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Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure. (English) Zbl 1130.13002

Let \((R, m, k)\) be a local ring of positive characteristic \(p\), \(p\) prime. The author studies systematically the \(R\)-modules \(H\) that admit a Frobenius action \(f\) with the goal of applying this theory to the top local cohomology of \(R\) with support in the maximal ideal \(m\). More precisely, the author considers the Frobenius skew polynomial ring over \(R\) denoted here by \(R[x, f]\) and studies the \(R[x,f]\)-left modules \(H\). He develops a theory of special annihilator submodules of \(H\) in full generality and then applies it to the case of \(R[x, f]\)-modules \(H\) that are \(x\)-torsion free (which means that the Frobenius action on \(H\) is injective). It should be noted that when \(H\) is \(x\)-torsion free, the author is led to the interesting notion of \(H\)-special \(R\)-ideal proving a number of remarkable results about the set of such ideals of \(R\).
As a consequence to this theory, the author obtains applications to tight closure theory via results on the weak test ideal of \(R\) and F-stable primes of \(R\). The authors proves the existence of weak parameter test elements for Cohen-Macaulay rings. M. Hochster and C. Huneke proved the existence of weak test elements for algebras of finite type over a local excellent ring [Trans. Am. Math. Soc. 346 1–62 (1994; Zbl 0844.13002)]. The application to F-stable primes relates the theory of special annihilators to a result of the reviewer [Proc. Am. Math. Soc. 131 3379–3386 (2003; Zbl 1039.13006)].

MSC:

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

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