getSchreyerMap -- get the Schreyer map of a free OI-module if it exists

Synopsis

Usage:

getSchreyerMap H
Inputs:
- H, an instance of the type FreeOIModule,
Outputs:
- an instance of the type FreeOIModuleMap,

Description

Let $G'$ be a non-empty Gröbner basis for the syzygy module of a finitely generated submodule $\mathbf{M}$ of a free OI-module $\mathbf{F}$ computed using oiSyz. Let H be the free OI-module obtained by applying getFreeOIModule to any element of $G'$. This method returns the canonical surjective map from H to $\mathbf{M}$.

i1 : P = makePolynomialOIAlgebra(2, x, QQ);

i2 : F = makeFreeOIModule(e, {1,1}, P);

i3 : installGeneratorsInWidth(F, 2);

i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);

i5 : G = oiGB {b}

o5 = {x   x   e        + x   x   e       , x   x   x   e        -
       1,2 1,1 2,{2},1    2,2 2,1 2,{1},2   2,3 2,2 1,1 3,{2},2  
     ------------------------------------------------------------------------
     x   x   x   e       }
      2,3 2,1 1,2 3,{1},2

o5 : List

i6 : G' = oiSyz(G, d)

o6 = {x   d              - x   d              - x   d             ,
       1,2 4,{1, 3, 4},2    1,1 4,{2, 3, 4},2    1,3 4,{1, 2, 4},2 
     ------------------------------------------------------------------------
     x   d           - x   d           + 1d             , x   d             
      1,2 3,{1, 3},1    1,1 3,{2, 3},1     3,{1, 2, 3},2   2,4 4,{1, 2, 3},2
     ------------------------------------------------------------------------
     - x   d             }
        2,3 4,{1, 2, 4},2

o6 : List

i7 : H = getFreeOIModule G'#0

o7 = Basis symbol: d
     Basis element widths: {2, 3}
     Degree shifts: {-2, -3}
     Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
     Monomial order: Schreyer

o7 : FreeOIModule

i8 : getSchreyerMap H

o8 = Source: (d, {2, 3}, {-2, -3}) Target: (e, {1, 1}, {0, 0})

o8 : FreeOIModuleMap

Caveat

If $G'$ is empty or if H does not have a Schreyer order, this method will throw an error.

Ways to use getSchreyerMap :

getSchreyerMap(FreeOIModule)

For the programmer

The object getSchreyerMap is a method function.