Infinite dimensional excellent rings

An excellent F-pure ring of prime characteristic has a big tight closure test element

In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if $R$ is local and the $R$-module structure on the injective envelope $E$ of the simple $R$-module can be extended to a structure as a torsion-free left module over the Frobenius skew polynomial ring, then $R$ is $F$-pure and has a tight closure test element. One…

Big tight closure test elements for some non-reduced excellent rings

This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big test element for $R$ is an element of $R^{\circ}$ which can be used in every tight closure membership test for every $R$-module, and not just the finitely generated ones. The main results of the paper are that, if $R$ is excellent and satisfies condition $(R_0)$, and $c \in R^{\circ}$ is such that $R_c$ is Gorenstein and weakly $F$-regular, then some power …

Excellent rings in transchromatic homotopy theory

The purpose of this note is to verify that several basic rings appearing in transchromatic homotopy theory are Noetherian excellent normal domains and thus amenable to standard techniques from commutative algebra. In particular, we show that the coefficients of iterated localizations of Morava $E$-theory at the Morava $K$-theories are normal domains and also that the coefficients in the transchromatic character map for a fixed group form a normal domain.

Associated graded rings of the filtration of tight closure of powers of parameter ideals

Let $I$ be an ideal generated by a system of parameters in an excellent Cohen-Macaulay local domain. We show that the associated graded ring $G^*(I)$ of the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ is Cohen-Macaulay. We prove that if $R$ is an excellent Buchsbaum local domain then $G^*(I)$ is a Buchsbaum module over the Rees ring $\mathcal R^*(I)=\oplus_{n\in \mathbb{N}}(I^n)^*.$ We provide quick proofs of well-known results of I. Aberbach, Huneke-Itoh and Huneke-Hochster about the filtration $\{(I^n)^*: n\in \mathbb{N}\}$ in excellent local domains. An important tool used in the proofs is a …

Groebner bases for the polynomial ring with infinite variables and their applications

We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.

有限Krull维数的优秀局部无分歧正则环的局部上同调模的相伴素数的有限性

Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension

本文证明了Huneke和Lyubeznik关于局部上同调模相伴素理想有限性的猜想，在有限Krull维数的优秀局部无分歧正则环等条件下取得了实质性新进展。

关于优异环的拟 $F$-纯性

On Quasi-$F$-Purity of Excellent Rings

本文提出适用于非 $F$-有限环的拟 $F$-纯性概念，证明其在完备局部 $F$-有限情形下等价于拟 $F$-分裂，并建立其在完备化、正向极限和 étale 扩张下的稳定性。

Asymptotic stability of depths of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. The asymptotic behavior of the quotient modules M/I^n M of M is an actively studied subject in commutative algebra. The main result of this paper asserts that the depth of the localization of M/I^n M at any prime ideal of R is stable for large integers n that do not depend on the prime ideal, if the module M or M/I^n M is Cohen-Macaulay for some n>0, or the ring R is one of the following: a homomorphic image of a Cohen-Macaulay ring, a semi-local ring, an excellent ring, a quasi-excellent and catenary…

Direct summands of infinite-dimensional polynomial rings

Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).

Local rings of countable Cohen--Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable Cohen--Macaulay type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

Hilbert series of symmetric ideals in infinite polynomial rings via formal languages

Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in \mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of the infinite symmetric group $S_{\infty}$. Nagel--Römer recently defined a Hilbert series $H_I(s,t)$ of $I$ and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.

Weak global dimension of Prufer-like rings

In 1969, Osofsky proved that a chained ring (i.e., local arithmetical ring) with zero divisors has infinite weak global dimension; that is, the weak global dimension of an arithmetical ring is 0, 1, or infinite. In 2007, Bazzoni and Glaz studied the homological aspects of Prufer-like rings, with a focus on Gaussian rings. They proved that Osofsky's aforementioned result is valid in the context of coherent Gaussian rings (and, more generally, in coherent Prufer rings). They closed their paper with a conjecture sustaining that ``the weak global dimension of a Gaussian ring is 0, 1, or infinite. …

Modules of infinite regularity over commutative graded rings

In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal of $R$ and by $M$ a finitely generated graded $R$-module, the nonzero modules of the form $\mathfrak{m} M$ have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of $k$ has infinite regularity. Finally, we relate the vanishing of the graded deviations of $R$ to having a nonzero direct summand of a syzygy of $k$ of finite regularity.

"Infinite" properties of certain local cohomology modules of determinantal rings

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $ω_R$ and the property that the socle dimensions of $H^{m+n-2}_I(ω_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(ω_R))$ are not finite. In the case of $m = n$, i.e. a Gorenstein ring, the socle dimensions provide further information about the $τ$-numbers as studied in \cite{MS}. Moreover, the endomorphism ring of $H^{n-1}_I(ω_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generat…

Tight closure of parameter ideals in local rings $F$-rational on the punctured spectrum

Let $(R, \mathfrak{m}, k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that $\ell_R(\mathfrak{q}^*/\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum.

Finite Groebner bases in infinite dimensional polynomial rings and applications

We introduce the theory of monoidal Groebner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Groebner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness theorems in commutative algebra and its applications. A major result of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. We use this machinery to give new proofs of some classical finiteness theorems in algebra…

Upper semi-continuity of the Hilbert-Kunz multiplicity

We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

On the depth of invariant rings of infinite groups

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly reductive. We show that there exists a faithful rational representation V of G (which we will give explicitly) such that cmdef K[\sum_i=1^k V]^G >= k-2 for all k. We give refinements in the case G = SL2.

Ramification of valuations and local rings

In this paper we consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a defectless extension of two dimensional excellent local rings. We also obtain general results on the structure of the extension of associated graded rings along a valuation, and show that the invariants alpha and beta of stable forms of two dimensional extensions in characteristic p of the author and Olivier Piltant are not eventually constant.

Extension of weakly and strongly F-regular rings by flat maps

Let (R,m) -> (S,n) be a flat local homomorphism of excellent local rings. We investigate the conditions under which the weak or strong F-regularity of R passes to S. We show that is suffices that the closed fiber S/mS be Gorenstein and either F-finite (if R and S have a common test element), or F-rational (otherwise).

构造非链状拟优局部预完备化

Constructing Noncatenary Quasi-Excellent Precompletions

本文研究局部环中两个素理想的粘合问题，给出存在拟优局部子环同时保持完备化与素谱性质的充分条件，并据此构造出素谱含非链状有限子集的拟优局部环。

Hilbert-Kunz Functions for Normal Rings

Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d + beta(M) q^{d-1} + O(q^{d-2}).

Local cohomology modules with infinite dimensional socles

Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous polynomial whose coefficients form a system of parameters for T. We show that the socle of H^n_I(R/fR) is infinite dimensional, generalizing an example due to Hartshorne.

Tate代数与优正则环的Frobenius非分裂性

Tate algebras and Frobenius non-splitting of excellent regular rings

本文利用刚性几何构造了特征p的优欧几里得整环，证明优F-纯环一般不是Frobenius分裂的，从而解决了F-奇点理论中长期悬而未决的问题，并首次区分了F-纯性与Frobenius分裂性。

Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type

Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime ideals P_1,...,P_s. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the mono…

Excellence, F-singularities, and solidity

An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following recent results of the first two authors on the Frobenius non-splitting of certain excellent $F$-pure rings, in this paper we use the notion of solidity to systematically study the notion of excellence, with an emphasis on $F$-singularities. We show that for rings $R$ essentially of finite type over complete local rings of characteristic $p$, reducedness implies t…

Asymptotic properties of infinite directed unions of local quadratic transforms

We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n. We previously showed the existence of a unique limit point V of the family of order valuation rings of the sequence. In this paper, we examine asymptotic properties of this family of order valuations. We link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. We construct examples of such S that are archimedean and completely integ…

Faltings' annihilator theorem and t-structures of derived categories

In this paper, we prove Faltings' annihilator theorem for complexes over a CM-excellent ring. As an application, we give a complete classification of the t-structures of the bounded derived category of finitely generated modules over a CM-excellent ring of finite Krull dimension.

An Algorithm for Finding Symmetric Gröbner Bases in Infinite Dimensional Rings

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.

Infinite prime avoidance

We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy prime avoidance. Examples and counterexamples are given throughout to illustrate the phenomena that can occur. As an application, we show how to use prime avoidance to construct counterexamples among rings essentially of finite type.

An uncountably infinite number of indecomposable totally reflexive modules

A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infi…

交换环无限乘积中的素理想

Prime ideals in infinite products of commutative rings

该研究刻画了交换环无限乘积中的素理想与极大理想，证明极大理想由特定布尔代数上的超滤子诱导，并针对两类环族完全刻画了对应乘积环的极大或全部素理想。

Noetherian环的分次性的有限性与无限性

Finiteness and infiniteness of gradings of Noetherian rings

本文刻画了无挠阿贝尔群上存在非零分次的Noetherian环的充要条件为群秩有限，构造了相应的PID例子，并利用分次除环与群上同调证明了第二上同调的消失性及Hilbert级数的良定义性。

On $G(A)_\mathbb{Q}$ of rings of finite representation type

Let $(A,\mathfrak{m})$ be an excellent Henselian Cohen-Macaulay local ring of finite representation type. If the AR-quiver of $A$ is known then by a result of Auslander and Reiten one can explicity compute $G(A)$ the Grothendieck group of finitely generated $A$-modules. If the AR-quiver is not known then in this paper we give estimates of $G(A)_\mathbb{Q} = G(A)\otimes_\mathbb{Z} \mathbb{Q}$ when $k = A/\mathfrak{m}$ is perfect. As an application we prove that if $A$ is an excellent equi-characteristic Henselian Gornstein local ring of positive even dimension with $\text{char} \ A/\mathfrak{m}…

Associated primes of Local cohomology modules over Regular rings

Let $R$ be an excellent regular ring of dimension $d$ containing a field $K$ of characteristic zero. Let $I$ be an ideal in $R$. We show that $Ass \ H^{d-1}_I(R)$ is a finite set. As an application we show that if $I$ is an ideal of height $g$ with $height \ Q = g$ for all minimal primes of $I$ then for all but finitely many primes $P \supseteq I$ with $height \ P \geq g +2$, the topological space $Spec^\circ(R_P/IR_P)$ is connected.

Symmetric ideals of the infinite polynomial ring

Let $R=\mathbf{C}[ξ_1,ξ_2,\ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the containments among these ideals, and describe the equivariant spectrum of $R$. We emphasize that $\mathfrak{S}$-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of $\mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the interest and applications of $\mathfrak{S}$-ideals seen in recent years.

伪有理和Du Bois奇点的Briançon-Skoda定理及优秀环中的一致性

The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings

本文证明了一个广义Briançon-Skoda型包含关系，并由此推出伪有理奇点（如正则环）和Du Bois奇点下的完整Briançon-Skoda包含，同时应用于证明有限维拟优秀环的一致Artin-Rees定理和一致Briançon-Skoda定理。

关于诺特局部环的扰动的注记

On the perturbations of Noetherian local domains

本文研究诺特局部环在小扰动下约化性、整环性和正规性的稳定性，证明了在因子优良 Hensel 局部环、特征零优良局部完全交及因子 Nagata 局部环等情形下扰动稳定性成立。

无限多个变量多项式环上的抛物等变模

Parabolic-equivariant modules over polynomial rings in infinitely many variables

本文研究无限变量多项式环上的抛物等变模范畴，通过组合分解建立有限性结果，并证明其等价于推广 FI 的组合范畴之表示范畴。

通过无限基数的幂等性研究群与环的基数

Cardinality of groups and rings via the idempotency of infinite cardinals

本文利用 ZFC 中无限基数幂等性这一集合论事实，研究群与环的基数性质，证明了无限阿贝尔群具有同基数真子群的充要条件，计算了幺半群环与多项式环的基数，并引入平衡环概念，刻画了诺特环与自内射环成为平衡环的条件。

$\mathbf{A}_{\text{inf}}$ is infinite dimensional

Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-$1$ nondiscrete valuation, we show that the ring $\mathbf{A}_{\text{inf}}$ of Witt vectors of $R$ has infinite Krull dimension.

Sp-equivariant modules over polynomial rings in infinitely many variables

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M fits into an exact triangle $T \to M \to F \to$ where T is a finite length complex of torsion modules and F is a finite length complex of "free" modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras ${\rm Sym}({\bf C}^{…

正则环中符号幂的一致界

Uniform bounds on symbolic powers in regular rings

本文在所有特征的正则环中，对任意理想的符号幂增长建立了统一上界，完全解决了Hochster-Huneke问题，并推广到混合特征情形。

凝聚环上Gorenstein平坦模的直积

Direct products of Gorenstein flat modules over coherent rings

本文构造了零Krull维数的交换凝聚局部代数及相关模，否定回答了右凝聚环上Gorenstein平坦模是否对直积封闭的问题，给出了相关等价条件并揭示了抽象初等类的性质。

Local rings of bounded Cohen-Macaulay type

Let (R,m,k) be a local Cohen-Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thra…

无限多项式环上的对称模 I：幂零商

Symmetric modules over the infinite polynomial ring I: nilpotent quotients

本文研究无限多项式环关于无限对称群作用下的幂零商模范畴，确定了其Grothendieck群、Krull–Gabriel维数及导出范畴生成元，为后续一般模研究奠定基础。

tame 和 wild 原素在交换环的直接积中的研究

Tame and wild primes in direct products of commutative rings

本文研究了交换环直接积中 tame 和 wild 素理想的特点，揭示了 tame 素理想集的结构及其与索引集的关系。

Brauer-Thrall for totally reflexive modules over local rings of higher dimension

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexiv…

B型与BC型李代数之间的一个无限维特征重合

An infinite-dimensional character coincidence between Lie algebras of type B and BC

本研究利用奇正交群与正辛超群特征环的同构，构建了关于B型二次超曲面环的性质良好的Schur模函子式构造，结合多领域思想研究相关表示，为纯自由分解构造提供关键支撑。

Ind-簇代数与无限格拉斯曼流形

Ind-cluster algebras and infinite Grassmannians

该研究证明有限格拉斯曼流形坐标环的特定余极限是无限秩簇代数，且无限秩簇代数等价于自然簇代数范畴的ind-对象，建立了无限秩簇代数与无限格拉斯曼流形的关联。