I1 = inverseSystem MM1 = inverseSystem IInverse systems are often used to construct artinian Gorenstein ideals and modules. For that application see Gorenstein.
Let S = k[x_1..x_n] be a standard graded polynomial ring, and let D be its dual, the divided power algebra, regarded as an S-module. Let M be an rxm matrix of polynomials, and let I be an ideal of S.
From a submodule of D^r to a submodule of S^r (or to an ideal, if r=1):
We think of the columns of M as generators of an S-submodule MM of D^r, and inverseSystem M returns the annihilator of MM in S^r = Hom_{graded}(D^r,k). In the default behavior a monomial $x^a$ in an entry of the matrix M is taken to represent $a!x^(a) \in D'$, where, $a = (a_1,\dots,a_n)$ then $a! = a_1!*\dots*a_n!$. Use
inverseSystem(M, DividedPowers => false)
to make the monomials of entries of M represent the dual basis of the monomial basis of S, that is, the divided powers of the generators of D as an algebra.
From an ideal of S to a submodule of D:
If $I$ is an ideal of $S$, homogeneous or not, we regard $I$ as an ideal of the localization $S'$ of $S$ at $(x_1,\dots,x_n)$. If $S'/I$ is of finite length then
M = inverseSystem I
and
M1 = inverseSystem(I, DividedPowers => false)
each return a 1 x m matrix whose entries are the minimal generators of the annihilator of $I$ in $D$. In the matrix $M$ a term $x^a$ is to be interpreted as $a! x^(a)$, while in the matrix $M'$ it is interpreted as $x^(a)$. Of course the first computation is only valid if all the powers of variables appearing in the generators of $I$ are < char k.
To make these computations it is necessary to represent some sufficiently large finitely generated S-submodule of $D$ (this will automatically be an $S'$-submodule. To do this we use the map of modules D-> S/(x_1^d,\dots, x_n^d) sending $x^{(a)}$ to contract(x^a, product(n, j-> x_i^{d-1})), defined only when the variables in $x^{(a)}$ appear only with powers < d.
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Here are some codimension 4 Gorenstein rings with different Betti tables, computed by inverseSystem from quartic polynomials
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Because inverseSystem involves a conversion between the bases of the dual, it should not be used in the default mode unless the characteristic is greater than the highest degree to which a variable appears. To make $x^a$ represent $x^(a)$, for example in small characteristics use
inverseSystem(Matrix, DividedPowers=>false)
(which was the default behavior of the old script "fromDual").
The object inverseSystem is a method function with options.