Characteristic p methods in characteristic zero via ultraproducts

摘要: In recent decades, by exploiting the algebraic properties of Frobenius in positive characteristic, many so-called homological conjectures and intersection have been established, culminating into powerful theory tight closure big Cohen–Macaulay algebras. present article, I give a survey how these methods also can be applied directly characteristic zero taking ultraproducts, rather than through cumbersome lifting/reduction techniques. This has led to some new results regarding rational log-terminal singularities, as well vanishing theorems. Even mixed we get results, albeit only asymptotically.