# initialModule -- compute the initial module of a given module

## Synopsis

• Usage:
initialModule M
• Inputs:
• M, a module over an exterior algebra
• Outputs:
• , the initial module of the module M with default monomial order

## Description

Let $F$ a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\}.$ The elements $e_{\sigma}g_i$ with $e_{\sigma}$ a monomial of $E$ are called monomials of $F$ and $\mathrm{deg}(e_{\sigma} g_i) = \mathrm{deg}(e_{\sigma}) + \mathrm{deg}(g_i).$ Any element $f$ of $F$ is a unique linear combination of monomials with coefficients in $K$. Let > be a monomial order on $E$. The largest monomial of $f$ is called the initial monomial of $f$ and it is denoted by $\mathrm{In}(f)$. If M is a graded submodule of $F$ then the submodule of initial terms of M, denoted by $\mathrm{In}(M)$, is the submodule of $F$ generated by the initial terms of elements of M.

Example:

 i1 : E=QQ[e_1..e_3,SkewCommutative=>true] o1 = E o1 : PolynomialRing, 3 skew commutative variables i2 : F=E^{0,0} 2 o2 = E o2 : E-module, free i3 : f_1=(e_1*e_2)*F_0 o3 = | e_1e_2 | | 0 | 2 o3 : E i4 : f_2=(e_1*e_3)*F_0+(e_2*e_3)*F_1 o4 = | e_1e_3 | | e_2e_3 | 2 o4 : E i5 : f_3=(e_1*e_2*e_3)*F_1 o5 = | 0 | | e_1e_2e_3 | 2 o5 : E i6 : M=image map(F,E^{-degree f_1,-degree f_2,-degree f_3},matrix {f_1,f_2,f_3}) o6 = image | e_1e_2 e_1e_3 0 | | 0 e_2e_3 e_1e_2e_3 | 2 o6 : E-module, submodule of E i7 : initialModule M o7 = image | 0 e_1e_2 e_1e_3 | | e_1e_2e_3 0 0 | 2 o7 : E-module, submodule of E