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--- status: draft
--- author(s): Stillman
--- notes:
doc ///
Node
Key
cokernel
Headline
cokernel of a map
SeeAlso
target
///
document {
Key => {
(cokernel, Matrix),
(cokernel, RingElement)
},
Headline => "cokernel of a map of modules",
Usage => "cokernel f",
Inputs => { { TT "f : A --> B", ofClass { Matrix, RingElement } } },
Outputs => { {"the object ", TT "B/(image f)"} },
PARA{ TT "coker", " is a synonym for ", TT "cokernel", "." },
PARA{ "The generators of the cokernel are provided by the generators of the target
of ", TT "f", ". In other words, ", TT "cover target f", " and ", TT "cover cokernel f", " are equal." },
PARA{ "An argument ", TT "f", " that is a ", TO RingElement, " is interpreted as a one by one matrix."},
EXAMPLE {
"R = ZZ[a..d];",
"M = cokernel matrix{{2*a-b,3*c-5*d,a^2-b-3}}"
},
"If ", TT "f", " is a matrix, and the target of ", TT "f", " is a submodule, the resulting module will be a ",
TO "subquotient", " module.",
EXAMPLE {
"f = map(a*M, M, a^3+a^2*b)",
"(target f,source f)",
"N = cokernel f",
"minimalPresentation N"
},
SeeAlso => { cover, image, kernel, coimage, comodule, minimalPresentation, "matrices to and from modules"}
}
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