Associated graded ring
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:
grIR=⊕n=0∞In/In+1{displaystyle 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 grIR{displaystyle operatorname {gr} _{I}R}:
grIM=⊕0∞InM/In+1M{displaystyle operatorname {gr} _{I}M=oplus _{0}^{infty }I^{n}M/I^{n+1}M}.
Contents
1 Basic definitions and properties
2 Examples
3 Generalization to multiplicative filtrations
4 See also
5 References
Basic definitions and properties
For a ring R and ideal I, multiplication in grIR{displaystyle operatorname {gr} _{I}R} is defined as follows: First, consider homogeneous elements a∈Ii/Ii+1{displaystyle ain I^{i}/I^{i+1}} and b∈Ij/Ij+1{displaystyle bin I^{j}/I^{j+1}} and suppose a′∈Ii{displaystyle a'in I^{i}} is a representative of a and b′∈Ij{displaystyle b'in I^{j}} is a representative of b. Then define ab{displaystyle ab} to be the equivalence class of a′b′{displaystyle a'b'} in Ii+j/Ii+j+1{displaystyle I^{i+j}/I^{i+j+1}}. Note that this is well-defined modulo Ii+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 fin M}, the initial form of f in grIM{displaystyle operatorname {gr} _{I}M}, written in(f){displaystyle mathrm {in} (f)}, is the equivalence class of f in ImM/Im+1M{displaystyle I^{m}M/I^{m+1}M} where m is the maximum integer such that f∈ImM{displaystyle fin I^{m}M}. If f∈ImM{displaystyle fin I^{m}M} for every m, then set 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 Nsubset M}, in(N){displaystyle mathrm {in} (N)} is defined to be the submodule of grIM{displaystyle operatorname {gr} _{I}M} generated by {in(f)|f∈N}{displaystyle {mathrm {in} (f)|fin N}}. This may not be the same as the submodule of grIM{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 grIR{displaystyle operatorname {gr} _{I}R} is an integral domain, then R is itself an integral domain.[1]
Examples
Let U be the enveloping algebra of a Lie algebra g{displaystyle {mathfrak {g}}} over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that grU{displaystyle operatorname {gr} U} is a polynomial ring; in fact, it is the coordinate ring k[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=I0⊃I1⊃I2⊃⋯{displaystyle R=I_{0}supset I_{1}supset I_{2}supset dotsb }
such that IjIk⊂Ij+k{displaystyle I_{j}I_{k}subset I_{j+k}}. The graded ring associated with this filtration is grFR=⊕n=0∞In/In+1{displaystyle 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
- Graded (mathematics)
- Rees algebra
References
^ Eisenbud, Corollary 5.5
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. MR 1322960..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
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. MR 1011461.