Tight Closure, F-purity, and Varieties of Nearly Commuting Matrices.

[摘要] The thesis focuses on the techniques and applications of the theory of tight closure in rings of positive prime characteristic. The first part studies the question of the localization in rings where every ideal is tightly closed; such rings are called weakly F-regular.The notions of quasi-parameter ideals and J-test ideals are defined. Moreover, it is proved that it is sufficient to have that all quasi-parameter ideals for the canonical ideal are tightly closed in order for the ring to be weakly F-regular. The second part concentrates on algebraic sets of pairs of matrices with a diagonal commutator. These are called algebraic sets of nearly commuting matrices. It is proved that their coordinate rings are reduced. Furthermore, in the case of 3 by 3 matrices we show that the coordinate rings of an algebraic set of nearly commuting matrices, of both of its irreducible components and of their intersection are F-pure. The last algebraic set is also proved to be irreducible for matrices of all sizes. In the third part we study algebras with straightening law. We show that one can do certain linear changes of basis with an induced partial order while preserving the property of being an algebra with straightening law.We apply this to the rings defined by the ideals associated with nearly commuting and commuting matrices.

[发布机构] University of Michigan

[效力级别] variety of commuting matrices

[关键词] Tight closure;variety of commuting matrices;F-purity;Mathematics;Science;Mathematics