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5.1.128 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 argument.
Let $0 \rightarrow\ \bigoplus_a K[x]e_{a,n}\ \rightarrow\ \dots \rightarrow\ \bigoplus_a K[x]e_{a,0}\ \rightarrow\ I\ \rightarrow\ 0$ be a minimal resolution of I considered with homogeneous maps of degree 0. The regularity is the smallest number $s$ with the property deg( $e_{a,i}) \leq s+i$ 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 behaviour:
  list jj=jr;
  regularity(jj);
==> 25
  jj=nres(j,0);
  regularity(jj);
==> 27
  jj=minres(jj);
  regularity(jj);
==> 25
See fres; list; minres; mres; option; res; resolution; sres.