Souvik Dey, Ryo Takahashi
Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In this paper, we study n-torsionfree modules in the sense of Auslander and Bridger, by comparing them with n-syzygy modules, and modules satisfying Serre's condition (S_n). We mainly investigate closedness properties of the full subcategories of mod R consisting of those modules.
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Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories
本文利用概形子集及特殊化闭子集分类了与线丛张量积封闭的无挠类和挠类，并据此推广 Gabriel 分类，刻画无挠类的 Serre 子范畴。
Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R-modules. In the present paper, we first provide various sufficient (and necessary) conditions for a full subcategory of mod R to be a Serre subcategory, which include several refinements of theorems of Stanley and Wang and of Takahashi with simpler proofs. Next we consider when an IKE-closed subcategory of mod R is a torsionfree class. We investigate certain modules out of which all modules of finite length can be built by taking direct summands and extensions, and then we apply it to show that the IKE…
Auslander and Bridger introduced the notions of n-spherical modules and n-torsionfree modules. In this paper, we construct an equivalence between the stable category of n-spherical modules and the category of modules of grade at least n, and provide its Gorenstein analogue. As an application, we prove that if R is a Gorenstein local ring of Krull dimension d>0, then there exists a stable equivalence between the category of (d-1)-torsionfree R-modules and the category of d-spherical modules relative to the local cohomology functor.
Let R be a commutative noetherian ring. We consider the question of when n-syzygy modules over R are n-torsionfree in the sense of Auslander and Bridger. Our tools include Serre's condition and certain conditions on the local Gorenstein property of R. Our main result implies the converse of a celebrated theorem of Evans and Griffith.
Let A be a Noetherian local ring with canonical module K. We characterize A when K is a torsionless, reflexive, or q-torsionfree module. If A is a Cohen-Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby's result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasi-normal rings, introduced by W. V. Vas…
Over a Cohen-Macaulay ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After describing these categories, we compare them and consider which properties they inherit from the maximal Cohen-Macaulay modules. We then consider some further properties of these classes and how they interact with the entire module category.
Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.
Let $Λ$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $Λ$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate totally reflexive modules and reducing Gorenstein dimension. Next, we introduce homological invariants for $Λ$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $Λ$ is commutative and local. Using it, we consider how upper reducing projective dimension relates to…
Let R be a Gorenstein local ring which is locally a hypersurface on the punctured spectrum. In this paper, we classify thick subcategories of the bounded derived category of finitely generated R-modules. Moreover, using this classification, we also classify thick subcategories of finitely generated R-modules, and find out the relationships with thick subcategories of Cohen-Macaulay R-modules.
Torsion pairs in subcategories of modules over a commutative ring
本文针对交换诺特环$R$上有限生成$R$-模范畴的特定全子范畴，研究其中挠对的分类问题。
In this paper, we consider subcategories consisting of the extensions of modules in two given Serre subcategories to find a method of constructing Serre subcategories of the category of modules. We shall give a criterion for this subcategory to be a Serre subcategory.
Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Kr…
Contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras. In this paper we study contravariantly finite resolving subcategories over commutative rings. The main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian Gorenstein local ring; in fact there exist only three ones. Our method to obtain this classification also recovers as a by-product the theorem of Christensen, Piepmeyer, Striuli and Takahashi concerning the relationship betwe…
We study two subcategories of the category of artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring.
On the finiteness of radii of resolving subcategories
本文研究交换诺特环上有限生成模范畴中分解子范畴相对于固定半对偶化模的半径有限性，并部分证明了 Dao-Takahashi 猜想：包含典范模和有限内射维数非 MCM 模的分解子范畴半径必为无限。
Contravariantly finite resolving subcategories over commutative local rings are trivial ones
该研究证明了亨泽尔局部环上反变有限预解子范畴的分类，消除了Takahashi定理的Gorenstein假设，还推导了有限CM型Cohen-Macaulay局部环的相关性质，强化了Takahashi猜想。
Let R be a commutative Noetherian local ring with residue field k. Let X be a resolving subcategory of finitely generated R-modules. This paper mainly studies when X contains k or consists of totally reflexive modules. It is proved that X does so if X is closed under cosyzygies. A conjecture of Dao and Takahashi is also shown to hold in several cases.
Let $R$ be a commutative ring. A full additive subcategory $\C$ of $R$-modules is triangulated if whenever two terms of a short exact sequence belong to $\C$, then so does the third term. In this note we give a classification of triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thick subcategories (triangulated subcategories closed under direct summands), wide subcategories (abelian subcategories closed under extensions) and Serre subcategories (wide subcategories closed …
Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher-dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite $p$-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of Cohen-Macaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of Cohen-Macau…
Contravariantly infinite resolving subcategories
本文研究了局部完全交換环上的反变无限解析子范畴的定义与判断标准。
Ext modules related to syzygies of the residue field
本文研究了交换noetherian环中模M的投射维数嵌入与其第n个合系的(n+1)-无挠性之间的关系，并计算了局部环剩余域k的合系相关Ext模的向量空间维数。
Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we characterize simple hypersurface singularities of type $(\a_1)$ as comple…
Let R be a Cohen-Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving subcategories of mod R in terms of specialization-closed subsets of Spec R. We give a classification of the resolving subcategories closed under tensor products and transposes. Under restrictive hypotheses, we also give better classification results.
We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving subcategories containing maximal Cohen-Macaulay modules are in bijection with grade consistent functions and thus are the precisely the dominant resolving subcategories.
Thick subcategories over weakly symmetric algebras with radical cube zero
本文分类了根立方零弱对称代数的厚子范畴，发现谱半径为2时可由Serre子范畴分类，否则仅平凡厚子范畴，并首次构造了非完全交的交换诺特Gorenstein优势局部环。
This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring.
Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the semidualizing modules, we define and study new classes of modules and homological dimensions and investigate the relations between them. In parallel, we obtain some necessary and sufficient condition for $ C $ to be dualizing.
Supports of intersections of thick subcategories and proxy smallness
研究厚子范畴交集支持的条件，探讨代理小子范畴的特征及特定厚子范畴的分类。
The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.
Let $R$ be a commutative noetherian ring and let $\frak a$ be an ideal of $R$. In this paper, we study a certain condition, namely $C_{\frak a}$, introduced by Aghapournahr and Melkersson, on the extension of two subcategories of $R$-modules. We extend and generalize some of the main results of Yoshizawa [Y1,Y2]. As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomolo…
For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking direct sums.
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。
Classifying thick subcategories over a Koszul complex via the curved BGG correspondence
本文利用弯曲 BGG 对应及完美弯曲 dg 模的厚张量子模分类，刻画了正则环中任意元素列表的 Koszul 复形上有界 dg 模导出范畴的厚子范畴，并恢复 Stevenson、Hopkins–Neeman 等结果。
Extension-closed subcategories over hypersurfaces of finite or countable CM-representation type
本文在有限或可数CM-表示型假设下，完全分类了超曲面上至多二维时CM_0(R)的扩张闭子范畴及任意维数时奇点范畴子范畴D^{sg}_0(R)的扩张闭子范畴。
Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing subcategori…
Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I, \rm{K_R})}$. We prove that if $R$ has multiplicity at most $6$, then $I$ is isomorphic to $R$ or $\rm{K_R}$ as an $R$-module, once $I\otimes_RI^{\vee}$ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.
Approximation of subcategories by abelian subcategories
该研究针对交换环与弱投射正则理想，证明了$L_0$-完备$R$-模范畴的泛性质，明确其为包含所有$I$-进完备$R$-模的最小满正合阿贝尔子范畴，推进了相关代数结构的刻画。
Given a good $n$-tilting module $T$ over a ring $A$, let $B$ be the endomorphism ring of $T$, it is an open question whether the kernel of the left-derived functor $T\otimes^L_B-$ between the derived module categories of $B$ and $A$ could be realized as the derived module category of a ring $C$ via a ring epimorphism $B\rightarrow C$ for $n\ge 2$. In this paper, we first provide a uniform way to deal with the above question both for tilting and cotilting modules by considering a new class of modules called Ringel modules, and then give criterions for the kernel of $T\otimes^L_B-$ to be equival…
In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of finitely presented modules when R is a quotient of a regular commutative coherent ring by a finitely generated ideal. This includes all finitely presented algebras over a principal ideal domain, as well as polynomial rings on infinitely many variable over a PID. The classification is in terms of subsets of Spec R, and depends heavily on Thomason's classification o…
Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\cS$ be an arbitrary Serre subcategory of $R$-modules and let $\cN$ be the subcategory of finitely generated $R$-modules. In this paper, we study $\cN\cS$-$\frak a$-cofinite modules with respect to the extension subcategory $\cN\cS$ when $\dim R/\frak a\leq 2$. We also study $\frak a$-cofiniteness with respect to a new dimension.
Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\mathcal{S}$ be an arbitrary Serre subcategory of $R$-modules satisfying the condition $C_{\frak a}$ and let $\mathcal{N}$ be the subcategory of finitely generated $R$-modules. In this paper, we define and study $\mathcal{NS}$-$\frak a$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$ as an generalization of the classical notion, namely $\frak a$-cofinite modules. For the lower dimensions, we show that the classical results of $\frak a$-cofiniteness hold for the new notion.
Cofiniteness of local cohomology modules and subcategories of modules
在交换 Noether 环上，本文证明当 $H_I^i(R)$ 均为 $I$-余有限、$I$ 外局部化正则且 $I$-余有限模构成 Abel 范畴时，任意有限生成模的所有局部上同调模均为 $I$-余有限。
Classifying KE-closed subcategories over a commutative noetherian ring
本文通过引入n-Bass函数，建立了交换诺特环上KE-闭子范畴与2-Bass函数间的双射，从而完成分类。
Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$, $\mathcal{S}$ a Serre subcategory of $R$-modules satisfying the condition $C_\mathfrak{a}$ and $\mathcal{N}$ the subcategory of finitely generated $R$-modules. In this paper, we continue the study of $\mathcal{NS}$-$\mathfrak{a}$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$, show that some classical results of $\mathfrak{a}$-cofiniteness hold for $\mathcal{NS}$-$\mathfrak{a}$-cofiniteness in the cases $\mathrm{dim}R=d$ or $\mathrm{dim}R/\mathfrak{a}=d-1$, where $d$ is a positive integer. We also s…
This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and M.Tousi (2010). Let R be a commutative Noetherian ring, I, J be two ideals of R and M be an R-module. Let a\in W(I; J) and t \in N_0 be such that Ext^t_R(R/a,M)\in S and Ext^j_R(R/a,H^i_I;J(M))\inS for all i < t and all j>=0. Then for any submodule N of H^t_I;J(M) such that Ext^1_R(R/a;N)\in,we obtain HomR(R=a;H^t_I;J(M)/N)\inS.
Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules
在Cohen-Macaulay环上的诺特代数上，证明有限生成Gorenstein投射模构成遗传cotorsion对的左半部分，并回答Takahashi的问题，刻画左弱Gorenstein性质。
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…
Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated R-modules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci and establishing an analogue of the notion of a level in a triangulated category which has been introduced by Avramov, Buchweitz, Iyengar and Miller. As an application, we prove a result on the dimension of the nonfree locus of a resolving subcategory having only countably many nonisomorphic indecomposable mo…
We classify certain resolving subcategories of finitely generated modules over a commutative noetherian ring R by using integer-valued functions on Spec R. As an application we give a complete classification of resolving subcategories when R is a locally hypersurface ring. Our results also recover a "missing theorem" by Auslander.
A generalization of the dimension and radius of a subcategory of modules and its applications
本文从有限生成模范畴满子范畴的维数与半径有限性出发，研究奇异轨迹及 Tor、Ext 零化子，并恢复及推广相关定理。
We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.
Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and $C_{\frak b}$ conditions. As an conclusion we show that over a artinian local ring, every Serre subcategory satisfies $C_{\frak a}$ condition. We also show that $\Ss_{\frak a}$ is closed under extension of modules. If $\Ss$ is a torsion subcategory, we prove that $S$ satisfies $C_{\frak a}$ condition. We prove that $C_{\frak a}$ condition can be transferred via rin…
Let A be a commutative noetherian ring. In this paper, we interpret localizing subcategories of the derived category of A by using subsets of Spec A and subcategories of the category of A-modules. We unify theorems of Gabriel, Neeman and Krause.
We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type $A$ cluster categories of Holm-Jorgensen, Fisher and Paquette-Yildirim. As a consequence, $\mathcal{C}_2$ has cluster tilting subcategories modelled by certain triangulations of the (completed) $\infty$-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategor…
We investigate the properties of categories of G_C-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G_C-flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G_C-flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G_C-flat R-modules yield only the modules in the original subcategories.
Exact subcategories, subfunctors of $\operatorname{Ext}$, and some applications
本文在正合范畴中利用数值函数可加性与子范畴限制识别 Ext 的子函子，并研究其在交换局部环上的应用，涵盖正则性检测与 Ulrich 模理解。
We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.
On the Auslander-Bridger-Yoshino theory for complexes of finitely generated projective modules
本文基于 Auslander-Bridger 稳定模理论与 Yoshino 的稳定复形理论，引入了 *torsionfreeness 和 *reflexivity 等概念的高阶版本，并证明了有限生成投射模复形的 Auslander-Bridger 逼近定理。
For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick subcategories of finitely generated modules over local complete intersections and produce generators for the category of coherent sheaves on a separated noetherian scheme with an ample family.
IE-closed subcategories of commutative rings are torsion-free classes
本文证明交换noetheian环上有限生成模范畴中在像和扩张下封闭的子范畴（IE闭子范畴）必在子模下封闭且为挠自由类，并完全刻画了其成为Serre子范畴的右noetherian环条件。
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