Studies in Commutative Algebra and Algebraic Geometry

Hochster建议继续研究（主要是局部）Noetherian环理论中的几个长期存在的问题，并探索这些问题与由Hochster和Huneke引入的爆炸性发展的紧闭理论之间的关系。一个主要的推力是探索不包含域的环中的闭包概念，例如整数上的有限生成代数，希望将紧闭包理论扩展到这样的环，从而解决许多悬而未决的问题。给出了证明大Cohen-Macaulay模在混合特性下存在性问题的方法。研究了一类局部上同模的准长度和准内容的新理论。这个理论提出了一些困难的核心问题，并且是提出的几种方法之一，以解决长期存在的重要问题，即规则环是否是其模有限扩展的直接和。拟议的研究涉及交换环，这是一个抽象的系统，其中一个可能被认为是人造数字。人们可以加、减、乘这些人造数字：称之为环元素。整数和实数是例子，但还有许多其他的例子，包括只包含有限多个元素的环。在一个例子中，只有0和1，并且1+1 = 0（类似于偶数和奇数的性质：奇数+奇数=偶数）。在研究多未知数的多方程系统时，人们可以强迫方程在一个抽象环中成立。对这个环的性质的研究给出了解的信息。我们也可以研究一种解的图，它存在于高维空间中。在这些代数和几何观点之间来回转换是非常有益的。第三种方法，将在提议的研究中经常使用，是研究有点像整数（但更一般）的解决方案，但在这样做时忽略素数的倍数：如果元素相差该素数的倍数，则认为它们相等。对许多不同的质数，可能是所有的质数做这个，可以提供大量关于方程解的信息。拟议的研究涉及一种称为紧密闭合的系统方法，用于使用这一思想，以及将其扩展到新的环境中。

Hochster proposes to continue investigating several long standing questions in the theory of (primarily local) Noetherian rings and to explore the relationship of these questions with the explosively developing theory of tight closure, which was introduced by Hochster and Huneke. One main thrust is to explore several notions of closure in rings that do not contain a field, such as finitely generated algebras over the integers, with the hope of extending tight closure theory to such rings, thereby solving many open questions. Approaches to the problem of proving existence of big Cohen-Macaulay modules in mixed characteristic are also given. A new theory of quasi-length and content related to certain local cohomology modules will be studied. This theory has raised some difficult, central problems, and is one of several methods proposed to attack the long standing and important question of whether regular rings are direct summands of their module-finite extensions. The proposed research deals with commutative rings, which are abstract systems in which one has what might be thought of as artificial numbers. One can add, subtract, and multiply these artificial numbers: call them ring elements. The integers and real numbers are examples, but there are many other examples, including rings that contain only finitely many elements. In one example, one has only 0 and 1, and 1+1 = 0 (like the properties of even and odd integers: odd + odd =even). In studying systems of many equations in many unknowns, one can force the equations to hold in an abstract ring. The study of the properties of this ring gives information about the solutions. One can also study instead a sort of graph of the solutions, that exists in a high dimensional space. It is of great benefit to go back and forth between these algebraic and geometric points of view. A third method, which will be used frequently in the proposed research, is to study solutions that are somewhat like integers (but more general), but to do so while ignoring multiples of a prime number: elements are considered equivalent if they differ by a multiple of this prime. Doing this for many different prime numbers, possibly all prime numbers, can provide a huge amount of information about the solutions of the equations. The proposed research deals with a systematic method, called tight closure, for using this idea, as well as its extensions into new contexts.