J. T. Stafford, J. J. Zhang
We provide examples of finitely generated noetherian PI algebras for which there is no finite dimensional filtration with a noetherian associated graded ring; thus we answer negatively a question raised by M. Lorenz.
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Resco and Small gave the first example of an affine Noetherian algebra which is not finitely presented. It is shown that their algebra has no finite-dimensional filtrations whose associated graded algebras are Noetherian, affirming their prediction. A modification of their example yields countable fields over which `almost all' (that is, a co-countable continuum of) affine Noetherian algebras lack such a filtration, and an answer to a question suggested by Irving and Small is derived.
Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over $k$ are finitely generated extensions of $k$. We give examples which show that $A\otimes_k K$ need not remain left Noetherian if the hypotheses are weakened.
Affineness on Noetherian graded rings, algebras and Hopf algebras
本文研究Noether分次环及Hopf代数的有限生成性，证明零次部分有限生成即可推出整体有限生成，并应用于分次Hopf代数及其辫子Hopf代数。
A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.
Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb{N}$-graded rings of finite global dimension; in particular, we see that a noetherian Arti…
It is proved in the paper that a Noetherian residually finite dimensional Hopf algebra is a flat module over any right Noetherian right coideal subalgebra. In the case of Hopf subalgebras we get faithful flatness. These results are obtained by verifying the existence of the classical quotient rings of those algebras. It is also proved that the antipode of either right or left Noetherian residually finite dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.
Noetherian enveloping algebras of simple graded Lie algebras
本文证明了单$\mathbb Z^n$-分级李代数的泛包络代数为Noetherian时该李代数必有限维。
We prove a version of a theorem of Auslander for finite group coactions on noetherian graded down-up algebras.
It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi-Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi-Yau algebras. As an application, we show that if $S$ is a 3-dimensional noetherian cubic Calabi-Yau algebra and $σ$ is a graded algebra automorphism of $S$, then the…
It is shown that Golod-Shaferevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators, and that these algebras can be homomorphically mapped onto prime, Noetherian algebras with linear growth. It is also shown that Golod-Shafarevich algebras of a reduced number of relations cannot be nil.
In this note we classify the non-Noetherian generalized Heisenberg algebras H(f) introduced by Rencai Lü and Kaiming Zhao [Linear Algebra Appl., 2015]. In case the polynomial f has degree greater than 1, we determine all locally finite and also all locally nilpotent derivations of H(f) and describe the automorphism group of these algebras.
We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conject…
This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian, extending a result of Sierra and Walton on the Witt (or classical Krichever-Novikov) algebra. As a subsidiary result, which may be of independent interest, we show that if $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$ of finite codimension, then the noetherianity of $U(\mathfrak{h})$ is equivalent to th…
Weakly Noetherian Lie Algebra and the Sierra-Walton Conjecture
本文研究了弱Noether Lie代数的结构，并证明了其与Sierra-Walton猜想的关系，改进了Sierra和Walton的著名定理。
We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property.
On central characteristic ideals and quasi-Noetherian Leibniz algebras
本文定义了 Leibniz 代数的中心特征理想并给出相关必要条件，同时引入拟诺特 Leibniz 代数类，探讨其封闭性并刻画诺特 Leibniz 代数。
We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.
Actions of Taft Algebras on Noetherian Down-Up Algebras
本文分类了Taft代数在Noetherian分次down-up代数上的作用，确定了不变环的交换性、Artin-Schelter正则与Gorenstein性质，并指出省略半单假设会使相关文献结论与猜想失效。
$\nabla$-good filtration dimensions of modules and of algebras are introduced by Parker for quasi-hereditary algebras. These concepts are now generalized to the setting of standardly stratified algebras. Let $A$ be a standardly stratified algebra. The $\bar{\nabla}$-good filtration dimension of $A$ is proved to be the projective dimension of the characteristic module of $A$. Several characterizations of $\bar{\nabla}$-good filtration dimensions and $\barΔ$-good filtration dimensions are given for properly stratified algebras. Finally we give an application of these results to the global dimens…
We solve the isomorphism problem for non noetherian down-up algebras $A(α,0,γ)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for $γ= 0$ or quantum versions of the Weyl algebra $A_1$ for non zero $γ$. In particular we obtain that no other down-up algebra is isomorphic to the monomial algebra $A(0,0,0)$. We prove in the second part of the article that this is the only monomial algebra within the family of down-up algebras. Our method uses homological invariants that determine the shape o…
In the past, it has been shown that the Leavitt path algebra $L(E)=L_K(E)$ of a graph $E$ over a field $K$ is left and right noetherian if and only if the graph $E$ is finite and no cycle of $E$ has an exit. If $Q(E)=Q_K(E)$ denotes the regular algebra over $L(E),$ we prove that these conditions are further equivalent with any of the following: $L(E)$ contains no infinite set of orthogonal idempotents, $L(E)$ has finite uniform dimension, $L(E)$ is directly finite, $Q(E)$ is directly finite, $Q(E)$ is unit-regular, and a few more equivalences. In addition, if the involution on $K$ is positive …
A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual coalgebra. If $A$ is Artin-Schelter regular, then the local cohomology of $A$ is isomorphic to a shift of twisted bimodule ${}_1C_{σ^*}$ with $σ$ a coalgebra automorphism. This yields that the balanced dualinzing complex of $A$ is a shift of the twisted bimodule ${}_{σ^*}A_1$. If $σ$ is an inner automorphism, then $A$ is Calabi-Yau.
Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group …
A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also refle…
Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is well-known: If $S$ is strongly-graded and $G$ is a polycyclic-by-finite group, then $S_e$ being noetherian implies that $S$ is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain…
Tilting complexes and codimension functions over commutative noetherian rings
本文在交换诺特环的导出范畴中，利用局部上同调构造与 sp-滤过关联的 silting 对象，并刻画其成为 tilting 的条件，特别与余维数函数及 Cohen-Macaulay 环同态像相关，同时给出 cosilting 对偶结果。
Noetherian criteria for dimer algebras
本文针对环面上的非退化 dimer 代数，通过循环收缩证明了 Noetherian 性、中心 Noetherian 性、非交换 crepant 解消、箭的完美匹配性质及角环同构等五个条件的等价性，为 dimer 代数的结构分类提供了统一判据。
We determine when a generalized down-up algebra is a Noetherian unique factorisation domain or a Noetherian unique factorisation ring.
This article is about equationally Noetherian and weak equationally Noetherian property of Ershov algebras. Here we show two canonical forms of the system of equations over Ershov algebras and two criteria of equationally Noetherian and weak equationally Noetherian properties.
In a series of papers \cite{BRV1}, \cite{BRV2}, \cite{BRV3} we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "$\bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $Γ$," i.e., a non-identity of the representable algebra $A$ arising from $Γ$, whose ideal of evaluations in $…
Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.
We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded l…
We classify the primitive ideals of generalized down-up algebras
We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_ω$-compact or $\mathbf{u}_ω$-compact. Also we solve the problem of geometric equivalence in the class of boolean algebras with distinguished elements.
The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelma…
Noetherianity of Diagram Algebras
本文针对Brauer代数、分块代数等相关图代数的线性范畴，在不对各类参数设限的条件下，证明了其局部Noether性质，拓展了图代数的结构理论研究。
We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\Ext^1_H(k,k)\neq 0$. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.
This short survey article reviews current understand- ing of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problems are listed.
This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (non…
We show that free algebras of the variety of algebras generated by the Witt algebra $W_n$, the left-symmetric Witt algebra $L_n$, and the symplectic Poisson algebra $P_n$ can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that $W_n$, $L_n$, $P_n$, and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian.
Cellular Noetherian algebras with finite global dimension are split quasi-hereditary
本文证明了有限整体维数的细胞诺特代数在正则交换诺特环上是分裂拟遗传的且结构唯一，给出了整体维数公式和有限维数上界，并应用于Schur代数。
This note concerns the still open question of representability of Noetherian PI-algebras. Extending a result of Rowen and Small (with an observation of Bergman) that every finitely generated module over a commutative Noetherian ring containing a field is representable, we provide a representability machinery for a Noetherian PI-algebra $R$ containing a field, which includes the case that $R$ is finite (as a module) over a commutative subalgebra isomorphic to $R/N$. We construct a family of non-representable PI-algebras demonstrating the sharpness of these results, as well as of some well known…
Noetherianity of twisted Zhu algebra and bimodules
本文证明了一大类自然顶点算子代数及其模的Zhu代数、双模及其$g$-扭类似具有Noetherian性，为应用非交换代数几何方法研究其表示理论与融合规则提供了可能。
We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine th…
We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.
Noetherian pointed Hopf algebras are affine
该研究引入词上的约化序与约化因子分解概念，证明域上右或左诺特有点Hopf代数是仿射的，部分回应了Wu和Zhang提出的诺特Hopf代数仿射性经典问题。
Canonical Filtrations of Finite-Dimensional Algebras
本文利用 GIT 最优失稳单参数子群定义有限维结合代数与李代数的典范滤过，刻画半单性并给出计算方法及相伴分次代数的结构结果。
Telescope conjecture for t-structures over noetherian path algebras
本文证明了诺特路径代数导出范畴中同伦破碎t-结构均为紧生成，并用素谱到非交叉划分滤过偏序集的同态完整刻画了紧生成t-结构，正则环下还描述了宽子范畴。
Let $k$ be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module $V$ is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules $V$ whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that $V$ induces a finite complex of filtered modules, and use it as well as a result of Church an…
We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.
Let $\mathcal A$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal A$ using subsets of the Rosenberg spectrum $\mathfrak Spec(\mathcal A)$. Along the way, we also develop results in local algebra with respect to the category $\mathcal A$ that we believe to be of independent interest.
In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain $S$-Noetherian modules with the following consequence in commutative algebra: If a module $M$ over a commutative ring is $S$-Noetherian with respect to a multiplicative set $S$ that contains no zero-divisors for $…
Let $A \subset H$ be a finite centralizing extension of noetherian Hopf algebras, and let $X$ denote the group of characters of $H$ that restrict to the augmentation map on $A$. Our main results provide necessary and sufficient conditions for the fibers of the canonical surjection from $spec H$ onto $spec A$ to coincide with the $X$-orbits in $spec H$. In particular, all of the fibers are $X$-orbits if and only if the fiber over the augmentation ideal of $A$ is an $X$-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.
On the geometry of spaces of filtrations on local rings
研究局部环上滤法空间的几何特性，引入一种度量并探讨其几何性质，结合牛顿-奥科诺夫体分析滤法空间结构，并研究对数canonical阈值函数的半连续性。
On Noetherian pointed Hopf algebras
本文针对点态霍普夫代数相关猜想，提出多循环-有限群上后Nichols代数的诺特性等价于仿射性与有限Gelfand-Kirillov维数，并给出部分支持结果及相关分析。
A noetherian Hopf algebra has bijective antipode
该研究证明，若任意域上的霍普夫代数的左或右正则模具有有限一致维数，则其具有双射反极，还解决了2006年斯克里亚宾提出的诺特霍普夫代数均具双射反极的猜想。
We define canonical and $n$-canonical modules on a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$-canonical modules, we generalize a theorem on $(n,C)$-syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a non-commutative version of Aoyama's theorem which states that a canonical module descends with respect to a flat local homomorphism. We also prove the codimension two-argument for modules over a coherent sheaf of algebras with a $2$-canonical module, generalizing a result o…
We show that many noetherian Hopf algebras A have a rigid dualising complex R with R isomorphic to ^νA^1 [d]. Here, d is the injective dimension of the algebra and νis a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call νthe Nakayama automorphism of A. We prove that ν= S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are s…
A Noetherian Hopf algebra is affine iff its Hopf coradical is affine
本文通过引入非交换约化序、因子分解理论和广义提升方法，建立了诺特Hopf代数仿射性的充要条件，即左或右诺特Hopf代数仿射当且仅当其Hopf余根仿射，从而推进了Wu-Zhang问题的解决。
We study associative graded algebras which have a ``complete flag'' of cyclic modules with linear free resolutions, i.e., algebras over which there is a cyclic Koszul module with every admissible number of relations (from zero up to the number of generators of the algebra). Commutative algebras with the same property has been studied in several papers by A. Conca and others. Here we present a non-commutative version. We introduce the concept of Koszul filtration in non-commutative algebras and study its connections with Koszul algebras and algebras with quadratic Groebner bases. Also, here are…
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