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
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
|
--- status: DRAFT
--- author(s): caviglia, kummini
--- notes:
doc ///
Key
[betti,Minimize]
Headline
minimal betti numbers of a non-minimal free resolution
Usage
betti(C, Minimize => true)
Inputs
C:ChainComplex
computed using @TO FastNonminimal@ (and therefore a
non-minimal free resolution of an ideal or
module in a polynomial ring
or skew commuting polynomial ring, over a finite prime field)
Outputs
:BettiTally
Description
Text
Given a chain complex computed using {\tt res(I, FastNonminimal => true)} (@TO FastNonminimal@),
returns the minimal graded Betti numbers of this complex.
To get the actual betti numbers of the non-minimal resolution, use @TO betti@.
If you simply want the minimal betti numbers of a module or ideal {\tt I},
use @TO "minimalBetti"@.
Example
I = Grassmannian(1,6, CoefficientRing => ZZ/101);
S = ring I
elapsedTime C = res(I, FastNonminimal => true)
Text
For a non-minimal resolution, @TO betti@ gives the actual ranks of the complex.
If the option @TT "Minimize => true"@ is given, the minimal Betti numbers are returned.
Example
betti C
betti(C, Minimize=>true)
Text
This command is useful if the non-minimal free resolution has already been computed.
However, as mentioned above, if one wants the minimal betti numbers of an ideal or module,
it is recommended to use the function @TO "minimalBetti"@ as that avoids much computation,
and allows the use of length and degree limits.
Caveat
Released in M2 1.9, still experimental. Only works over finite prime field.
If the complex is the resolution of a non-homogeneous or multi-homogeneous object,
then this function will result in an error.
SeeAlso
minimalBetti
betti
resolution
FastNonminimal
///
document {
Key => [betti, Weights],
Usage => "betti(..., Weights => w)",
Inputs => {
"w" => List => { "a list of integers" }
},
PARA {
"The module or chain complex provided should be over a ring whose degree length
(see ", TO "degreeLength", ") is the same as the length of the list ", TT "w", ".
The dot products of w with the multi-degrees occurring will be used to construct
the resulting betti tally (see ", TO "BettiTally", "). If the option is not
specified, then the heft vector of the underlying ring is used, see ", TT "heft vectors", ".
If the ring has no heft vector, then the weights are taken to be all zero."
}
}
document {
Key => [regularity, Weights],
Usage => "regularity(..., Weights => w)",
Inputs => {
"w" => List => { "a list of integers" }
},
PARA {
"The module or chain complex provided should be over a ring whose degree length
(see ", TO "degreeLength", ") is the same as the length of the list ", TT "w", ".
The dot products of w with the multi-degrees occurring will be used in
the resulting computation."
}
}
document {
Key => {betti},
Headline => "display degrees",
"The function ", TT "betti", " displays the degrees of generators and
relations of graded modules and ideals."
}
document {
Key => (betti,GroebnerBasis),
Headline => "diagram of the degrees of a groebner basis",
Usage => "betti G",
Inputs => { "G" => GroebnerBasis },
Outputs => { { "a diagram showing the degrees of the generators of the
source and target modules of the matrix of generators of ", TT "G" } },
"Here is an example:",
EXAMPLE {
"S = ZZ/10007[x,y];",
"G = gb ideal(x^3+y^3, x*y^4);",
"gens G",
"betti G"
},
"For an explanation of the diagram, see ", TO (betti, GradedModule), ".",
}
document {
Key => (betti,Matrix),
Headline => "display of the degrees of a map",
Usage => "betti f",
Inputs => { "f" => Matrix },
Outputs => { { "a diagram showing the degrees of the generators of the source and target modules of ", TT "f" } },
EXAMPLE {
"S = ZZ/10007[x,y];",
"betti matrix{{x^3,x*y^2},{y*x,y^2}}"
},
"For an explanation of the diagram, see ", TO (betti, GradedModule), ".",
Caveat => {"The diagram ignores the degree of the map itself."},
"For an explanation of the diagram, see ", TO (betti, GradedModule), ".",
}
document {
Key => {(betti,Module),(betti,CoherentSheaf)},
Headline => "show the degrees of the generators and relations of a module or a coherent sheaf",
Usage => "betti M",
Inputs => { "M" => Module => " graded." },
Outputs => {{ "a diagram showing the degrees of the generators and the relations in the module ", TT "M" }
},
"The diagram lists the zeroth and first graded and total Betti numbers of the module.",
EXAMPLE {
"S = ZZ/10007[x,y];",
"betti coker matrix{{x^3,x*y^2},{y*x^2,y^3}}",
},
"Another example:",
EXAMPLE {
"betti coker map(S^{0,-1},,matrix{{x^2,y},{y^3,x^2}})",
},
"For an explanation of the diagram, see ", TO (betti,GradedModule), ".",
}
document {
Key => (betti, GradedModule),
Headline => "display of degrees in a graded module",
Usage => "betti C",
Inputs => {
"C" => GradedModule,
Weights => List => {
"a list of integers whose dot product with the multidegree of a basis
element is enumerated in the display returned. The default is the
heft vector of the ring. See ", TO "heft vectors", "."
}
},
Outputs => { { "a diagram showing the degrees of the generators of the modules in ", TT "C"} },
PARA {
"The diagram can be used to determine the degrees of the entries in the matrices
of the differentials in a chain complex (which is a type of graded module) provided they are homogeneous maps
of degree 0."
},
PARA{"Here is a sample diagram."},
EXAMPLE {
"R = ZZ/101[a..h]",
"p = genericMatrix(R,a,2,4)",
"q = generators gb p",
"C = resolution cokernel leadTerm q",
"betti C",
},
PARA {
"Column ", TT "j", " of the top row of the diagram gives the rank of the
free module ", TT "C_j", ". (Columns are numbered from 0.) The entry in
column ", TT "j", " in the row labelled ", TT "i", " is the number
of basis elements of (weighted) degree ", TT "i+j", " in the free module", TT " C_j",
". When the chain complex is the resolution of a module the entries are
the total and the graded Betti numbers of the module."
},
PARA{
"If the numbers are needed in a program, then they are accessible, because
the value returned is ", ofClass BettiTally, ", and the diagram you see
on the screen is just the way it prints out."
},
PARA {
"The heft vector is used, by default, as the weight vector for weighting the
components of the degree vectors of basis elements."
},
EXAMPLE lines ///
R = QQ[a,b,c,Degrees=>{-1,-2,-3}];
heft R
betti res coker vars R
betti(oo, Weights => {1})
R = QQ[a,b,c,d,Degrees=>{{1,0},{2,1},{0,1},{-2,1}}];
heft R
b = betti res coker vars R
betti(b, Weights => {1,0})
betti(b, Weights => {0,1})
///
}
document {
Key => {(betti,Ideal),(betti,MonomialIdeal)},
Headline => "gives the degrees of generators.",
Usage => "betti I",
Inputs => { "I" => Ideal },
Outputs => { { "a diagram showing the degrees of the generators of the modules in ", TT "C" } },
"The output is the degrees of generators and relations of the quotient of
the ambient ring by the ideal.",
EXAMPLE {
"S = ZZ/10007[x,y];",
"I = ideal(x^2,y^3);",
"betti I",
},
"For an explanation of the diagram, see ", TO (betti,GradedModule), ".",
}
|
