Nilpotent algebras and affinely homogeneous surfaces

Abstract

To every nilpotent commutative algebra N of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ⊂ N can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A of N . In case N admits a grading, the surface S is affinely homogeneous. More can be said if A has dimension 1, that is, if N is the maximal ideal of a Gorenstein algebra. In this case two such algebras N , N are isomorphic if and only if the associated hypersurfaces S, S are affinely equivalent. If one of S, S even is affinely homogeneous, 'affinely equivalent' can be replaced by 'linearly equivalent'. In case the nil-index of N does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.

A complex-analytic proof of a criterion for isomorphism of Artinian Gorenstein algebras

“…Numerous examples of hypersurfaces S π explicitly computed for particular algebras can be found in [2], [6], [7] (see also Section 6 below). We will now state the criterion for isomorphism of Artinian Gorenstein algebras obtained in [6], [11]. Theorem 2.1.…”

“…http://www.casjournal.com/content/1/1/1 Currently, there are two different proofs of the above criterion. The one given in [11] is purely algebraic, whereas the one proposed in [6] reduces the case of an arbitrary field to that of k = C. A proof of the criterion in the latter case is contained in our earlier article [7] and, quite surprisingly, is based on a complex-analytic argument. It turns out that one can give an independent complex-analytic proof of the criterion for k = R as well.…”

“…In the case when the field k has characteristic zero, in articles [6], [11], a surprising criterion for isomorphism of Artinian Gorenstein algebras was found. The criterion is given in terms of a certain algebraic hypersurface S π in the maximal ideal m of A associated to a linear projection π on m with range Soc(A), where we assume that dim k A > 1.…”

“…The criterion is given in terms of a certain algebraic hypersurface S π in the maximal ideal m of A associated to a linear projection π on m with range Soc(A), where we assume that dim k A > 1. The hypersurface S π passes through the origin and is the graph of a polynomial map P π : ker π → Soc(A) k. It is shown in [6], [11] that two Artinian Gorenstein algebras A and A are isomorphic if and only if any two hypersurfaces S π and S π arising from A and A, respectively, are affinely equivalent, that is, there exists a bijective affine map f : m → m such that f (S π ) = S π . http://www.casjournal.com/content/1/1/1 Currently, there are two different proofs of the above criterion.…”

Classifying local Artinian Gorenstein algebras

“…The results usually rest on additional assumptions: small length [5,[7][8][9]37] or being a codimension two complete intersection [3,15,18]. See also [11,12,20,26] for other approaches.…”