##
**Graded rings of complete intersection local rings with finite residue fields.**
*(English)*
Zbl 0862.13002

Summary: In this paper we prove that there are only finitely many graded rings arising as the associated graded rings of complete intersection local rings, with a given finite residue field \(k\), which have a given dimension and multiplicity. More generally, if \((R, {\mathfrak m})\) is a Cohen-Macaulay (in fact a generalized Cohen-Macaulay) local ring with finite residue field, then given an integer \(e>0\) the set
\[
S_e(R)= \{\overline {\text{gr}_{\mathfrak m} (R/I)}|\;I\text{ generated by a part of a systems of parameters of }R\text{ and }e(R/I) = e\}
\]
is finite, where \(e(R/I)\) denotes the multiplicity of the ring \(R/I\), and \(\overline {\text{gr}_{\mathfrak m} (RI)}\) denotes the isomorphism class of graded rings determined by \(\text{gr}_{\mathfrak m} (R/I)\). In particular, we see that only finitely many numerical functions can arise as Hilbert functions of such quotient rings \(R/I\). This question, for arbitrary residue fields, was raised by V. Srinivas and the author [J. Algebra 186, No. 1, 1-19, Art. No. 0358 (1996)].