Noncommutative generalizations of theorems of Cohen and Kaplansky

Endomorphism rings of finite global dimension

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $Λ$ of the form $Λ= End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $Λ$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $Spec R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $C$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $Λ$ with finite global dimension and which is maximal Cohen--Macaulay…

Noncommutative Knörrer's periodicity theorem and noncommutative quadric hypersurfaces

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Knörrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory since it reduces the number of variables in computing the stable category $\underline{\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Knörrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computi…

Commutative Local Rings whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian lo…

非交换Crepant解的对数中心是Kawamata对数终端：关于Stafford和van den Bergh论文的注记

Log Centres of Noncommutative Crepant Resolutions are Kawamata Log Terminal: Remarks on a paper of Stafford and van den Bergh

本文证明了满足特定极大Cohen-Macaulay模和全局维数条件的有限生成素代数，其中心与分歧除子构成的配对是Kawamata对数终端的。

Pre-resolutions of noncommutative isolated singularities

We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version of quasi-resolutions introduced by Qin-Wang-Zhang. We prove that right quasi-resolutions for noetherian bounded below and locally finite graded algebra with right injective dimension 2 are always Morita equivalent. When we restrict to noncommutative quadric hypersurfaces, we prove that a noncommutative quadric hypersurface, which is a noncommutative isolated singularity, always admits a right pre-resolution. Besides, we provide a method to verify whether a n…

Finitary Galois extensions over noncommutative bases

We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a Frobenius Hopf algebroid. We obtain generalizations of various theorems in Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by the Brzezinski-Militaru Theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's Theorem and t…

Generalized Knörrer's Periodicity Theorem

Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $f\in A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformation to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if $B/(g)$ is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories $\underline{\text{mcm}}\,A/(f)\cong\underline{\text{mcm}}\,(A\otimes B)/(f+g)$ of…

正则局部环的纯子环、自同态环与Frobenius态射

Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

本文研究正则局部环纯子环的自同态环，证明其在极大Cohen-Macaulay条件下为非交换crepant消解，并利用Frobenius态射构造Morita等价的另一消解，同时探讨非交换Frobenius态射与整体维数有限性的关系，并应用于wild商奇点的强F-正则性判定。

An analogue of Hilbert's Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$

Gorenstein modifications and $\mathbb{Q}$-Gorenstein rings

Let $R$ be a Cohen--Macaulay normal domain with a canonical module $ω_R$. It is proved that if $R$ admits a noncommutative crepant resolution (NCCR), then necessarily it is $\mathbb{Q}$-Gorenstein. Writing $S$ for a Zariski local canonical cover of $R$, then a tight relationship between the existence of noncommutative (crepant) resolutions on $R$ and $S$ is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of…

Trace ideals, normalization chains, and endomorphism rings

In this paper we consider reduced (non-normal) commutative noetherian rings $R$. With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication with scalars in an integral extension of $R$. Using results of Greuel and Knörrer this yields a characterization of plane curves of finite Cohen--Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings $(R,\mathfrak{m})$ such that that their normalization is isomorphic to the endomorphism ring $\mathrm{End}_R(\mathfrak{m})$: we gi…