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

## Synopsis

• Usage:
oiSyz(G, d)
• Inputs:
• Optional inputs:
• Strategy => ..., default value Minimize
• Verbose => ..., default value false
• Outputs:

## Description

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.

## Ways to use oiSyz :

• oiSyz(List,Symbol)

## For the programmer

The object oiSyz is .