Yongwei Yao

Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular

We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to 1 must be regular.

Modules with Finite F-Representation Type

Finitely generated modules with finite -representation type over Noetherian (local) rings of prime characteristic are studied. If a ring has finite -representation type or, more generally, if a faithful -module has finite -representation type, then tight closure commutes with localizations over . -contributors are also defined, and they are used as an effective way of characterizing tight closure. Then it is shown that always exists under the assumption that satisfies the Krull-Schmidt condition and has finite -representation type by , in which all the are indecomposable -modules that belong to distinct isomorphism classes and .

Some extensions of Hilbert–Kunz multiplicity

Let R be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do not assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures give truly different numerical results) and explore special cases where equivalence with matching tight closure can be shown. All of our measures derive ultimately from Hilbert–Kunz multiplicity.

The lower semicontinuity of the Frobenius splitting numbers

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

Criteria for flatness and injectivity

Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity.

Infinite Rings with Planar Zero-Divisor Graphs

For any commutative ring R that is not a domain, there is a zero-divisor graph, denoted Γ(R), in which the vertices are the nonzero zero-divisors of R and two distinct vertices x and y are joined by an edge exactly when xy = 0. Smith (2007) characterized the graph structure of Γ(R) provided it is infinite and planar. In this article, we give a ring-theoretic characterization of R such that Γ(R) is infinite and planar.

The Compatibility, Independence, and Linear Growth Properties

The first part is about primary decomposition. After reviewing the basic definitions, we survey the compatibility, independence, and linear growth properties that have been known. Then, we prove the linear growth property of primary decomposition for a new family of modules. In the remaining sections, we study secondary representation, which can be viewed as a dual of primary decomposition. Correspondingly, we study the compatibility, independence, and linear growth properties of secondary representations.