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Node
Key
annihilator
(annihilator, Ideal)
(annihilator, Module)
(annihilator, RingElement)
(annihilator, CoherentSheaf)
[annihilator, Strategy]
Headline
the annihilator ideal
Usage
ann M
annihilator M
Inputs
M:{Ideal,Module,RingElement,CoherentSheaf}
Strategy=>Symbol
either @TT "Quotient"@ or @TT "Intersection"@
Outputs
:Ideal
the annihilator ideal $\mathrm{ann}(M) = \{ f \in R | fM = 0 \}$ where $R$ is the ring of $M$
Description
Text
You may use @TT "ann"@ as a synonym for @TT "annihilator"@.
As an example, we compute the annihilator of the canonical module of the rational quartic curve.
Example
R = QQ[a..d];
J = monomialCurveIdeal(R,{1,3,4})
M = Ext^2(R^1/J, R)
annihilator M
Text
For another example, we compute the annihilator of an element in a quotient ring.
Example
A = R/(a*b, a*c, a*d)
ann a
Text
Currently two algorithms to compute annihilators are implemented.
The default is to compute the annihilator of each generator of the module @TT "M"@ and to intersect
these two by two. Each annihilator is done using a submodule quotient. The other algorithm computes
the annihilator in one large computation and is used if @TT "Strategy => Quotient"@ is specified.
Example
annihilator(M, Strategy => Quotient)
SeeAlso
(quotient, Module, Module)
monomialCurveIdeal
Ext
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