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

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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.