Nicholas Baeth
We determine, up to isomorphism, the indecomposable maximal Cohen-Macaulay modules over certain complete one-dimensional local rings of finite Cohen-Macaulay type. We then investigate the direct sum relations of maximal Cohen-Macaulay modules over non-complete rings of finite Cohen-Macaulay type and give a Krull-Schmidt theorem for such rings.
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Krull--Gabriel dimension and Cohen-Macaulay representations
本文针对R-序Λ的极大Cohen-Macaulay模稳定范畴的函子范畴，研究其Krull–Gabriel维数，结合Herzog与Krause的不存在定理，证明了Gorenstein序及不可数表示型孤立奇点序的相关维数性质。
We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is $0$ if the hypersurface is of finite Cohen--Macaulay representation type and that is $2$ if the hypersurface is of countable but not finite Cohen--Macaulay representation type.
The Krull-Remak-Schmidt decomposition of commutative group algebras
本文研究了域$k$和有限阿贝尔群$G$构成的群代数$k[G]$，通过提出循环坐标环这一几何等价概念，成功给出了该群代数的Krull-Remak-Schmidt分解。
Injective modules over pseudo-krullian orders
本文引入伪Krull序，研究其模范畴关于伪同构的Serre商，描述了商范畴中的内射对象及全局同调维数，推广了I. Beck关于Krull环的结论，并确立了特定条件下极大Cohen-Macaulay模范畴的全局同调维数。
In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.
We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras. In the first part of this thesis we use a K-theory recipe of Thomason to classify these subcategories. This recipe when applied to the category of finite p-local spectra gives a refinement of the ``chromatic tower''. This refinement has some interesting consequences. In particular, it gives new evidence to a conjecture of Frank Adams that the thick subcategory C_2 can be generated b…
This paper main goal is to measure the defect of Cohen-Macaulayness, Gorensteiness, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components.
Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson.
Categorical Krull-Remak-Schmidt for triangulated categories
本文证明了三角化范畴的范畴Krull-Remak-Schmidt定理，即若两个由连通三角化范畴构成的直和等价，则其连通分量间存在双射且逐一等价，并给出了相关示例。
Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on projective curves. In these Auslander-Reiten theories, the number `2' is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple functors of projective dimension `2'. For another, Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most beautiful situation in representation theory, which is closely r…
Arithmetic Invariant Theory of Reductive Groups
本文定义了交换含幺环上的线性reductive group，研究了其不变量环的Cohen-Macaulay性质，并证明了Krull维数为2的正则环上reductivity概念的等价性。
Restricted injective dimensions over Cohen-Macaulay rings
本文证明有限 Krull 维数 Cohen-Macaulay 环上大小受限内射维数一致，并借助倾斜理论提出 Cohen-Macaulay Hom 内射维数新定义，证明其构成完美余挠对的右半部分。
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\ldots,x_{n-1}]$. Under the hypothesis that $n$ is prime we : 1) compute the regularity index of $R/I(G)$; 2) compute the Castelnuovo-Mumford regularity when $R/I(G)$ is Cohen-Macaulay; 3) prove that the circulant graphs with $S=\{1,\ldots,s\}$ are sequentially $S_2$ . We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension $2$ and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity.
Polynomial Extensions and Localization of Non-Noetherian Cohen--Macaulay Rings
研究非诺特环中Hamilton-Marley意义下Cohen-Macaulay（HMCM）性质的多项式扩张保持性，以及无挠模的局部化现象，给出了稳定凝聚环有限弱整体维数下的保持性，并刻画了局部化保持无挠性的条件。
The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by …
Let A be a Noetherian ring. It is shown that any finite A--module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying (S_2) which is over a local ring with Cohen--Macaulay formal fibres.
On Matlis reflexive modules
本文研究交换环上模的Matlis自反性，证明Matlis自反模构成Krull-Schmidt范畴，并指出在诺特环上无无限直和是其核心特征，同时给出若干小例子的分类。
In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over a Cohen-Macaulay local ring, uniform bounds on the Betti numbers of a Cohen-Macaulay module in terms of dimension and multiplicity, and similar bounds for the Bass numbers of a finitely generated module…
Govorov--Lazard and finite deconstructibility for Gorenstein and restricted homological dimensions
本文在 Cohen–Macaulay 环及有限 Krull 维数交换诺特环上证明受限投射/平坦维数模类的有限可解构性与 Govorov–Lazard 性质，并推广 Holm 关于平衡大 Cohen–Macaulay 模的结构结果。
We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring $R$ through the factorization theory of the corresponding monoid $T(R)$. Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of $T(R)$. The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations--a natural class of monoids serving as combinatorial models for the factorization theory of $T(R)$. As a consequence, the monoid $T(R)$ is transfer…
Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.
Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that $Γ_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective $R$-module $G$. Let $(R,\frak m)$ be a local ring and $M$ be a finitely generated $R$-module. We show that ${\rm Gid}{\bf R}Γ_{\frak m}(M)<\infty$ if and only if ${\rm Gid}_{\hat{R}}(M\otimes_R\hat{R})<\infty$. We also show that if ${\rm Gfd}_R{\bf R}Γ_{\frak m}(M)<\infty$, then ${\rm Gfd}_RM<\infty$. Let $(R,\frak m)$ be a Cohen-Macaulay local ring and $M$ be a Cohen-Macaulay…
A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity-II
本文推广了此前针对CM局部环$\u200b\mathfrak{m}$-进滤过提出的$T$-分裂序列概念，研究相关函数与序列性质，并将技术拓展到整闭包滤过，对非分歧CM局部环证明了Brauer-Thrall-II的一个版本。
Cohen--Macaulayness of $\mathrm{Inc}(\mathbb{N})$-Invariant Chains of Edge Ideals
本文针对五类经典图族，研究Inc(ℕ)-不变边理想链对应的商环的Cohen--Macaulay性，给出线图的完整分类，证明三类图的链无条件Cohen--Macaulay，确定循环图的Cohen--Macaulay范围并提出完整分类猜想。
Intersection Theorem for DG-modules
本文在具有有界上同调的交换诺特局部 DG-环上研究 DG-模的交截定理，并利用它证明若干经典定理的 DG 版本，同时完全解决了 Minamoto 猜想。
Auslander-Reiten annihilators
本文研究 Ext 模不消失时其零化子，推广 Auslander-Reiten 猜想，并在多类环上证明，同时为 Tachikawa 猜想提供证据。
This paper deals with computing the global dimension of endomorphism rings of maximal Cohen--Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type $A_n$ for all $n$, $D_n$ for $n \leq 13$ and $E_{6,7,8}$ and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [Dao-Faber-Ingalls]. Moreover, we determine the centre of an endomorphism ring of a MC…
Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of smal…
Quasi-injective dimension
本文引入拟内射维数以推广内射维数，并证明其在局部环上关于深度、Cohen-Macaulay、Gorenstein 性质及与拟投射维数等价性的多个经典结论。
Affine semigroups of maximal projective dimension-II
该研究聚焦极大投射维数仿射半群，给出单纯半群为Buchsbaum的充要条件、几类半群的刻画，证明部分半群满足扩展Wilf猜想，还发现Betti型可不受嵌入维数有界约束，并探索满足Arf性质的子幺半群。
A full subcategory of modules over a commutative ring $R$ is wide if it is abelian and closed under extensions. Hovey \cite{wide} gave a classification of wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialisation closed subsets of $\Spec(R)$. We use this classification theorem to study K-theory and Krull-Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide subcategory $\W$ of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcate…
$2$-Periodic complexes over regular local rings
本文研究正则局部环上2-周期复形的同伦范畴与导出范畴的等价性，并在完备情形下证明有限长度上同调子范畴为Krull-Schmidt且具AR三角，同时计算维数1时的AR箭图。
In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi-Yau type property. If C is such a category, we say that C is Calabi-Yau with dimension at most one. We extend the notion of Calabi-Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categor…
The local homological structure of generalized splines
本文研究了广义样条的局部同调结构，通过两个精确序列揭示其与局部环同调机器的协调性，并证明了特定条件下广义样条模块的自由性。
On the Matlis Reflexive Modules
本文系统研究Matlis自反模，证明其构成Krull-Schmidt范畴，并在完备Noetherian局部环上等价于minimax模，建立了完全闭包性质，综述了相关应用与开放问题。
Serre depth and local cohomology
本文提出塞雷深度作为同调不变量，研究其在不同环上的性质及应用，包括等维性、稳定性等贡献。
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