Modules over Local Rings

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This award supports research in commutative algebra. The principal investigator will study the rigidity question for finitely generated modules over a noetherian local ring. He will apply this work to the problem of determining when the tensor product is torsion-free. He will also work on the classification of local rings of finite Cohen-Macaulay type and will try to determine the ranks of the indecomposable maximal Cohen-Macaulay modules over these rings. Finally, he will study the arithmetical properties of Picard groups of algebraic varieties and several related questions concerning the multiplicative structure of fields. 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.