A finiteness theorem for the Hilbert functions of complete intersection local rings (Q1284351)

For a homogeneous graded ring \(A=\bigoplus_{n\geq 0}A_n= k[A_1]\) over a field \(k\) or for a Noetherian local ring \((A,m,k)\), we denote by \(H(A,n)\) and \(h(A,t)\) the Hilbert function \(\dim_k A_n\) or \(\dim_k m^n/m^{n+1}\) of \(A\) and its associated Hilbert polynomial, respectively. \textit{S. Kleiman} [in Sém. Géométrie algébrique 1966/67, SGA 6, Lect. Notes Math. 225, 616-666 (1971; Zbl 0227.14007)] showed that if \(A\) is a polynomial ring, then for any positive integers \(d\) and \(e\), the set \[ \{h(A/p,t)\mid p\text{ is a homogeneous prime ideal of }A \text{ with } \dim(A/p)=d \text{ and } e(A/p)=e\} \] is finite. In the similar vein, the main theorem of this paper states that if \(A\) is an equicharacteristic regular local ring, then the set \[ \{H(A/p,n)\mid p\text{ is a complete intersection ideal of } A \text{ with }\dim(A/p)=d\text{ and }e(A/p)=e\} \] is finite. Also the authors construct a sequence \(p_k\) of prime ideals of \(A=k [[X_1, \dots, X_6]]\) with \(h(A/p_k,t) =4t+k\). Hence the set \[ \{h(A/p,n)\mid p\text{ is a prime ideal of }A \text{ with }\dim(A/p)=2\text{ and }e(A/p)=4\} \] is infinite.