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--- status: DRAFT
--- author(s): MES
--- notes:
document {
Key => {isInjective,
(isInjective, Matrix),
(isInjective, RingMap)},
Headline => "whether a map is injective",
Usage => "isInjective f",
Inputs => {
"f" => Nothing => {ofClass Matrix, ", or ", ofClass RingMap}
},
Outputs => {
Boolean => "whether the kernel is zero"
},
"This function computes the kernel, and checks whether it is zero.",
EXAMPLE lines ///
R = QQ[a..d];
F = matrix{{a,b},{c,d}}
isInjective F
G = substitute(F, R/(det F))
isInjective G
///,
PARA{},
"Similarly for ring maps:",
EXAMPLE lines ///
S = QQ[r,s,t];
phi = map(S,R,{r^3, r^2*s, r*s*t, s^3})
isInjective phi
S' = coimage phi
phi' = phi * map(R,S')
isInjective phi'
///,
Caveat => "One could imagine a faster routine for this. If you write one,
please send it to us!",
SeeAlso => {kernel, isSurjective, (coimage,RingMap), det, "substitution and maps between rings"}
}
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