hilbertSequence -- the Hilbert sequence of a multi-graded module

Synopsis

Usage:

hilbertSequence M
Inputs:
- M, a module, or ideal
Optional inputs:
- DoSaturate => ..., default value false
Outputs:
- a hash table, the Hilbert sequence of M

Description

Given an $\NN^p$-graded module M, this function computes the coefficients of the pth sum transform of the $\NN^p$-graded Hilbert function of M in its Macaulay expansion. If the input is an ideal I, then the Hilbert sequence of comodule I is returned.

i1 : R = QQ[a..e, DegreeRank => 5]

o1 = R

o1 : PolynomialRing

i2 : I = monomialIdeal "de,abe,ace,abcd"

o2 = monomialIdeal (a*b*c*d, a*b*e, a*c*e, d*e)

o2 : MonomialIdeal of R

i3 : hilbertSequence I

o3 = HashTable{{0, 0, 0, 0, 1} => -1}
               {0, 0, 0, 1, 0} => 1
               {0, 0, 1, 0, 0} => 1
               {0, 0, 1, 1, 0} => -1
               {0, 1, 0, 0, 0} => 1
               {0, 1, 0, 1, 0} => -1
               {0, 1, 1, 0, 0} => -2
               {0, 1, 1, 0, 1} => 1
               {0, 1, 1, 1, 0} => 1
               {1, 0, 0, 0, 1} => 1
               {1, 0, 0, 1, 0} => -1
               {1, 0, 1, 0, 0} => -1
               {1, 0, 1, 1, 0} => 1
               {1, 1, 0, 0, 0} => -1
               {1, 1, 0, 1, 0} => 1
               {1, 1, 1, 0, 0} => 1

o3 : HashTable

For singly-graded modules, one can read off the Hilbert polynomial from the Hilbert sequence:

i4 : R = QQ[a..e]

o4 = R

o4 : PolynomialRing

i5 : I = monomialIdeal "de,abe,ace,abcd"

o5 = monomialIdeal (a*b*c*d, a*b*e, a*c*e, d*e)

o5 : MonomialIdeal of R

i6 : hilbertSequence I

o6 = HashTable{{1} => 2 }
               {2} => -6
               {3} => 5

o6 : HashTable

i7 : hilbertPolynomial I

o7 = 2*P  - 6*P  + 5*P
        0      1      2

o7 : ProjectiveHilbertPolynomial

A convenient expression for the Hilbert sequence is provided via printHilbertSequence.

Caveat

In general, to retain a connection to the Hilbert polynomial (as opposed to the pth sum transform) it is necessary to saturate with respect to the irrelevant ideal, cf. page 235 of Conca-De Negri-Gorla, "Cartwright–Sturmfels ideals associated to graphs and linear spaces", 2018. This is handled by the optional argument DoSaturate.

Ways to use hilbertSequence :

hilbertSequence(Ideal)
hilbertSequence(Module)
hilbertSequence(Ring)

For the programmer