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--- status: DRAFT
--- author(s): MES, taken from before
--- notes:
undocumented {
(presentation,QuotientRing,QuotientRing),
(presentation,PolynomialRing),
(presentation,PolynomialRing,PolynomialRing),
(presentation,QuotientRing,PolynomialRing)
}
document {
Key => presentation,
Headline => "presentation of a module or ring",
SeeAlso => {minimalPresentation}
}
document {
Key => {
(presentation,PolynomialRing,QuotientRing),
(presentation,QuotientRing)
},
Headline => "presentation of a quotient ring",
Usage => "presentation B\npresentation(A,B)",
Inputs => {
"A",
"B" => "a quotient ring of A"
},
Outputs => {
Matrix => {"whose image is the ideal of ", TT "A", " defining ", TT "B"}
},
"If A is not present, then it is understood to be the ultimate ambient polynomial ring of B.
In general, A may be any ring of which B is a quotient.",
PARA{},
"In the examples below, A is the ultimate ambient polynomial ring of A, B and C.",
EXAMPLE lines ///
A = QQ[a..d];
B = A/(a^2,b^3);
C = B/(a*b*c,b*c*d, b^2);
presentation A
presentation B
presentation C
presentation(B,C)
presentation(A,C)
minimalPresentation C
///,
PARA{},
Caveat => "The given presentation is often not minimal",
SeeAlso => {minimalPresentation}
}
document {
Key => (presentation,Module),
Headline => "presentation of a module",
Usage => "presentation M",
Inputs => {
"M"
},
Outputs => {
Matrix => {"a presentation matrix of ", TT "M"}
},
"A presentation of ", TT "M", " is a map ", TT "p", " so that ", TT "coker p", " is
isomorphic to ", TT "M", ". The presentation obtained is expressed in
terms of the given generators, i.e., the modules ", TT "cover M", " and
", TT "target p", " are identical.
The isomorphism can be obtained as ", TT "map(M,coker p,1)", ".",
PARA{},
"Since a module M may be described as a submodule or a subquotient
module of a free module, some computation may be required to produce
a presentation. See also ",
TO "trim", ", or ",
TO "minimalPresentation", ",
which do a bit more work to try to
eliminate redundant generators.",
EXAMPLE lines ///
R = QQ[a,b,c];
I = ideal"a2-b2,abc"
M = I/(I^2+a*I)
presentation M
///,
SeeAlso => {minimalPresentation, trim, generators, relations, cover}
}
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