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This dissertation focuses on the following topics: (1) asymptotic primedivisors over complete intersection rings, (2) asymptotic stability ofcomplexities over complete intersection rings, (3) asymptotic linear bounds ofCastelnuovo-Mumford regularity in multigraded modules, and (4)characterizations of regular local rings via syzygy modules of the residuefield.

Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules