Mathematical Sciences: Rings with Specified Completions and Cohen-Macaulay Algebras

该奖项支持交换代数的研究。 一个目标是证明对于任何局部环，都有一个代数是Cohen-Macaulay。 这一结果意味着大科恩-麦考利模猜想，并因此产生了许多其他猜想。 它是通过减少到一个唯一的分解域的情况下，孤立的奇点，然后执行一个建设，这是普遍的性质。 还将寻找这种减少可能有用的其他问题。 此外，本地域将被构建为具有指定的通用形式纤维。 在不相关的工作中，将研究刚性猜想的一种弱化形式，其中假设两个模都具有有限的投射维数。 本研究涉及交换代数和代数几何中的一些问题。 代数几何研究多项式方程族的解。 人们既可以研究解集的几何，也可以通过研究形成交换环的解集上的某些函数来代数地处理问题。 这种双重视角在交换代数和代数几何之间建立了密切的联系。

This award supports research in commutative algebra. One goal is to show that for any local ring, there is an algebra over it which is Cohen-Macaulay. This result would imply the Big Cohen-Macaulay Modules Conjecture and consequently a number of other conjectures. It is to be achieved by reducing to the case of a unique factorization domain with an isolated singularity and then performing a construction which is universal in nature. Other problems where this reduction may be useful will also be sought out. In addition, local domains will be constructed to have specified generic formal fibres. In unrelated work, a weakened form of the Rigidity Conjecture will be studied wherein both modules are assumed to have finite projective dimension This study will primarily consist of looking for a counterexample. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.