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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,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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