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ExteriorModules :: isHilbertSequence(List,Module)

isHilbertSequence(List,Module) -- whether the given sequence is a Hilbert sequence

Synopsis

Description

Let $F$ a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\},$ with $deg(g_i)=f_i,\ i=1, \ldots, r.$ If $M$ is a graded submodule of F, and $H_{F/M}(t) =\sum_{i=f_1}^{f_r+n}H_{F/M}(i)t^i$ is the Hilbert series of $F/M,$ then the sequence $(H_{F/M}(f_1), H_{F/M}(f_1+1), \ldots, H_{F/M}(f_r+n))\in \mathbb{N}_0^{f_r+n-f_1+1}$ is called the Hilbert sequence of $F/M$ and we denote it by $Hs_{F/M}.$ The integers $f_1, f_1+1, \ldots, f_r+n$ are called the $Hs_{F/M}$-degrees.

Example:

i1 : E=QQ[e_1..e_4,SkewCommutative=>true]

o1 = E

o1 : PolynomialRing, 4 skew commutative variables
i2 : F=E^{0,0}

      2
o2 = E

o2 : E-module, free
i3 : isHilbertSequence({2,8,3,1,0},F)

o3 = true
i4 : isHilbertSequence({2,8,3,2,0},F)

o4 = false

See also