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--- status: done
--- author(s): dan
--- notes:
document {
Key => homology,
Headline => "general homology functor",
"Most applications of this functor are dispatched through ", TT "HH", ".
If it is intended that ", TT "i", " be of class ", TO "ZZ", ",
", TT "M", " be of class ", TT "A", ", and ", TT "N", " be of
class ", TT "B", ", then the method for computing ", TT "HH_i(M,N)", " can be installed with
code of the following form.",
PRE " homology(ZZ, A, B) := opts -> (i,M,N) -> ...",
SeeAlso => {"cohomology", "HH", "ScriptedFunctor"}
}
document {
Key => (homology,ZZ,ChainComplexMap),
Headline => "homology of a chain complex map",
Usage => "HH_i f",
Inputs => { "i", "f" },
Outputs => { {"the map on the ", TT "i", "-th homology module induced by the map ", TT "f", " of chain complexes" } }
}
document {
Key => (homology,ChainComplexMap),
Headline => "homology of a chain complex map",
Usage => "HH f",
Inputs => { "i" },
Outputs => { {"the map on the homology induced by the map ", TT "f", " of chain complexes" } }
}
document {
Key => (homology,ChainComplex),
Headline => "homology of a chain complex",
Usage => "HH C",
Inputs => { "C" },
Outputs => {
{"the homology of ", TT "C"}
},
EXAMPLE {
"R = QQ[x]/x^5;",
"C = res coker vars R",
"M = HH C",
"prune M"
}
}
document {
Key => (homology,ZZ,ChainComplex),
Headline => "homology of a chain complex",
Usage => "HH_i C",
Inputs => { "i", "C" },
Outputs => {
{"the homology at the i-th spot of the chain complex ", TT "C", "."}
},
EXAMPLE {
"R = ZZ/101[x,y]",
"C = chainComplex(matrix{{x,y}},matrix{{x*y},{-x^2}})",
"M = HH_1 C",
"prune M",
}
}
document {
Key => (homology,Matrix,Matrix),
Headline => "homology of a pair of maps",
Usage => "M = homology(f,g)",
Inputs => { "f", "g" },
Outputs => {
"M" => {"computes the homology module ", TT "(kernel f)/(image g)", "."}
},
"Here ", TT "g", " and ", TT "f", " should be composable maps with ", TT "f*g", "
equal to zero.",
PARA {
"In the following example, we ensure that the source of ", TT "f", " and the target of
", TT "f", " are exactly the same, taking even the degrees into account, and we ensure
that ", TT "f", " is homogeneous."},
EXAMPLE {
"R = QQ[x]/x^5;",
"f = map(R^1,R^1,{{x^3}}, Degree => 3)",
"M = homology(f,f)",
"prune M"
}
}
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