# part(List,Complex) -- extract a graded component of a complex

## Synopsis

• Function: part
• Usage:
part(d, C)
• Inputs:
• d, a list, or ZZ, if the underlying ring $R$ is singly graded.
• C, , that is homogeneous over $R$
• Outputs:
• , a complex over the coefficient ring of $R$

## Description

If $C$ is a graded (homogeneous) complex over a ring $R$, and $d$ is a degree, this method computes the degree $d$ part of the complex over the coefficient ring of $R$.

Taking parts of a graded (homogeneous) complex commutes with taking homology.

 i1 : R = QQ[a,b,c,d]; i2 : I = ideal(a*b, a*c, b*c, a*d) o2 = ideal (a*b, a*c, b*c, a*d) o2 : Ideal of R i3 : C = freeResolution I 1 4 4 1 o3 = R <-- R <-- R <-- R 0 1 2 3 o3 : Complex i4 : D = part(4,C) 35 40 16 1 o4 = QQ <-- QQ <-- QQ <-- QQ 0 1 2 3 o4 : Complex i5 : prune HH D == part(4, HH C) o5 = true i6 : prune HH D == part(4, complex(R^1/I)) o6 = true

Given a squarefree monomial ideal corresponding to a simplicial complex, in a polynomial ring equipped with the fine grading, parts of the dual of the free resolution of the monomial ideal are the chain complexes of the induced simplicial subcomplexes.

 i7 : S = QQ[a..d, DegreeRank=>4]; i8 : I = intersect(ideal(a,b), ideal(c,d)) o8 = ideal (b*d, a*d, b*c, a*c) o8 : Ideal of S i9 : C = dual freeResolution I 1 4 4 1 o9 = S <-- S <-- S <-- S -3 -2 -1 0 o9 : Complex i10 : prune HH (part({-1,-1,-1,-1}, C)) -- empty quadrilateral 1 o10 = QQ -3 o10 : Complex i11 : prune HH part({-1,-1,0,0}, C) -- 2 points 1 o11 = QQ -2 o11 : Complex i12 : prune HH part({0,0,-1,-1}, C) -- 2 points 1 o12 = QQ -2 o12 : Complex i13 : prune HH part({0,0,0,0}, C) -- solid quadrilateral o13 = 0 0 o13 : Complex