The structure of generic subintegrality

Abstract

In order to give an elementwise characterization of a subintegral extension of Q-algebras, a family of generic Q-algebras was introduced in [3]. This family is parametrized by two integral parameters p ≥ 0,N ≥ 1, the member corresponding to p, N being the subalgebra R = Q [{γ_n |n ≥ N}] of the polynomial algebra Q[x₁,...,x_p, z] in p + 1 variables, where γ_n = zⁿ + Σ^p_{i = 1}(_iⁿ ) x_i z^{n - i}. This is graded by weight (z) = 1, weight (x_i) =i, and it is shown in [2] to be finitely generated. So these algebras provide examples of geometric objects. In this paper we study the structure of these algebras. It is shown first that the ideal of relations among all the γ_n's is generated by quadratic relations. This is used to determine an explicit monomial basis for each homogeneous component of R, thereby obtaining an expression for the Poincaré series of R. It is then proved that R has Krull dimension p+1 and embedding dimension N + 2p, and that in a presentation of R as a graded quotient of the polynomial algebra in N + 2p variables the ideal of relations is generated minimally by (^{N + p}₂) elements. Such a minimal presentation is found explicitly. As corollaries, it is shown that R is always Cohen-Macaulay and that it is Gorenstein if and only if it is a complete intersection if and only if N + p ≤ 2. It is also shown that R is Hilbertian in the sense that for every n ≥ 0 the value of its Hilbert function at n coincides with the value of the Hilbert polynomial corresponding to the congruence class of n.