randomSocleModule(List,ZZ) -- random finite length module with prescribed number of socle elements in single degree

Synopsis

Function: randomSocleModule
Usage:

randomSocleModule(L,m)
Inputs:
- L, a list, a strictly increasing degree sequence of integers
- m, an integer
Optional inputs:
- CoefficientRing (missing documentation) => ..., default value ZZ/101, The base field for the resulting module
Outputs:
- a module, randomly generated having m b_c socle generators in degree L_c and m b_0 generators, where b = pureBetti L, and c = length L

Description

There are many cases where these produce pure resolutions of the minimal size.

i1 : L={0,2,3,7}

o1 = {0, 2, 3, 7}

o1 : List

i2 : B = pureBetti L

o2 = {10, 42, 35, 3}

o2 : List

i3 : betti res randomSocleModule(L,1)

             0  1  2 3
o3 = total: 10 42 35 3
         0: 10  .  . .
         1:  . 42 35 .
         2:  .  .  . .
         3:  .  .  . .
         4:  .  .  . 3

o3 : BettiTally

i4 : betti res randomModule(L,1)

             0  1  2  3
o4 = total: 10 42 50 18
         0: 10  .  .  .
         1:  . 42 26  .
         2:  .  . 24 18

o4 : BettiTally

The method used is roughly the following: Given a strictly increasing degree sequence L and a number of generators m, this routine produces a generic module of finite length with the m generators and number of socle elements and regularity corresponding to the pure resolution with degree sequence L. The module is constructed by taking a certain number of generic elements inside an appropriate direct sum of copies of a zero-dimensional complete intersection. We use the fact that in a polynomial ring in c variables, modulo the r+1 st power of each variable, the part of generated in degree (c-1)r looks like the part of the injective hull of the residue class field generated in degree -r.

Ways to use this method: