operatorname\{gr\}\_I\; M$, written $\backslash mathrm\{in\}(f)$, is the equivalence class of f in $I^mM/I^\{m+1\}M$ where m is the maximum integer such that $f\backslash in\; I^mM$. If $f\; \backslash in\; I^mM$ for every m, then set $\backslash mathrm\{in\}(f)\; =\; 0$. The initial form map is only a map of sets and generally not a homomorphism. For a submodule $N\; \backslash subset\; M$, $\backslash mathrm\{in\}(N)$ is defined to be the submodule of $\backslash operatorname\{gr\}\_I\; M$ generated by $\backslash \{\backslash mathrm\{in\}(f)\; |\; f\; \backslash in\; N\backslash \}$. This may not be the same as the submodule of $\backslash operatorname\{gr\}\_IM$ 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 $\backslash operatorname\{gr\}\_I\; R$ is an integral domain, then R is itself an integral domain.{{harvnb|Eisenbud|1995|loc=Corollary 5.5}}

Let $N\; \backslash subset\; M$ be left modules over a ring R and I an ideal of R. Since

:$\{I^n(M/N)\; \backslash over\; I^\{n+1\}(M/N)\}\; \backslash simeq\; \{I^n\; M\; +\; N\; \backslash over\; I^\{n+1\}M\; +\; N\}\; \backslash simeq\; \{I^n\; M\; \backslash over\; I^n\; M\; \backslash cap\; (I^\{n+1\}\; M\; +\; N)\}\; =\; \{I^n\; M\; \backslash over\; I^n\; M\; \backslash cap\; N\; +\; I^\{n+1\}\; M\}$

(the last equality is by modular law), there is a canonical identification:{{harvnb|Zariski|Samuel|1975|loc=Ch. VIII, a paragraph after Theorem 1.}}

:$\backslash operatorname\{gr\}\_I\; (M/N)\; =\; \backslash operatorname\{gr\}\_I\; M\; /\; \backslash operatorname\{in\}(N)$

where

:$\backslash operatorname\{in\}(N)\; =\; \backslash bigoplus\_\{n=0\}^\{\backslash infty\}\; \{I^n\; M\; \backslash cap\; N\; +\; I^\{n+1\}\; M\; \backslash over\; I^\{n+1\}\; M\},$

called the ''submodule generated by the initial forms of the elements of $N$.

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

The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.

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

:$R\; =\; I\_0\; \backslash supset\; I\_1\; \backslash supset\; I\_2\; \backslash supset\; \backslash dotsb$

such that $I\_jI\_k\; \backslash subset\; I\_\{j\; +\; k\}$. The graded ring associated with this filtration is $\backslash operatorname\{gr\}\_F\; R\; =\; \backslash bigoplus\_\{n=0\}^\backslash infty\; I\_n/\; I\_\{n+1\}$. Multiplication and the initial form map are defined as above.

• Graded (mathematics)
• Rees algebra

{{reflist}}

• {{cite book|last=Eisenbud|first=David|authorlink=David Eisenbud|title=Commutative Algebra|series=Graduate Texts in Mathematics|volume=150|publisher=Springer-Verlag|year=1995|isbn=0-387-94268-8|doi=10.1007/978-1-4612-5350-1|mr=1322960|location=New York}}
• {{cite book|last=Matsumura|first=Hideyuki|title=Commutative ring theory|others=Translated from the Japanese by M. Reid|edition=Second|series=Cambridge Studies in Advanced Mathematics|volume=8|publisher=Cambridge University Press|location=Cambridge|year=1989|isbn=0-521-36764-6|mr=1011461}}
• {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}

Category:Ring theory