# universalPresentation -- A presentation of the Stanley-Reisner ring over its universal parameter ring

## Synopsis

• Usage:
universalPresentation D
universalPresentation F
• Inputs:
• D, ,
• F, a list, A list of facets defining a simplicial complex D
• Outputs:
• , A graded presentation of the Stanley-Reisner ring over its universal parameter ring

## Description

 i1 : S = QQ[a..e]; i2 : F = {a*b*c,c*d,e} o2 = {a*b*c, c*d, e} o2 : List i3 : D = simplicialComplex F o3 = | e cd abc | o3 : SimplicialComplex i4 : universalPresentation D o4 = cokernel {0} | 0 0 0 0 | {1} | 0 0 0 0 | {2} | 0 0 0 0 | {3} | 0 0 0 0 | {1} | 0 0 0 0 | {2} | 0 0 0 0 | {1} | 0 d 0 0 | {2} | 0 0 d 0 | {1} | c 0 0 d | 9 o4 : QQ[b..d]-module, quotient of (QQ[b..d]) i5 : M = universalPresentation F o5 = cokernel {0} | 0 0 0 0 | {1} | 0 0 0 0 | {2} | 0 0 0 0 | {3} | 0 0 0 0 | {1} | 0 0 0 0 | {2} | 0 0 0 0 | {1} | 0 d 0 0 | {2} | 0 0 d 0 | {1} | c 0 0 d | 9 o5 : QQ[b..d]-module, quotient of (QQ[b..d]) i6 : degrees M o6 = {{0}, {1}, {2}, {3}, {1}, {2}, {1}, {2}, {1}} o6 : List

• Module -- the class of all modules
• universalSOP -- Universal System of Parameters (USOP)

## Ways to use universalPresentation :

• "universalPresentation(List)"
• "universalPresentation(SimplicialComplex)"

## For the programmer

The object universalPresentation is .