Abstract
Let A be an Artinian Gorenstein algebra over a field k of characteristic zero with dimk A>1. To every such algebra and a linear projection π on its maximal ideal with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface , which is the graph of a polynomial map P π : kerπ→Soc(A)≃k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A and are isomorphic if and only if any two hypersurfaces S π and arising from A and , respectively, are affinely equivalent. In the present paper, we focus on the cases and and explain how in these situations the above criterion can be derived by complex-analytic methods. The complex-analytic proof for has not previously appeared in print but is foundational for the general result. The purpose of our paper is to present this proof and compare it with that for , thus highlighting a curious connection between complex analysis and commutative algebra. 13H10; 32V40
Highlights
1 Introduction This paper concerns a result in commutative algebra but is intended mainly for experts in complex analysis and CR-geometry
The criterion is given in terms of a certain algebraic hypersurface Sπ in the maximal ideal m of A associated to a linear projection π on m with range Soc(A), where we assume that dimk A > 1
The hypersurface Sπ passes through the origin and is the graph of a polynomial map Pπ : ker π → Soc(A) k
Summary
This paper concerns a result in commutative algebra but is intended mainly for experts in complex analysis and CR-geometry. Set K := ker π ∩ m and let Sπ be the hypersurface in m given as the graph of the polynomial map Pπ : K → Soc(A) of degree ν defined as follows: Pπ (x) := π(exp(x)) = π ν 1 xm m=2 m!
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