On unramified finitely generaed extensions of polynomial rings over a field

Syzygy Theorems via Comparison of Order Ideals on a Hypersurface

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring $R$ versus a hypersurface ring $R/(x^n)$. Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic $p$, with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by $p^…

Finite Generation of Extensions of Associated Graded Rings Along a Valuation

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{ν^*}(S)$, is a finite ${\rm gr}_{ν^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and $ν^*$ is a valuation of the quotient field of $S$ which dominates $S$. We obtain a very general result in equicharacteristic zero in Theorem 1.5. We also obtain general results for unramified extensions of excellent local rings in Proposition 1.7.

Normal Hilbert Polynomials : A survey

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moral{é}s and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials …

Finitely generated and not finitely generated rings

Let $R_1$ be a commutative ring, let $R_2$ be a finitely generated extension ring of $R_1$, and let $S$ be a ring that is intermediate between $R_1$ and $R_2$. For $R_1 = R[x]$ and $R_2 = R[x,y]$, this paper gives simple combinatorial constructions of intermediate rings $S$ that are not finitely generated over $R[x]$.

On rings for which finitely generated ideals have only finitely many components

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over R as well as to R-algebras which are finitely presented as R-modules.

检测和量化离散赋值环上的孤立奇点

Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings

本文在混合特征(0,p)下发展孤立超曲面奇点理论，研究离散赋值环上的商环，引入经典Tjurina和Milnor数的类比，证明确定性定理和Mather-Yau定理的推广，并定义数值不变量以区分非可分和可分情况下的孤立奇点。

Trace ideal and annihilator of Ext and Tor of regular fractional ideals, and some applications

Given a commutative Noetherian ring $R$ with total ring of fractions $Q(R)$, and a finitely generated $R$-submodule $M$ of $Q(R)$, we prove an equality between trace ideal, and certain annihilator of Ext and Tor of $M$. As a consequence, we answer in one-dimensional local analytically unramified case, a question raised by the present author and R. Takahashi. As another application, we give an alternative proof of a recent result of Ö. Esentepe that for one-dimensional analytically unramified Gorenstein local rings, the cohomology annihilator of Iyengar and Takahashi coincides with the conducto…

分歧正则局部环上的局部上同调模

On local cohomology modules over ramified regular local rings

本文将分歧正则局部环视为非分歧正则局部环的艾森斯坦扩张，利用迈耶-维托里斯谱序列构造出具有有限相伴素理想集合的非扩张理想局部上同调模例子，其中部分例子的相伴素理想还包含素数p。

Ideals defining Gorenstein rings are (almost) never products

This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.

Finite generation of Cox rings

In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely generated, the variety admits a quotient presentation by a quasitorus, which resembles the quotient construction of the projective space. We discuss the problem of the finite generation of Cox rings from a geometric perspective and provide examples of both the finitely and non-finitely generated cases.

When is the intersection of two finitely generated subalgebras of a polynomial ring also finitely generated?

We study two variants of the following question: "Given two finitely generated subalgebras R_1, R_2 of C[x_1, \ldots, x_n], is their intersection also finitely generated?" We show that the smallest value of $n$ for which there is a counterexample is 2 in the general case, and 3 in the case that R_1 and R_2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. The counterexample for the general case is a simple modification of a construction of Neena Gupta,…

Commutative rings with every non-maximal ideal finitely generated

In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely generated doesn't imply all ideals be finitely generated. In his article we ask the question that what happens if we assume all non-maximal ideals are finitely generated, and we answer the question by showing that indeed then all maximal ideals are also finitely generated i.e. the ring becomes Noetherian.

Rings over which all (finitely generated) strongly Gorenstein projective modules are projective

The main aim of this paper is to investigate rings over which all (finitely generated strongly) Gorenstein projective modules are projective. We consider this propriety under change of rings, and give various examples of rings with and without this propriety.

局部上同调模的第二消失定理

The Second Vanishing Theorem for Local Cohomology Modules

本文证明了非分歧正则局部环局部上同调模的第二消失定理并给出素特征$p$下的新证明，将等特征局部环最高Lyubeznik数的拓扑刻画推广到混合特征，还得到了混合特征下局部上同调维数的上界。

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.

Some observations on local and projective hypersurfaces

Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of $\Tor^R(M,N)$, based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connections and applications to projective hypersurfaces such as intersection dimension of subvarieties and cohomological criterion for splitting of vector bundles are discussed.

Finitely generated symbolic Rees rings of ideals defining certain finite sets of points in P^2

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let $R$ be a Noetherian ring, $B$ a finitely generated $R$-algebra, and $A$ a pure $R$-subalgebra of $B$. Then $A$ is finitely generated over $R$.

Test ideals in rings with finitely generated anti-canonical algebras

Many results are known about test ideals and $F$-singularities for ${\bf Q}$-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus ...$ is finitely generated (or more generally, in the log setting for $-K_X - Δ$). In particular, we show that the $F$-jumping numbers of $τ(X, a^t)$ are discrete and rational. We show that test ideals $τ(X)$ can be described by alterations as in Blickle-Schwede-Tucker (and hence show that splinters are strongly $F$-regular in this setting -- recovering a result …

Minimal free resolution of a finitely generated module over a regular local ring

Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{\mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Am…

The space of finitely generated rings

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor-Bendixson rank of any member of this space. For instance, the Cantor-Bendixson rank of the free commutative ring on n generators is omega^n, where omega is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

Cosupport computations for finitely generated modules over commutative noetherian rings

We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring has full cosupport. We also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite Krull dimension. Finally, we give an example showing that the cosupport of a finitely generated module need not be a closed subset of $\operatorname{Spec}(R)$, providing a negative answer to a question of Sathe…

环的严格闭包何时是有限生成的？

When is the strict closure of rings finitely generated?

本文研究任意维数环的严格闭包的有限生成性，给出了诺特局部环的严格闭包作为模有限生成的充分条件，并刻画了优良环上严格闭包的有限生成性。

Notes on finitely generated flat modules

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a finitely many minimal primes or a finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions then every finitely generated flat module over this ring is projective. These results general…

关于多项式扩张有限生成环上单模行的完成

On completion of unimodular rows over polynomial extension of finitely generated rings over $\mathbb{Z}$

本文证明在有限生成环R上，若R的维数d≥2且1/d!∈R，则R[X]上的单模行可通过初等变换映射为因子行。

Finitely generated powers of prime ideals

Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension $\le$ 2. And if P is a prime ideal of R whose a power is finitely generated then we show that P is finitely generated if R is either a reduced coherent ring or a polynomial ring over a reduced arithmetical ring. These results extend a theorem of Roitman, published in 2001, on prime ideals of coherent integral domains.

The logical complexity of finitely generated commutative rings

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring $A$ is bi-interpretable with $(\mathbb N,{+},{\times})$ if and only if the space of non-maximal prime ideals of $A$ is nonempty and connected in the Zariski topology and the nilradical of $A$ has a nontrivial annihilator in $\mathbb Z$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb Z$ is not bi-interpretable with $\mathbb N$.

当每个有限生成理想都是S-主理想时

When every finitely generated ideal is S-principal

本文提出S-Bézout环概念以推广Bézout环，研究其与相关环类的关系，并在直积、局部化、平凡扩张和融合环等背景下给出刻画，同时引入非零S-Bézout环并建立相应特征。

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also e…

Equivalent generating vectors of finitely generated modules over commutative rings

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $μ$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We denote by $\operatorname{E}_n(R)$ the subgroup of $\operatorname{SL}_n(R)$ generated by the the matrices which differ from the identity by a single off-diagonal coefficient. Given $n \ge μ$ and $G \in \left\{\operatorname{SL}_n(R),\operatorname{E}_n(R)\right\}$, we study the action of $G$ by matrix right-multiplication on $\operatorname{V}_n(M)$, the set of elemen…

The existence of finitely generated modules of finite Gorenstein injective dimension

In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.

On The Integral Closure of Radical Towers in Mixed Characteristic

We study the Cohen-Macaulay property of a particular class of radical extensions of an unramified regular local ring having mixed characteristic.

Reflexive modules over the endomorphism algebras of reflexive trace ideals

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong and T. Kobayashi concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified …

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely ge…

On the formal grade of finitely generated modules over local rings

Let \fa be an ideal of a local ring (R,\fm) and M a finitely generated R-module. This paper concerns the notion \fgrade(\fa,M), the formal grade of M with respect to \fa (i.e. the least integer i such that {\vpl}_nH^i_{\fm}(M/\fa^n M)\neq 0). We show that \fgrade(\fa,M)\geq \depth M-\cd_{\fa}(M), and as a result, we establish a new characterization of Cohen-Macaulay modules. As an application of this characterization, we show that if M is Cohen-Macaulay and L a pure submodule of M with the same support as M, then \fgrade(\fa,L)=\fgrade(\fa,M). Also, we give a generalization of the Hochster-Eag…

Gorenstein n-X-injective and n-X-flat modules with respect to a special finitely presented module

Let R be a ring, X a class of R-modules and n>1 an integer. In this paper, via special finitely presented modules, we introduce the concepts of Gorenstein n-X-injective and n-X-flat modules. And aside, we obtain some equivalent properties of these modules on n-X-coherent rings. Then, we investigate the relations among Gorenstein n-X-injective, n-X-flat, injective and flat modules on X-FC-rings (i.e., self n-X-injective and n-X-coherent rings). Several known results are generalized to this new context

一维几乎Gorenstein环上上同调的湮灭

Annihilation of cohomology over one dimensional almost Gorenstein rings

本文研究一维几乎Gorenstein环的上同调湮灭理想，给出其与稳定范畴湮灭理想相等的充分条件，并证明其与导子理想一致。

Finitely Supported *-Simple Complete Ideals in a Regular Local Ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for some of the factors. The existence of negative exponents occurs if the dimension of R is at least 3 because of the existence of finitely supported *-simple ideals that are not special. We consider properties of special *-simple complete ideals such as their Rees valuations and point basis. Let (R, m) be a d-dimensional equicharacterstic regular local ring with …

Gorenstein 投射预覆盖与有限表现模

Gorenstein projective precovers and finitely presented modules

针对任意环上 Gorenstein 投射预覆盖的存在性问题，本文证明可归约到有限表现模：若其 Gorenstein 投射维数一致有界，则该模类特殊预覆盖。

On rings each of whose finitely generated modules is a direct sum of cyclic modules

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring $R$ is a left FGC-ring if and only if $R$ is a right FG…

Multiplicative closure operations on ring extensions

Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes the classes of star, semistar and semiprime operations. We study how the set $\mathrm{Mult}(A,B,\mathcal{G})$ of these closure operations vary when $A$, $B$ or $\mathcal{G}$ vary, and how $\mathrm{Mult}(A,B,\mathcal{G})$ behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dime…

The Auslander-Bridger formula and the Gorenstein property for coherent rings

The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely generated modules over a Noetherian ring, is studied in the context of finitely presented modules over a coherent ring. A generalization of the Auslander-Bridger formula is established and is used as a cornerstone in the development of a theory of coherent Gorenstein rings.

On finite generation of Noetherian algebras over two-dimensional regular local rings

Let $R$ be a complete regular local ring with an algebraically closed residue field and let $A$ be a Noetherian $R$-subalgebra of the polynomial ring $R[X]$. It has been shown in \cite{DO2} that if $\dim R=1$, then $A$ is necessarily finitely generated over $R$. In this paper, we give necessary and sufficient conditions for $A$ to be finitely generated over $R$ when $\dim R=2$ and present an example of a Noetherian normal non-finitely generated $R$-subalgebra of $R[X]$ over $R= {\mathbb C}[[u, v]]$.

Local cohomology modules of polynomial or power series rings over rings of small dimension

Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has dimension one and $π$ is an nonzero divisor, then the same properties hold for prime ideals that contain $π.$ These results do not require that $A$ contains a field. As a consequence, we give a different proof for the finiteness properties of local cohomology over unramified regular local rings. In addition, we extend previous results on the injective dimension of…

Blowing Up Finitely Supported Complete Ideals in a Regular Local Ring

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). We consider singularities of the normalization of the blow-up Proj R[It] of I. A theorem of Lipman implies that the ideal I has a unique factorization as a star-product of special star-simple complete ideals with possibly negative exponents for some of the factors. If the normalization of the projective model Proj R[It] is regular, we prove that it is the regular model obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke-Sally in dimension 2, we prove that every loca…

The structure of balanced big Cohen-Macaulay modules over Cohen-Macaulay rings

Over a Cohen-Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a "structure theorem" for balanced big CM modules. (2) Every finitely generated module has a preenvelope with respect to the class of fi…

何时是规范导数最小？

When is the canonical conductor minimal?

研究一维Cohen-Macaulay局部环中导数最小的条件，探讨其与almost Gorenstein环的关系。

Symmetry in vanishing of Tate cohomology over Gorenstein rings

We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.

Symmetry in the vanishing of Ext over Gorenstein rings

We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $Ext^i_R(M,N)=0$ for all $i >> 0$ if and only if $Ext^i_R(N,M)=0$ for all $i >> 0$.

分歧离散赋值环上代数的奇异轨迹

Singular loci of algebras over ramified discrete valuation rings

本文针对混合特征分歧离散赋值环上的代数，推广了经典的雅可比准则，用以刻画其奇异轨迹。

一般二次余定理的构造性证明：对于Jacobian环

A constructive proof of the general Nullstellensatz for Jacobson rings

本文通过构造性方法证明了Jacobian环的一般二次余定理，指出单变量多项式环在Jacobian环上也是Jacobian环，并解决了有限生成代数在零维环或整数环上的Jacobian性问题。

强离散凝聚环上的广义Buchberger与Schreyer算法

Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings

本文在强离散凝聚环上推广Buchberger算法，证明首项模可数生成并给出计算生成集的算法，同时给出Hilbert合冲定理的构造性版本。

On a new invariant of finitely generated modules over local rings

Let $M$ be a finitely generated module on a local ring $R$ and $\F: M_0\subset M_1\subset...\subset M_t=M$ a filtration of submodules of $M$ such that $ d_o<d_1< ... <d_t=d$, where $d_i=\dim M_i$. This paper is concerned with a non-negative integer $p_\mathcal F(M)$ which is defined as the least degree of all polynomials in $n_1, ..., n_d$ bounding above the function $$\ell(M/(x_1^{n_1}, ..., x_d^{n_d})M)-\sum_{i=0}^tn_1...n_{d_i}e(x_1,..., x_{d_i};M_i).$$ We prove that $p_\mathcal F(M)$ is independent of the choices of good systems of parameters $\underline x=x_1, ..., x_d$. When $\F$ is the …

交换凝聚环上的宽子范畴

Wide subcategories over commutative coherent rings

本文证明交换凝聚环上有限表示模的宽子范畴与完美导出范畴的厚子范畴格同构，构造了逆映射并研究了有限扩张滤过。