p⁻¹-linear maps in algebra and geometry

الأصل · EN

In this article we survey the basic properties of p⁻ᵉ-linear endomorphisms of coherent Øₓ-modules, i.e. of Øₓ-linear maps F* → where, are Øₓ-modules and F is the Frobenius of a variety of finite type over a perfect field of characteristic p > 0. We emphasize their relevance to commutative algebra, local cohomology and the theory of test ideals on the one hand, and global geometric applications to vanishing theorems and lifting of sections on the other.