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\(F\)-modules: Applications to local cohomology and \(D\)-modules in characteristic \(p>0\). (English) Zbl 0904.13003

Let \(R\) be a commutative Noetherian regular ring containing a field of characteristic \(p>0\). The notion of \(F_R\)-modules (or simply \(F\)-module) is defined by using the Frobenius homomorphism. A typical example of an \(F\)-module is the local cohomology module \(H^i_I (R)\) where \(I\) is an arbitrary ideal of \(R\). In the present paper the author establishes the fundamentals of \(R\)-modules and uses it to study the structure of local cohomology modules – for example injective dimension, Bass numbers and vanishing – and \(D\)-modules.

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13D45 Local cohomology and commutative rings
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