# stronglyStableModule -- compute the smallest strongly stable module containing a given monomial module

## Synopsis

• Usage:
stronglyStableModule M
• Inputs:
• M, a monomial module over an exterior algebra
• Outputs:
• , the smallest strongly stable submodule of the ambient module containing M

## Description

Let $F$ be a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\}$ and let $M$ be a monomial submodule of F. This method allows the construction of the smallest strongly stable submodule of F containing M. It is useful, although it does not preserve invariants. In fact, the computation by hand of a strongly stable submodule implies some tedious calculations overall in the case when the elements of the homogeneous basis of F have different degrees. Furthermore, it is worth pointing out that such methods are analogous to the Macaulay2 function borel that computes the smallest borel ideal containing a given ideal.

Example:

 i1 : E = QQ[e_1..e_4, SkewCommutative => true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : F=E^{0,0} 2 o2 = E o2 : E-module, free i3 : I_1=ideal(e_1*e_2) o3 = ideal(e e ) 1 2 o3 : Ideal of E i4 : I_2=ideal(e_1*e_2*e_3,e_1*e_2*e_4,e_1*e_3*e_4) o4 = ideal (e e e , e e e , e e e ) 1 2 3 1 2 4 1 3 4 o4 : Ideal of E i5 : M=createModule({I_1,I_2},F) o5 = image | e_1e_2 0 0 0 | | 0 e_1e_3e_4 e_1e_2e_4 e_1e_2e_3 | 2 o5 : E-module, submodule of E i6 : isStronglyStableModule M o6 = false i7 : Mss=stronglyStableModule M o7 = image | e_1e_2 e_1e_3e_4 0 0 0 | | 0 0 e_1e_3e_4 e_1e_2e_4 e_1e_2e_3 | 2 o7 : E-module, submodule of E i8 : isStronglyStableModule Mss o8 = true