Families of Artinian and one-dimensional algebras.

Jan O. Kleppe

The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg^H(R) be the scheme parametrizing graded quotients of R with Hilbert function H. Let B -> A be any graded surjection of quotients of R with Hilbert function H_B=(1,h₁,...,h_j,...) and H_A respectively. If dim A = 0 (resp. dim A = depth A = 1) and A is a ``truncation'' of B in the sense that H<sub>A</sub>=(1,h<sub>1</sub>,...,h<sub>j-1</sub>,s,0,0,...) (resp. H<sub>A</sub>=(1,h<sub>1</sub>,...,h<sub>j-1</sub>,s,s,s,...)) for some s < h<sub>j</sub>+1, then we show there is a close relationship between GradAlg<sup>H<sub>A</sub></sup>(R) and GradAlg<sup>H<sub>B</sub></sup>(R) concerning e.g. smoothness and dimension at the points (A) and (B) respectively, provided B is a complete intersection or provided the Castelnuovo-Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to ``non-truncated'' Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg^H(R), H=(1,3,6,10,14,10,6,2), whose general elements are Artinian level algebras of type 2.