# On the number of generators of ideals defining Gorenstein Artin algebras with Hilbert function \( \left( 1,n+1 , 1+{n+1\atopwithdelims ()2}, \ldots ,{n+1\atopwithdelims ()2}+1, n+1 ,1\right) \)

Original Paper

First Online:

- 81 Downloads

## Abstract

Let \(R = k[w, x_1, \ldots , x_n]/I\) be a graded Gorenstein Artin algebra. \(I = \mathrm{ann }F\) for some \(F\) in the divided power algebra \(k[W, X_1,\ldots , X_n]\). Suppose that \(RI_2\) is a height one ideal generated by \(n\) quadrics so that \(I_2 \subset (w)\) after a possible change of variables. Let \(J = I \cap k[x_1, \ldots , x_n]\). Then \(\mu (I) \le \mu (J)+n+1\) and \(I\) is said to be \(\mu \)-generic if \(\mu (I) = \mu (J) + n+1\). In this article we prove necessary conditions, in terms of \(F\), for an ideal to be \(\mu \)-generic. With some extra assumptions on the exponents of terms of \(F\), we obtain a characterization for height four ideals \(I\) to be \(\mu \)-generic.

## Keywords

Gorenstein Artin Algebras Hilbert function \(\mu \)-generic## Mathematics Subject Classification

13E10 13E15 13C05## References

- Buchsbaum, D.A., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math.
**99**(3), 447–485 (1977)CrossRefMathSciNetzbMATHGoogle Scholar - Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)Google Scholar
- El Khoury, S., Srinivasan, H.: A class of Gorenstein Artinian algebras of embedding codimension four. Commun. Algebra
**37**(9), 3259–3277 (2009)CrossRefMathSciNetzbMATHGoogle Scholar - Iarrobino, A., Srinivasan, H.: Artin Gorenstein algebras of embedding dimension four: components of \(\mathbb{P}\)Gor(H) for \(H = (1,4,7, \ldots, 1)\). J. Pure Appl. Algebra
**201**, 62–96 (2005)CrossRefMathSciNetzbMATHGoogle Scholar - Kustin, A., Miller, M.: Structure theory for a class of grade four Gorenstein ideals. Trans. Am. Math. Soc.
**270**(1), 287–307 (1982)CrossRefMathSciNetzbMATHGoogle Scholar - Macaulay, F.H.S.: The Algebraic Theory of Modular Systems. Cambridge University Press, Cambridge (1916). (reprinted with a foreword by P. Roberts. Cambridge University Press, London (1994))Google Scholar
- Migliore, J., Nagel, U., Zanello, F.: A characterization of Gorenstein Hilbert functions in codimension four with small initial degree. Math. Res. Lett.
**15**(2), 331–349 (2008)CrossRefMathSciNetzbMATHGoogle Scholar - Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math.
**28**(1), 57–83 (1978)Google Scholar

## Copyright information

© The Managing Editors 2014