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--- status: Draft
--- author(s): Giulio
--- notes: updated November 2021
-- TODO: this is currently a duplicate one, remove one of them
doc ///
Node
Key
regularity
(regularity, Ideal)
(regularity, Module)
[regularity, Weights]
Headline
compute the Castelnuovo-Mumford regularity
Usage
r = regularity C
r = regularity(C, Weights => w)
Inputs
C: -- an Ideal, a Module, or a ChainComplex
Weights=>List -- a weight vector @TT "w"@, see @TO [betti, Weights]@
Outputs
r:ZZ
Description
Text
For a free chain complex @TT "C"@, the regularity @TT "r"@ is the smallest number so that each
basis element of @TT "C_i"@ has degree at most @TT "i+r"@. For an ideal @TT "I"@, regularity is
one plus the regularity of the minimal free resolution of the quotient of the ambient ring by @TT "I"@.
For a module @TT "M"@, regularity is the regularity of a minimal free resolution of @TT "M"@.
Example
R = ZZ/32003[a..d];
I = ideal(a^20, b^20, a*c^19-b*d^19);
C = resolution I
regularity C
regularity comodule I
regularity I
regularity module I
Text
The regularity is the label of the last row in the @TO2 {betti, "Betti diagram"}@ of a chain complex.
However, this depends on the total degree weights in the Betti tally, which are computed based on the
@TO2 {"heft vectors", "heft vector"}@ of the underlying ring. To adjust this vector, a vector @TT "w"@
whose length is the same as the @TO2 {degreeLength, "degree length"}@ of the ring can be provided using
the option @TT "Weights"@. The dot products of @TT "w"@ with the multidegrees in the tally will be used
in the resulting computation.
Example
C = resolution ideal(a^3, a^2*b, a*b^6, a^2*c);
betti C
regularity C
betti(C, Weights => {2})
regularity(C, Weights => {2})
SeeAlso
"OldChainComplexes :: resolution"
betti
comodule
"VirtualResolutions :: multigradedRegularity"
///
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