# 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 [{γn¦n ⩾ N}] of the polynomial algebra ℚ[x1,…,x p, z] inp + 1 variables, where γn=zn+∑pi=1(ni)xizn−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 γn’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 (N+p2) 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.

## Department(s)

Mathematics

## Document Type

Article

## DOI

https://doi.org/10.1007/bf02840585

## Keywords

subintegral extensions, subrings of polynomial rings

## Publication Date

1995

## Recommended Citation

Reid, Les, Leslie G. Roberts, and Balwant Singh. "The structure of generic subintegrality." In Proceedings of the Indian Academy of Sciences-Mathematical Sciences 105, no. 1 1-22. India, 1995.

## Journal Title

Proceedings of the Indian Academy of Sciences - Mathematical Sciences