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}}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}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}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 grI⁡R{displaystyle operatorname {gr} _{I}R}operatorname {gr}_{I}R is defined as follows: First, consider homogeneous elements a∈Ii/Ii+1{displaystyle ain I^{i}/I^{i+1}}ain I^{i}/I^{{i+1}} and b∈Ij/Ij+1{displaystyle bin I^{j}/I^{j+1}}bin I^{j}/I^{{j+1}} and suppose a′∈Ii{displaystyle a'in I^{i}}a'in I^{i} is a representative of a and b′∈Ij{displaystyle b'in I^{j}}b'in I^{j} is a representative of b. Then define ab{displaystyle ab}ab to be the equivalence class of a′b′{displaystyle a'b'}a'b' in Ii+j/Ii+j+1{displaystyle I^{i+j}/I^{i+j+1}}I^{{i+j}}/I^{{i+j+1}}. Note that this is well-defined modulo Ii+j+1{displaystyle I^{i+j+1}}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}fin M, the initial form of f in grI⁡M{displaystyle operatorname {gr} _{I}M}operatorname {gr}_{I}M, written in(f){displaystyle mathrm {in} (f)}{mathrm {in}}(f), is the equivalence class of f in ImM/Im+1M{displaystyle I^{m}M/I^{m+1}M}I^{m}M/I^{{m+1}}M where m is the maximum integer such that f∈ImM{displaystyle fin I^{m}M}fin I^{m}M. If f∈ImM{displaystyle fin I^{m}M}fin I^{m}M for every m, then set in(f)=0{displaystyle mathrm {in} (f)=0}{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}Nsubset M, in(N){displaystyle mathrm {in} (N)}{mathrm {in}}(N) is defined to be the submodule of grI⁡M{displaystyle operatorname {gr} _{I}M}operatorname {gr}_{I}M generated by {in(f)|f∈N}{displaystyle {mathrm {in} (f)|fin N}}{{mathrm {in}}(f)|fin N}. This may not be the same as the submodule of grI⁡M{displaystyle operatorname {gr} _{I}M}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}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}}}{mathfrak {g}} over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that gr⁡U{displaystyle operatorname {gr} U}operatorname {gr}U is a polynomial ring; in fact, it is the coordinate ring k[g∗]{displaystyle k[{mathfrak {g}}^{*}]}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 }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}}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}}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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