Common threads in the theories of Local Cohomology, D-modules and Tight Closure and their interactions

许多交换代数定理可以证明表明:(1)如果定理失败,可以找到一个反例的环p主要特征(例如,一个环包含整数环模素数p),和(2)没有这样的反例存在于p.Step特点(2)往往比在更容易证明以上特点存在的零,因为弗罗贝尼乌斯函数f (r),提出了r的甲状旁腺素的力量。这个函数是环的自同态，即它具有f(r+s)=f(r)+f(s)的性质，令人惊奇的是，它很好地处理了特征p中的许多问题。在素特征交换环研究的发展过程中，引入了各种概念和技术，如理想的某种紧闭运算，“大”对象上称为局部上同模的某些结构，以及作用于这些环上的微分算子。这些对象及其相关技术已被证明在解决代数和几何问题方面非常成功，并且这些概念之间的相互作用被证明是特别丰富的。我建议在研究助理的帮助下进一步研究这些相互作用，并将由此产生的技术应用于解决我所在领域的几个突出问题。

Many theorems in Commutative Algebra can be proved by showing that:(1) if the theorem fails, one can find a counter-example in a ring of prime characteristic p (i.e., a ring which contains the ring of integers modulo a prime number p), and(2) no such counter-example exists in characteristic p.Step (2) above is often much easier to prove than in characteristic zero because of the existence of the Frobenius function f(r) which raises r to the pth power. This functon is an endomorphism of the rings, i.e., it has the property that f(r+s)=f(r)+f(s), and surprisingly, gives a good handle on many problems in characteristic p.During the course of development of the study of commutative rings of prime characteristic, various notions and techniques were introduced, e.g., a certain tight-closure operation of ideals, certain structures on ``large'' objects called local cohomology modules, and differential operators acting on these rings. The objects and their associated techniques have proved to be very successful in tackling algebraic and geometric problems, and the interactions between these concepts turned out to be especially fertile.I propose to study these interactions further with the aid of a research assistant, and to apply the resulting techniques to the solution of several outstanding problems in my field.