oiSyz -- compute a Gröbner basis for the syzygy module of a submodule of a free OI-module

Given a non-empty Gröbner basis G for a submodule $\mathbf{M}$ of a free OI-module $\mathbf{F}$, this method computes a Gröbner basis $G'$ for the syzygy module of $\mathbf{M}$ with respect to the Schreyer order induced by $G$; see OIMonomialOrder.

The new Gröbner basis $G'$ lives in an appropriate free OI-module $\mathbf{G}$ with basis symbol d whose basis elements are mapped onto the elements of $G$ by a canonical surjective map $\varphi:\mathbf{G}\to\mathbf{M}$ (see Definition 4.1 of [1]). Moreover, the degrees of the basis elements of $\mathbf{G}$ are automatically shifted to coincide with the degrees of the elements of $G$, so that $\varphi$ is homogeneous if $G$ consists of homogeneous elements. If $G'$ is not empty, then one obtains $\mathbf{G}$ by applying getFreeOIModule to any element of $G'$. One obtains $\varphi$ by using getSchreyerMap.

The Verbose option must be either true or false, depending on whether one wants debug information printed.

The Strategy option has the following permissible values:

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 : oiSyz(G, d)

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

o6 : List

References:

[1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of OI-Modules, Preprint, arXiv:2303.06725, 2023.