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 variable(s)
|
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
|