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
|
--- status: DRAFT
--- author(s): MES
--- notes:
document {
Key => chainComplex,
Headline => "make a chain complex",
TT "chainComplex", " -- a method for creating chain complexes.",
}
document { Key => (chainComplex,Ring), Headline => "make an empty chain complex over a ring",
Usage => "chainComplex R",
Inputs => {"R"},
Outputs => {{"an empty chain complex over ", TT "R"}},
"The resulting chain complex can be further populated with modules and maps.",
EXAMPLE lines ///
C = chainComplex ZZ
C.dd_101 = random(ZZ^2,ZZ^3)
C.dd_105 = id_(ZZ^2)
C
C.dd
///}
document {
Key => {(chainComplex,List),(chainComplex,Sequence)},
Headline => "make a chain complex",
Usage => "C = chainComplex{f1,f2,f3,...}\nC = chainComplex(f1,f2,f3,...)",
Inputs => {
Nothing => { TT "f1,f2,f3,...", ", homomorphisms over the same ring, forming a complex" },
},
Outputs => { "C" => ChainComplex => {"the given complex, where ", TT "f1 == C.dd_1", ", ", TT "f2 == CC.dd_2", ", etc."} },
"The maps f1, f2, ... must be defined over the same base ring, and they must form
a complex: the target of f(i+1) is the source of fi.",
PARA{},
"The following example illustrates how chainComplex adjusts the degrees of
the modules involved to ensure that sources and targets of the differentials
correspond exactly.",
EXAMPLE {
"R = ZZ/101[x,y]",
"C = chainComplex{matrix{{x,y}},matrix{{x*y},{-x^2}}}",
},
"We check that this is a complex:",
EXAMPLE {
"C.dd^2 == 0"
},
"The homology of this complex:",
EXAMPLE {
"HH C"
},
Caveat => {},
SeeAlso => {"chain complexes"}
}
document {
Key => (chainComplex,GradedModule),
Headline => "make a chain complex from a graded module",
Usage => "chainComplex M",
Inputs => { "M" },
Outputs => {
{"the chain complex made from ", TT "M", " by installing the zero map as differential" }
},
SeeAlso => {"chain complexes"}
}
document {
Key => (chainComplex,Matrix),
Headline => "make a small chain complex",
Usage => "chainComplex f",
Inputs => { "f" },
Outputs => {
{"a chain complex with the map ", TT "f", " serving as the differential ", TT "C.dd_1"}
},
EXAMPLE {
"R = QQ[x,y];",
"f = vars R",
"chainComplex f",
"HH oo"
},
SeeAlso => {"chain complexes"}
}
|
