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doc ///
Key
minimalBetti
(minimalBetti,Ideal)
(minimalBetti,Module)
[minimalBetti,DegreeLimit]
[minimalBetti,LengthLimit]
[minimalBetti,Weights]
[minimalBetti,ParallelizeByDegree]
Headline
minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
Usage
B = minimalBetti I
B = minimalBetti(I, DegreeLimit=>d, LengthLimit=>len, Weights=>h)
Inputs
I:{Ideal,Module}
a homogeneous ideal or module in a singly graded polynomial ring or skew commuting polynomial ring over a finite prime field
DegreeLimit=>ZZ
if given, only compute enough to determine the Betti diagram up to and including the row labelled {\tt d}
LengthLimit=>ZZ
if given, only compute enough to determine the Betti diagram up to and including the column labelled {\tt len}
Weights=>List
see @TO [betti, Weights]@
ParallelizeByDegree => Boolean
Use additional parallelism to compute different (homological degree, internal degree) pairs in
parallel when possible. To control parallelism, see @TO "parallelism in engine computations"@
Outputs
:BettiTally
Description
Text
Given a singly-graded module, this function computes the minimal betti numbers of the module.
If the input is an ideal $I \subset S$, it computes the minimal betti numbers of $S^1/I$.
The algorithm used is based on the @TO FastNonminimal@ algorithm,
except that the complex is not constructed, resulting in a smaller memory footprint and often reduced computation time.
Example
I = Grassmannian(1,6, CoefficientRing => ZZ/101);
S = ring I
elapsedTime C = minimalBetti I
Text
One can compute smaller parts of the Betti table, by using @TO DegreeLimit@ and/or @TO LengthLimit@.
Example
I = ideal I_*;
elapsedTime C = minimalBetti(I, DegreeLimit=>2)
I = ideal I_*;
elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
I = ideal I_*;
elapsedTime C = minimalBetti(I, LengthLimit=>5)
Text
This function computes only as much of the non-minimal resolution as needed
to compute the desired Betti numbers. Further calls will generally not recompute
previously computed parts of the resolution, except that if you ask for
a longer resolution than previously, it currently will recompute the resolution.
This behavior might change in later releases.
Text
If one has already computed the non-minimal free resolution using
@TO FastNonminimal@, then one can use @TO [betti,Minimize]@, except that it doesn't currently
have support for {\tt DegreeLimit} and {\tt LengthLimit}, and probably still computes more
than is needed (it is still experimental).
Caveat
Only works over finite prime field.
If the ideal or module is a non-homogeneous or multi-homogeneous object,
then this function will result in an error.
SeeAlso
betti
"OldChainComplexes :: betti(...,Minimize=>...)"
"OldChainComplexes :: resolution"
"OldChainComplexes :: resolution(...,FastNonminimal=>...)"
"Complexes :: Nonminimal"
///
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