A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
Cohen-Macaulayness in graded algebras
Cohen-Macaulayness of Rees algebras and associated graded rings of ideals has been actively investigated in recent years, cf. e.g., [AH], [AHT], [GN], [JK], and their references. We illustrate here how some methods introduced by Sancho de Salas in [SS] yield additional insights in this area. For one thing, they provide another approach to a basic result of TrungIkeda (Theorem 4.4). They also lead to new results. Theorem 4.1 relates the Cohen-Macaulay property of a Rees algebra to vanishing of cohomology of its Proj. As a special case, we get that in a normal local ring R, there exists an ideal I whose blowup desingularizes R and whose Rees algebra R[It] is Cohen-Macaulay (CM) if and only if R has a rational singularity—at least in characteristic zero, and more generally if desingularization theorems such as have been announced by Spivakovsky hold up. (In contrast, Theorem 4.3, due to Sancho de Salas, implies that for any CM local ring on a variety over C, there exists a desingularizing I whose associated graded ring is CM [SS, Thm. 2.8].) Theorem 5 gives an affirmative answer to a question of Huneke: it states that if R is pseudo-rational (for instance, R regular [LT, §4]) and if I is a non-zero R-ideal whose associated graded ring is CM, then R[It] is CM too. We begin with a most useful exact sequence due essentially to Sancho de Salas. Consider a noetherian graded ring
Distribution-free monitoring of univariate processes
We consider statistical process control (SPC) of univariate processes when observed data are not normally distributed. Most existing SPC procedures are based on the normality assumption. In the literature, it has been demonstrated that their performance is unreliable in cases when they are used for monitoring non-normal processes. To overcome this limitation, we propose two SPC control charts for applications when the process data are not normal, and compare them with the traditional CUSUM chart and two recent distribution-free control charts. Some empirical guidelines are provided for practitioners to choose a proper control chart for a specific application with non-normal data.