The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen–Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of F-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold).

In this paper, we investigate the lower bound s HK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that s HK (p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurface k[[X, Y, Z,W]]/(X 2 + Y 2 + Z 2 + W 2) under mild conditions.

Furthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

We study the behavior of Hilbert-Kunz multiplicity for powers of an ideal, especially the case of stable ideals and ideals in local rings of dimension 2. We can characterize regular local rings by certain equality between Hilbert-Kunz multiplicity and usual multiplicity.

We show that rings with “minimal” Hilbert-Kunz multiplicity relative to usual multiplicity are “Veronese subrings” in dimension 2.

Surface oxidation of an iridium film and the possibility of epitaxial growth of PZT spin-coated film on sputtered iridium were investigated. The free surface on the iridium film oxidized over 400 °C with random orientation. Nevertheless, both the PZT(111)/Ir(111) and PZT(100)/Ir(100) interfaces were realized using the highly oriented iridium layer. These results suggest that PZT nucleation has priority over iridium surface oxidation.

In this paper, we prove that for any ideal I of dimension one is I-cofinite for all i and for any finite A-module M. Furthermore, for any ideal I over any regular local ring A, we investigate the relationship between I-cofiniteness and vanishing for local cohomology modules .