Gorenstein -- Constructing Gorenstein Rings and Modules

Description

Each artinian graded (or local) Gorenstein ring is the inverse system of a unique element of D, and inverse systems are often used to construct such examples. (Higher-dimensional Gorenstein rings also correspond to special inverse systems, though these are not finitely generated submodules of D. See "The structure of the inverse system of Gorenstein k-algebras" by Joan Elias and Maria Evelina Rossi, Adv. Math. (2017) 306-327, for a recent treatment with computational intent.)

For example, studying artinian Gorenstein rings of codimension 4, one might consider those corresponding to the sum of n d-th powers of linear forms. For example with n= 4,5 and d=3:

 i1 : S = ZZ/101[a,b,c,d] o1 = S o1 : PolynomialRing i2 : nPowers = (S, n,d) ->sum(apply(n, j->(random(1,S))^d)) o2 = nPowers o2 : FunctionClosure i3 : minimalBetti inverseSystem nPowers(S,4,3) 0 1 2 3 4 o3 = total: 1 9 16 9 1 0: 1 . . . . 1: . 6 8 3 . 2: . 3 8 6 . 3: . . . . 1 o3 : BettiTally i4 : minimalBetti inverseSystem nPowers (S,5,3) 0 1 2 3 4 o4 = total: 1 6 10 6 1 0: 1 . . . . 1: . 6 5 . . 2: . . 5 6 . 3: . . . . 1 o4 : BettiTally

One can also construct self-dual modules with more generators by taking the inverseSystem of a submodule that is isomorphic to its dual, for example the image of a symmetric or skew-symmetric matrix:

 i5 : Msymm = matrix"0,a,b;a,0,c;b,c,0" o5 = | 0 a b | | a 0 c | | b c 0 | 3 3 o5 : Matrix S <--- S i6 : Mskew = matrix"0,a,b;-a,0,c;-b,-c,0" o6 = | 0 a b | | -a 0 c | | -b -c 0 | 3 3 o6 : Matrix S <--- S i7 : minimalBetti coker gens inverseSystem Msymm 0 1 2 3 4 o7 = total: 3 10 14 10 3 0: 3 9 7 1 . 1: . 1 7 9 3 o7 : BettiTally i8 : minimalBetti coker gens inverseSystem Mskew 0 1 2 3 4 o8 = total: 3 9 12 9 3 0: 3 9 6 . . 1: . . 6 9 3 o8 : BettiTally

For an interesting series of examples, consider the d-th Hessian matrices obtained by taking the d-th mixed partials of a form of some degree e>2d. The ranks of such matrices are connected to the Lefschetz properties of the corresponding artinian Gorenstein rings, as explained in "The Lefschetz properties", Springer Lecture Notes in Math. 2080, by T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe.

 i9 : Hessian = (d,f) ->( S = ring f; B = basis(d,S); diff(transpose B, diff(B,f)) ) o9 = Hessian o9 : FunctionClosure i10 : S = ZZ/101[x_1..x_4] o10 = S o10 : PolynomialRing i11 : f = nPowers (S,6,4) 4 3 2 2 3 4 3 2 2 o11 = 33x - 20x x - 44x x + 33x x + 32x - 19x x - 37x x x + x x x - 1 1 2 1 2 1 2 2 1 3 1 2 3 1 2 3 ----------------------------------------------------------------------- 3 2 2 2 2 2 3 3 4 3 6x x + 37x x - 16x x x - x x - 2x x - 40x x + 10x - 47x x + 2 3 1 3 1 2 3 2 3 1 3 2 3 3 1 4 ----------------------------------------------------------------------- 2 2 3 2 2 x x x - 16x x x - 27x x + 13x x x + 41x x x x + 40x x x - 1 2 4 1 2 4 2 4 1 3 4 1 2 3 4 2 3 4 ----------------------------------------------------------------------- 2 2 3 2 2 2 2 2 2 18x x x - 49x x x + 38x x + 39x x - 7x x x - 32x x - 24x x x + 1 3 4 2 3 4 3 4 1 4 1 2 4 2 4 1 3 4 ----------------------------------------------------------------------- 2 2 2 3 3 3 4 48x x x - x x + 7x x - 10x x + 3x x + 16x 2 3 4 3 4 1 4 2 4 3 4 4 o11 : S i12 : minimalBetti coker gens inverseSystem f 0 1 2 3 4 o12 = total: 1 6 10 6 1 0: 1 . . . . 1: . 4 2 . . 2: . 2 6 2 . 3: . . 2 4 . 4: . . . . 1 o12 : BettiTally i13 : minimalBetti coker gens inverseSystem Hessian(1, f) 0 1 2 3 4 o13 = total: 4 13 18 13 4 1: 4 10 6 1 . 2: . 2 6 2 . 3: . 1 6 10 4 o13 : BettiTally i14 : minimalBetti coker gens inverseSystem Hessian(2, f) 0 1 2 3 4 o14 = total: 6 24 36 24 6 2: 6 24 36 24 6 o14 : BettiTally