# hilbertSequence(Module) -- compute the Hilbert sequence of a given module over an exterior algebra

## Synopsis

• Function: hilbertSequence
• Usage:
hilbertSequence M
• Inputs:
• M, , a module over an exterior algebra E
• Outputs:
• a list, nonnegative integers representing the Hilbert sequence of the quotient F/M

## Description

Let $\{g_1,g_2,\ldots,g_r\}$ be a graded basis of F with $deg(g_i)=f_i,\ i=1, \ldots, r.$ Given $\sum_{i=f_1}^{n+f_r}{h_i t^i}$ the Hilbert series of a graded E-module $F/M$, the sequence $(h_{f_1},\ldots,h_{n+f_r})$ is called the Hilbert sequence of $F/M.$

Example:

 i1 : E = QQ[e_1..e_4, SkewCommutative => true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : M=image matrix {{e_1*e_2,e_3*e_4,0,0,0},{0,0,e_1*e_2,e_2*e_3*e_4,0},{0,0,0,0,e_2*e_3*e_4}} o2 = image | e_1e_2 e_3e_4 0 0 0 | | 0 0 e_1e_2 e_2e_3e_4 0 | | 0 0 0 0 e_2e_3e_4 | 3 o2 : E-module, submodule of E i3 : hilbertSequence M o3 = {3, 12, 15, 4, 0} o3 : List