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
Gambling and Problem Gambling in Sweden: Changes Between 1998 and 2009
Gambling participation and rates of problem gambling change over time in response to a variety of factors including gambling availability, demographic changes and adaptation at individual and societal levels. These relationship are complex and only partially understood. The major aim of the present study was to provide general population estimates of gambling participation and problem gambling for Sweden and compare these estimates with estimates from a previous national study. The study was also designed to identify risk factors for problem gambling including change in these factors over time. Data are from the first phase of the Swedish Longitudinal Gambling Study (Swelogs) in which a representative sample of 8,165 people was assessed using validated problem gambling and other measures to facilitate comparison with findings from the 1997/1998 Swedish Gambling Study (Swegs). Overall, it was found that gambling participation reduced markedly, although in some population sectors increases were evident for some forms including poker and electronic gaming machines. Lifetime prevalence of probable pathological gambling increased; however, past 12 months probable pathological and problem gambling prevalence did not. Males, younger adults and people born outside Sweden were at high risk in both studies. Significant prevalence increases were evident for people aged 18–24 and those with low levels of education. The results indicate that relationships between gambling exposure, participation and problems are dynamic with shifting implications for public health and social policy.