Associated graded ring

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

grI⁡R=⊕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 grI⁡R{displaystyle operatorname {gr} _{I}R}:

grI⁡M=⊕0∞InM/In+1M{displaystyle operatorname {gr} _{I}M=oplus _{0}^{infty }I^{n}M/I^{n+1}M}.

Contents

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  1 Basic definitions and properties
  


• 
  2 Examples
  


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  3 Generalization to multiplicative filtrations
  


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  4 See also
  


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  5 References
  

Basic definitions and properties

For a ring R and ideal I, multiplication in grI⁡R{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 grI⁡M{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 grI⁡M{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 grI⁡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 grI⁡R{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 gr⁡U{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 grF⁡R=⊕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

1. 
   ^ 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.
  

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