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 top, N being the subalgebraR = ℚ [{γ_n¦n ⩾ N}] of the polynomial algebra ℚ[x₁,…,x _p, z] inp + 1 variables, where$\gamma _n = z^n + \sum\nolimits_{i = 1}^p {(_i^n )} x_i z^{n - i}$. 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 ofR, thereby obtaining an expression for the Poincaré series ofR. It is then proved thatR has Krull dimension p+1 and embedding dimensionN + 2p, and that in a presentation ofR as a graded quotient of the polynomial algebra inN + 2p variables the ideal of relations is generated minimally by$\left( \begin{gathered} N + p \\ 2 \\ \end{gathered} \right)$ elements. Such a minimal presentation is found explicitly. As corollaries, it is shown thatR is always Cohen-Macaulay and that it is Gorenstein if and only if it is a complete intersection if and only ifN + p ⩽ 2. It is also shown thatR is Hilbertian in the sense that for everyn ⩾ 0 the value of its Hilbert function atn coincides with the value of the Hilbert polynomial corresponding to the congruence class ofn.