# hfModuleAsExt -- predict betti numbers of moduleAsExt(M,R)

## Synopsis

• Usage:
seq = hfModuleAsExt(numValues,M,numgensR)
• Inputs:
• numValues, an integer, number of values to compute
• M, , module over the ring of operators
• numgensR, an integer, number of generators of the target ring
• Outputs:
• seq, , sequence of numValues integers, the expected total Betti numbers

## Description

Given a module M over the ring of operators $k[x_1..x_c]$, the call $N = moduleAsExt(M,R)$ produces a module N over the ring R whose ext module is the exterior algebra on n=numgensR generators tensored with M. This script computes numValues values of the Hilbert function of $$M \otimes \wedge k^n,$$ which should be equal to the total betti numbers of N.

 i1 : kk = ZZ/101; i2 : S = kk[a,b,c]; i3 : ff = matrix{{a^4, b^4,c^4}}; 1 3 o3 : Matrix S <--- S i4 : R = S/ideal ff; i5 : Ops = kk[x_1,x_2,x_3]; i6 : MM = Ops^1/(x_1*ideal(x_2^2,x_3)); i7 : N = moduleAsExt(MM,R); i8 : betti res( N, LengthLimit => 10) 0 1 2 3 4 5 6 7 8 9 10 o8 = total: 36 27 29 31 33 35 37 39 41 43 45 -6: 18 6 . . . . . . . . . -5: . . . . . . . . . . . -4: 18 21 21 7 . . . . . . . -3: . . . . . . . . . . . -2: . . 8 24 24 8 . . . . . -1: . . . . . . . . . . . 0: . . . . 9 27 27 9 . . . 1: . . . . . . . . . . . 2: . . . . . . 10 30 30 10 . 3: . . . . . . . . . . . 4: . . . . . . . . 11 33 33 5: . . . . . . . . . . . 6: . . . . . . . . . . 12 o8 : BettiTally i9 : hfModuleAsExt(12,MM,3) o9 = (23, 25, 27, 29, 31, 33, 35, 37, 39, 41) o9 : Sequence

• moduleAsExt -- Find a module with given asymptotic resolution

## Ways to use hfModuleAsExt :

• "hfModuleAsExt(ZZ,Module,ZZ)"

## For the programmer

The object hfModuleAsExt is .