Studies in Commutative Algebra and Algebraic Geometry

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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.