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5.1.107 regularity
Syntax:
regularity ( list_expression )
regularity ( resolution_expression )
Type:
int
Purpose:
computes the regularity of a homogeneous ideal, resp. module, from a
minimal resolution given by the list expression.
Let
0 → ⊕
aK[x]ea,n → … → ⊕
aK[x]ea,0 → I → 0 be a minimal resolution of I considered with
homogeneous maps of degree 0. The regularity is the smallest number s with the property deg(ea,i) ≤ s + i for all i.
Note:
If applied to a non minimal resolution only an upper bound is returned.
If the input to the commands res and mres is homogeneous
the regularity is computed and used as a degree bound during
the computation unless option(notRegularity); is given.
Example:
ring rh3=32003,(w,x,y,z),(dp,C);
poly f=x11+y10+z9+x5y2+x2y2z3+xy3*(y2+x)^2;
ideal j=homog(jacob(f),w);
def jr=res(j,0);
regularity(jr);
→ 25
// example for upper bound behavior:
list jj=jr;
regularity(jj);
→ 25
jj=nres(j,0);
regularity(jj);
→ 27
jj=minres(jj);
regularity(jj);
→ 25
See
list;
minres;
mres;
option;
res;
resolution;
sres.
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