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Graded Betti Number

A graded Betti number beta_(i,j)(M) counts the copies of S(-j) in homological position i of a minimal graded free resolution of a finitely generated module M that is also a graded module over the standard graded polynomial ring S=k[x_1,...,x_n], where k is a field and the grading is by total degree. The shift is S(-j)_a=S_(a-j), and the maps in the minimal resolution have entries in <x_1,...,x_n> (Bruns and Herzog 1998). Equivalently, using Tor,

beta_(i,j)(M)=dim_kTor_i^S(M,k)_j.

The total Betti number in position i is sum_(j)beta_(i,j)(M).

For example, for n>=2, S/<x_1^2,x_2^3> has graded Betti numbers beta_(0,0)=beta_(1,2)=beta_(1,3)=beta_(2,5)=1, with all others 0. They record the two generators of the ideal and their relation in total degree 5. The lex-plus-powers conjecture proposes simultaneous upper bounds for graded Betti numbers.

See also

Betti Number, Graded Free Resolution, Graded Module, Hilbert Function, Lex-Plus-Powers Conjecture, Tor

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.

Cite this as:

Weisstein, Eric W. "Graded Betti Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GradedBettiNumber.html

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