Module Multiplicity
Module multiplicity is a number associated with every nonzero finitely generated graded module M over a graded ring R for which the Hilbert series is defined. If dim(M)=d, the Hilbert series of M can be written in the form
and the multiplicity of M is the integer
| e(M)=Q_M(1). |
If R is the polynomial ring K[X_1,...,X_n] over the field K, the multiplicity of the quotient ring S=R/<f>, where f is a polynomial of degree delta>0, is equal to delta. This example shows the geometric origin of the notion. The number delta is in fact the so-called intersection multiplicity of the algebraic variety V of K^n defined by the equation f=0, of which S is the coordinate ring (i.e., a line of K^n chosen in a sufficiently general way intersects V in delta distinct points).
The definition of multiplicity can be extended to nonzero finitely generated modules over a Noetherian local ring R. If M is the maximal ideal of R, one can define the multiplicity of M as the multiplicity of the associated graded module of M with respect to M.
See also
Associated Graded Module, MultiplicityThis entry contributed by Margherita Barile
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References
Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Referenced on Wolfram|Alpha
Module MultiplicityCite this as:
Barile, Margherita. "Module Multiplicity." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ModuleMultiplicity.html