On the number of generators of ideals in polynomial rings


For an ideal $I$ in a noetherian ring $R$, let $\mu(I)$ be the minimal number of generators of $I$. It is well known that there is a sequence of inequalities $\mu(I/I^2)\leq \mu(I)\leq \mu(I/I^2)+1$ that are strict in general. However, Murthy conjectured in 1975 that $\mu(I/I^2)=\mu(I)$ for ideals in polynomial rings whose height equals $\mu(I/I^2)$. The purpose of this article is to prove a stronger form of the conjecture in case the base field is infinite of characteristic different from $2$: Namely, the equality $\mu(I/I^2)=\mu(I)$ holds for any ideal $I$, irrespective of its height.


Jean Fasel

Institut Fourier, Université Grenoble Alpes, Grenoble, France