Asymptotic Behaviors in Commutative Algebra and Algebraic Geometry

We study three types of asymptotic behaviors in commutative algebra: limit of length of local cohomology of thickenings, degrees of generators of socle of local cohomology of thickenings and limit of F-pure thresholds. In Chapter 1 and 2 we will review necessary background and motivations for the questions that we study. Chapter 3 will be devoted to the study of limit of length of local cohomology of determinantal thickenings, where the author proves the existence of such limits and find closed formulae for the generalized multiplicities of interest (this is the content of (1)). Chapter 4 is joint work with Perlman (this is the content of (2)). We will continue the investigation of local cohomology modules of determinantal thickenings, this time we turn our attention to their socle. In particular, precise formulas of socle degrees are proved. Finally, in Chapter 5 we will study singularities of pairs in both zero and positive characteristic, where the main object of study is linkage of ideals, and the relation of least critical exponents of a pair and its linkage has been proved. The author also applies this result to deduce corresponding results on F-pure thresholds and log canonical thresholds (this is the content of (3)).