In this note, we use the theory of test ideals and Cartier algebras to examine the interplay between the tight and integral closures in a local ring of positive characteristic. Using work of Schwede, we prove the abundance of strong test ideals, recovering some older fundamental results, and use this approach in concrete computations. In the second part of the paper, the case of Stanley-Reisner rings is fully examined.
The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius operators of order e on the injective hull of the residue field of R. It is known that, for Stanley-Reisner rings, the Frobenius complexity is either −∞ or 0. This invariant is determined by the complexity sequence {c e } e of the ring of Frobenius operators on the injective hull of the residue field. We will show that {c e } e is constant for e ≥ 2, generalizing work of Àlvarez Montaner, Boix and Zarzuela. Our result settles an open question mentioned by Àlvarez Montaner in [1].
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