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--- status: DRAFT
--- author(s): Caviglia, Kummini
--- notes:
document {
Key => koszul,
Headline => "Koszul complex or specific matrix in the Koszul complex",
"Returns either the entire Koszul complex, or a specific matrix in the Koszul complex.",
SeeAlso => {eagonNorthcott}
}
document {
Key => (koszul,Matrix),
Headline => "the Koszul complex",
Usage => "g = koszul f",
Inputs => {
"f" => {"a ", TT "1", " by ", TT "n", " matrix"},
},
Outputs => {
"g" => { "the Koszul complex of the matrix ", TT "f"}
},
EXAMPLE lines ///
R = QQ[x_1..x_4];
f = matrix{{x_1..x_4}}
C = koszul f
C.dd^2
///,
SeeAlso => {eagonNorthcott}
}
document {
Key => (koszul,ZZ,Matrix),
Headline => "a differential in a Koszul complex",
Usage => "g = koszul(i,f)",
Inputs => {
"i",
"f" => {"a ", TT "1", " by ", TT "n", " matrix"},
},
Outputs => {
"g" => { "the ", TT "i", "-th differential in the Koszul complex of the matrix ", TT "f"}
},
EXAMPLE {
"R = QQ[x_1..x_4];",
"f = matrix{{x_1..x_4}}"
},
"To see the second differential in the Koszul complex of the matrix ", TT "f", " look at:",
EXAMPLE {
"koszul(2,f)"
}
}
///
document {
Key => (koszul,Matrix,Matrix),
Headline => "a differential in a Koszul complex",
Usage => "koszul(M,N)",
Inputs => {
"M" => {"a ", TT "1", " by ", TT "m", " matrix"},
"N" => {"a ", TT "1", " by ", TT "n", " matrix"},
},
Outputs => {
Matrix => { "the ", TT "i", "-th differential in the
Koszul complex of the matrix ", TT "f"}
},
EXAMPLE {
R = QQ[a..d]
M = matrix{{a*b, a*c, b*d, b^2}}
N = matrix{{a,b,c,d}}
koszul(N,M)
},
"To see the second differential in the Koszul complex of the matrix ", TT "f", " look at:",
EXAMPLE {
"koszul(2,f)",
},
SeeAlso => {}
}
///
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