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# subcomplex -- the subcomplex induced by a degree or monomial

## Synopsis

• Usage:
subcomplex(C,m)
subcomplex(C,d)
C_m
C_d
• Inputs:
• C, ,
• m, , a monomial
• d, an integer or a list, a degree
• Optional inputs:
• LabelRing => a ring, default value null, for the labels on the output
• Outputs:
• ,

## Description

This function computes the subcomplex of the input cell complex induced by a monomial or a degree. The returned cell complex will be labeled over the coefficient ring of the label ring of the original cell complex, or the ring given by the LabelRing optional parameter if provided.

When given a monomial, the function expects all labels to be monomials, ideals, or submodules of the ring associated to the cell complex. In that case, the returned cell complex will be the subcomplex of the original complex containing the cells whose labels either divide or contain the provided monomial. The labels on the cells in the returned cell complexes will be 1.

 i1 : R = QQ[x,y,z]; i2 : C = taylorComplex monomialIdeal {x,y,z} o2 = C o2 : CellComplex i3 : subcomplex(C,x*y) o3 = CellComplex over QQ of dimension 1 with 3 total cells {Cell of dimension 0 with label 1, Cell of dimension 0 with label 1} {Cell of dimension 1 with label 1} o3 : CellComplex i4 : subcomplex(C,x*y,LabelRing=>R) o4 = CellComplex over R of dimension 1 with 3 total cells {Cell of dimension 0 with label 1, Cell of dimension 0 with label 1} {Cell of dimension 1 with label 1} o4 : CellComplex

When given a degree, the function will use basis to find the appropriate free module over the new label ring representing the elements of that degree in the module associated to the label

 i5 : subcomplex(C,1) o5 = CellComplex over R of dimension 0 with 3 total cells 1 1 1 {Cell of dimension 0 with label QQ , Cell of dimension 0 with label QQ , Cell of dimension 0 with label QQ } o5 : CellComplex

In general, the degree version is intended to be used with fine-graded polynomial rings

 i6 : S = QQ[a,b,c,DegreeRank=>3]; i7 : D = taylorComplex monomialIdeal {a,b,c} o7 = D o7 : CellComplex i8 : subcomplex(D,{1,1,0}) o8 = CellComplex over S of dimension 1 with 3 total cells 1 1 {Cell of dimension 0 with label QQ , Cell of dimension 0 with label QQ } 1 {Cell of dimension 1 with label QQ } o8 : CellComplex

## Ways to use subcomplex :

• subcomplex(CellComplex,List)
• subcomplex(CellComplex,RingElement)
• subcomplex(CellComplex,ZZ)

## For the programmer

The object subcomplex is .