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Graded Free Resolution

A graded free resolution is a resolution by graded free modules of a finitely generated graded module M over a commutative Noetherian Z-graded ring R, with module homomorphisms preserving degree. When R is a standard graded polynomial ring over a field, a minimal graded free resolution is written in the form

...-> direct sum _(j in Z)R(-j)^(beta_(sj))->...-> direct sum _(j in Z)R(-j)^(beta_(1j))-> direct sum _(j in Z)R(-j)^(beta_(0j))->M->0,
(1)

where R(-j) indicates the ring R with the shifted grading such that, for all a in Z,

R(-j)_a=R_(a-j).
(2)

For all nonnegative integers i and all integers j, beta_(ij) is the number of copies of R(-j) appearing in the ith module of the minimal resolution, and is a graded Betti number. The total Betti number in position i is beta_i=sum_(j in Z)beta_(ij).

For example, if R is the polynomial ring K[X_1,X_2,X_3] over a field K, with the usual grading, the graded free resolution of M=R/<X_1^2,X_2^3> is

0->R(-5)-->^(1|->(-X_2^3,X_1^2))R(-2) direct sum R(-3) -->^((1,0)|->X_1^2; (0,1)|->X_2^3)R->^(1|->1^_)M->0.
(3)

In R(-2), the constant polynomials have degree 2. It follows that -X_2^3 has degree 5. Similarly, X_1^2 has degree 5 in R(-3).

The graded free resolution can be used to compute the Hilbert function.

See also

Graded Betti Number, Graded Module, Hilbert Function, Lex-Plus-Powers Conjecture, Projective Dimension

Portions of this 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, 1998.

Referenced on Wolfram|Alpha

Graded Free Resolution

Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Graded Free Resolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GradedFreeResolution.html

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