Related papers: Regular rings and perfectoid towers

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

Let R be a commutative ring with unit and let I be an ideal generated by a regular sequence. Then it is known that the natural sequences 0-> Tor_*^R(R/I,I^s)-> Tor_*^R(R/I,I^s/I^{s+1})-> Tor_{*-1}^R(R/I,I^{s+1})-> 0 are short exact…

We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our…

In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these…

Let R be a commutative Noetherian d-dimensional complete equicharacterisitc regular local ring and let I be an ideal of R such that every minimal prime over I has height at most c. Let v=d - [(d-2)/c]-1 and v'=d - [(d-1)/c]. It has been…

We prove a generalization of the Hochster-Roberts-Boutot-Kawamata Theorem conjectured by Aschenbrenner and the author: let $R\to S$ be a pure homomorphism of equicharacteristic zero Noetherian local rings. If $S$ is regular, then $R$ is…

Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…

For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally…

We provide a proof of the Alpern multi-tower theorem for Z^d actions. We reformulate the theorem as a problem of measurably tiling orbits of a Z^d action by a collection of rectangles whose corresponding sides have no non-trivial common…

A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates…

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$:…

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The…

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of…

The aim of this paper is to extend Cohen structure theorem beyond local rings. Both Cohen structure theorem and Nagata's generalization of it are special cases of our results. We investigate for which rings $R$ there exists a maximal ideal…

Let $R$ be a commutative Noetherian local ring. We characterize when its completion has an isolated singularity, thereby strengthening the Dao-Takahashi refinement of the Auslander-Huneke-Leuschke-Wiegand theorem. We investigate the ascent…

We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals…

We propose a unified framework for random locations exhibiting some probabilistic symmetries such as stationarity, self-similarity, etc. A theorem of Noether's type is proved, which gives rise to a conservation law describing the change of…