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# buchbergerResolution -- make a Buchberger resolution of a monomial ideal

## Synopsis

• Usage:
buchbergerResolution(L)
buchbergerResolution(I)
• Inputs:
• I, ,
• L, a list, a minimal set of generators for a monomial ideal $I$
• Outputs:
• , the free resolution of the monomial ideal $I$ we get by homogenizing the Buchberger complex of $I$.

## Description

The Buchberger resolution of a monomial ideal is obtained by homogenizing the Buchberger complex of the ideal.

 i1 : R = ZZ/101[x_0..x_4]; i2 : L = {x_1^2, x_2^2, x_3^2, x_1*x_3, x_2*x_4}; i3 : BRes = (buchbergerResolution L); i4 : BRes.dd 1 5 o4 = 0 : R <--------------------------------------- R : 1 | x_1^2 x_2^2 x_3^2 x_1x_3 x_2x_4 | 5 9 1 : R <------------------------------------------------------------------------ R : 2 {2} | -x_2^2 -x_3 -x_2x_4 0 0 0 0 0 0 | {2} | x_1^2 0 0 -x_3^2 -x_1x_3 -x_4 0 0 0 | {2} | 0 0 0 x_2^2 0 0 -x_1 -x_2x_4 0 | {2} | 0 x_1 0 0 x_2^2 0 x_3 0 -x_2x_4 | {2} | 0 0 x_1^2 0 0 x_2 0 x_3^2 x_1x_3 | 9 7 2 : R <----------------------------------------------------------- R : 3 {4} | x_3 x_4 0 0 0 0 0 | {3} | -x_2^2 0 x_2x_4 0 0 0 0 | {4} | 0 -x_2 -x_3 0 0 0 0 | {4} | 0 0 0 x_1 x_4 0 0 | {4} | x_1 0 0 -x_3 0 x_4 0 | {3} | 0 x_1^2 0 0 -x_3^2 -x_1x_3 0 | {3} | 0 0 0 x_2^2 0 0 x_2x_4 | {4} | 0 0 0 0 x_2 0 -x_1 | {4} | 0 0 x_1 0 0 x_2 x_3 | 7 2 3 : R <--------------------- R : 4 {5} | -x_4 0 | {5} | x_3 0 | {5} | -x_2 0 | {5} | 0 -x_4 | {5} | 0 x_1 | {5} | x_1 -x_3 | {5} | 0 x_2 | o4 : ChainComplexMap i5 : BRes == chainComplex(buchbergerSimplicialComplex(L,R), Labels => L) o5 = true

When the Buchberger resolution is a minimal free resolution, it agrees with the Scarf complex.

 i6 : Scarf = scarfChainComplex L 1 5 9 7 2 o6 = R <-- R <-- R <-- R <-- R 0 1 2 3 4 o6 : ChainComplex i7 : BRes == Scarf o7 = true