Chapter 1 Characteristic p methods in characteristic zero via ultraproducts
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Characteristic p methods in characteristic zero via ultraproducts
In recent decades, by exploiting the algebraic properties of the Frobenius in positive characteristic, many so-called homological conjectures and intersection conjectures have been established, culminating into the powerful theory of tight closure and big Cohen–Macaulay algebras. In the present article, I give a survey of how these methods also can be applied directly in characteristic zero by ...
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A canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over C is obtained by taking ultraproducts of absolute integral closures in positive characteristic. Among the applications are a new tight closure characterization of rational singularities in characteristic zero, and a necessary condition for Q-Gorenstein logterminal singularities. In particular, it ...
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Using a tight closure argument in characteristic p and then lifting the argument to characteristic zero with aid of ultraproducts, I present an elementary proof of the Briançon-Skoda Theorem: for an m-generated ideal a of C[[X1, . . . ,Xn]], the m-th power of its integral closure is contained in a. It is well-known that as a corollary, one gets a solution to the following classical problem. Let...
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