1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
|
doc ///
Key
minimalBetti
(minimalBetti,Ideal)
(minimalBetti,Module)
(minimalBetti,MonomialIdeal)
[minimalBetti,DegreeLimit]
[minimalBetti,LengthLimit]
[minimalBetti,Weights]
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,MonomialIdeal}
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]@
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
Released in M2 1.9.1, still experimental. 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
[betti, Minimize]
resolution
[resolution, FastNonminimal]
///
|
