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--- status: Draft
--- author(s): Amelia Taylor
--- notes: Be sure to note that trim I and trim R^1/I do
--- the same thing as the minimal relations for
--- R^1/I are the minimal generators for I.
doc ///
Node
Key
MinimalGenerators
Headline
whether to compute minimal generators and return a trimmed set of generators
Description
Text
The following returns two minimal generators (Serre's Theorem: a codim 2 Gorenstein ideal is a complete intersection.)
Example
S = ZZ/101[a,b]
i = ideal(a^4,b^4)
quotient(i, a^3+b^3)
Text
Without trimming we would get 4 generators instead.
Example
quotient(i, a^3+b^3, MinimalGenerators => false)
Text
Sometimes the extra time to find the minimal generators is too large.
This allows one to bypass this part of the computation.
Example
needsPackage "Truncations"
R = ZZ/101[x_0..x_4]
I = monomialCurveIdeal(R, {1,4,5,9});
time J = truncate(8, I, MinimalGenerators => false);
time K = truncate(8, I, MinimalGenerators => true);
numgens J
numgens K
SeeAlso
"Truncations::truncate(ZZ,Ideal)"
"Saturation :: quotient(Ideal,Ideal)"
"Saturation :: saturate(Ideal,Ideal)"
monomialCurveIdeal
-- this is the old version
Node
Key
trim
(trim, Ideal)
(trim, Ring)
(trim, Module)
(trim, MonomialIdeal)
(trim, QuotientRing)
Headline
minimize generators and relations
Description
Text
There are two ways to present an $R$-module $M$. One way is to take a free module $F$
(whose generators are called the @TO generators@) and form the quotient $M = F/H$ by a submodule
$H\subset F$ (whose generators are called the @TO relations@).
Another way is take a free module $F$,
a submodule $G\subset F$ (whose generators are called the @TO generators@),
a submodule $H\subset F$ (whose generators are called the @TO relations@), and
form the @TO2 {"subquotient modules", "subquotient module"}@ $M = (G+H)/H$,
obtained also as the image of $G$ in $F/H$.
The purpose of @TT "trim"@ is to minimize presentations of the latter type.
This applies also to rings and ideals.
Example
R = ZZ/101[x,y,z,u,w]
I = ideal(x^2-x^2-y^2, z^2+x*y, w^2-u^2, x^2-y^2)
trim I
trim (R^1/I)
Example
R = ZZ/32003[a..d]
M = coker matrix {{a,1,b},{c,3,b+d}}
trim M
prune M
SeeAlso
prune
mingens
"subquotient modules"
generators
relations
subquotient
///
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