Sally's question and a conjecture of Shimoda (Q2843355)

Let \((R,\mathfrak m,k)\) be a (noetherian) local ring, and set \(d:=\dim(R)\). It is well-known that \(d\leq1\) iff there exists \(n\in\mathbb N\) such that \(\mu(I)\leq n\) for every ideal \(I\) of \(R\) (cf.\ [\textit{J. D. Sally}, Numbers of generators of ideals in local rings. Lecture Notes in Pure and Applied Mathematics. Vol. 35. New York - Basel: Marcel Dekker, Inc. (1978; Zbl 0395.13010)]). Sally showed [\,loc.~cit., p.\ 52, Theorem 2.1] that \(d\leq2\) iff there exists \(n\in\mathbb N\) such that \(\mu(I)\leq n\) for every ideal \(I\) of \(R\) satisfying \(\mathfrak m\notin{\text{Ass}}(I)\). If \(d\leq2\), then this result implies that there exists \(n\in\mathbb N\) such that \(\mu(\mathfrak p)\leq n\) for every \(\mathfrak p\in \text{Spec}(R)\), \(\mathfrak p\neq\mathfrak m\). Sally remarked that it is an open question if the converse is true. The authors call \(R\) a Shimoda ring if \(\mu(\mathfrak p)=\text{ht}(\mathfrak p)\) for every \(\mathfrak p\neq\mathfrak m\); then the question asked by \textit{Y. Shimoda} [``On power stable ideals and symbolic powers'', In: Proc. third Japan-Vietnam Joint Seminar on Commutative Algebra, Hanoi, 90--95 (2007)] can be put in the following form: If \(R\) is a Shimoda ring, does this imply that \(d\leq 2\)? The authors remark that though the question of Shimoda seems easier than that of Sally, it has proved to be difficult to answer.NEWLINENEWLINEIn section 2 they study this question. First of all, it is enough to consider only local UFD's of dimension at most \(3\). If \(R\) is regular of dimension \(3\), they construct even a family \(\mathfrak p\) of prime ideals of \(R\) with \(\text{ht}=2\), and \(\mu(\mathfrak p)=3\), hence such an \(R\) is not Shimoda (cf. Proposition 2.6). Their main result is Theorem 2.3: Let \(R\) be Shimoda and \(2\leq d\leq 3\). If \(R\) is complete, contains its residue field \(k\), \(|k|=\infty\) and \(e(R)\leq 3\), then \(d=2\).NEWLINENEWLINEIn section 3 the authors study the Shimoda property for standard graded algebras \(A=k[\,x_1,\ldots,x_n\,]\) where \(k\) is an infinite field, \(x_1,\ldots,x_n\) are homogeneous of degree \(1\), and \(n\) is minimal with respect to the property to generate the irrelevant ideal \(\mathcal M\) of \(A\); the assume that \(d:=\dim(A)\geq2\). They call \(A\) \textit{gr-Shimoda} if every relevant homogeneous prime ideal \(P\) of \(A\) can be generated by \(\text{ht}(P)\) elements. Their main result is Theorem 3.6, answering definitely Shimoda's question in this situation: Let \(A\) be a gr-Shimoda ring; if \(k\) has characteristic \(0\), then \(\dim(\text{Proj}(A))=2\).NEWLINENEWLINEIn section 4 the authors show -- adapting results of \textit{M.\ Miller} [J. Algebra 64, 29--36 (1980; Zbl 0467.13004)] -- under some technical conditions: If \(\text{dim}(\text{Proj})(A)\geq 3\), then \(A\) possesses height 2 homogeneous prime ideals requiring an arbitrarily large number of generators (cf.\ Theorem 4.1) and that \(A\) admits a homogeneous prime ideal \(P\) of height \(2\) with \(\mu(P)=3\).