Families of Artinian and one-dimensional algebras

Abstract

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, h 1 , . . . , h j , . . .) and H A , respectively. If dim A = 0 (respectively dim A = depth A = 1) and A is a "truncation" of B in the sense that H A = (1, h 1 , . . . , h j −1 , α, 0, 0, . . .) (respectively H A = (1, h 1 , . . . , h j −1 , α, α, α, . . .)) for some α h j , then we show there is a close relationship between GradAlg H A (R) and GradAlg H B (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 graded 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.

Reducible family of height three level algebras

“…Recently J.O. Kleppe has shown that L (H) and LevAlg(H) give the same topological structures [Kle4,Theorem 44], extending his earlier result that there is an isomorphism between the tangent spaces to LevAlg(H) and to L(H) at corresponding closed points.…”

“…Kleppe has given an example where, proving a conjecture of the second author about the Hilbert function H = (1, 3, 6, 10, 14, 10, 6, 2), he shows that LevAlg(H) has at least two irreducible components. He also notes that by linking one can construct further such examples [Kle4,Example 49,Remark 50(b)]. …”

“…The consecutive cancellation portion was shown by I. Peeva [Pe, Remark after Theorem 1.1], based on a result of K. Pardue [Par]. For further discussion see [M1,Theorem 1.1] and [Kle4,Remark 7].…”

“…The condition (1.6) is just that the sequence of vector spaces forms the degree two to j graded components of a graded ideal of R. For further detail see [Kle1,Kle3,Kle4], or the discussion of the "postulation Hilbert scheme" in [IK], or [I2,Definition 1.9]. …”

“…The length n(Z) of a punctual subscheme Z of P n is that of a minimal Artinian reduction of its coordinate ring. The family LevAlg(H) of level algebra quotients of R having Hilbert function H forms an open subscheme of the family of graded algebras [Kle1,Kle4,I2] or, via Macaulay duality, of a Grassmannian [IK,ChGe,Kle4].…”