Zhaoyong Huang, Osamu Iyama
We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.
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Auslander conditions and tilting-like cotorsion pairs
研究满足Auslander条件的模类的同调性质，证明其预解性与Gorenstein投射模的包含关系，并给出Artinian代数下类倾斜余挠对与Gorenstein性质的等价判据。
Motivated by its links to $τ$-tilting theory, we introduce a generalization of cotorsion pairs in module categories. Such pairs are also linked to co-t-structures in corresponding triangulated categories, and to cotorsion pairs in certain extension closed (and hence extriangulated) subcategories, which we call two-term categories.
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.
Two classes $\mathcal A$ and $\mathcal B$ of modules over a ring $R$ are said to form a cotorsion pair $(\mathcal A, \mathcal B)$ if $\mathcal A={\rm Ker Ext}^1_R(-,\mathcal B)$ and $\mathcal B={\rm Ker Ext}^1_R(\mathcal A,-)$. We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of $R$. We show that $\Findim R<\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\mathcal A, \mathcal B)$ in $\mathrm{Mod} R$: the relative projective dimension of $\A$ with respect to itself, and…
Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(ω, \Ext^i_{R}(ω,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(ω, \Tor_i^{S}(ω,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $ω$.
Let $(\mathcal{A,B})$ be a complete and hereditary cotorsion pair in the category of left $R$-modules. In this paper, the so-called Gorenstein projective complexes respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$. As applications, we prove that both Gorenstein projective modules with respect to cotorsion pairs and Gorenstein projective complexes with respect to cotorsion pairs possess of stability.
In the paper we first construct a new cotorsion pair, in the category of chain complexes, from two given cotorsion pairs in the category of modules, and then we consider completeness of such pairs under certain conditions.
Gluing of cotorsion pairs via recollements of abelian categories
本文通过abelian范畴的recollement构造新的cotorsion对，并研究其heredity和completeness，应用于Morita环中cotorsion对的构造。
Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples
本文研究了环R中平衡对与余 torsion 三元组的关系，证明了右几乎格罗蒙德环的条件，并通过无限维翻转理论分类了所有余 torsion 三元组。
Higher Cotorsion Pairs on Image-Nilpotent \(η\)-Extensions
本文针对图像幂零 η-扩张引入一致图像幂零性，证明右 n-余挠对的提升定理与完备性定理，并统一多种环与范畴构造。
Relative Auslander--Gorenstein Pairs
本文引入相对Auslander--Gorenstein对与相对Auslander对的概念，用倾斜-余倾斜模的相对支配/余支配维数刻画其性质，推广统一已有结果并给出范畴对应应用与实例。
In this paper, we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module. Applications go in three directions. The first is to characterize when the little finitistic dimension is finite. The second is to obtain equivalent formulations for a Wakamatsu tilting module to be a tilting module. The third is to give some new characterizations of Gorenstein rings.
For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of $\mathcal{T}$, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that the Gabriel …
Enochs Conjecture for cotorsion pairs and more
该研究针对一般情形下仍未解决的Enochs猜想，证明了由固定n<ω的ℵₙ-表现模构成的类对应的Filt(S)类满足猜想，还证明所有由模集合生成的Filt(S)类满足猜想与ZFC相容，推进了该猜想的证明进程。
Cotorsion pairs and Tor-pairs over commutative noetherian rings
本文对交换诺特环上的遗传余挠对进行分类，利用环谱上的整值函数和局部深度条件描述左类，并通过对偶对应到遗传 Tor-对。
Construction of $n$-cotorsion pairs in some abelian categories
本文在逗号范畴、平凡环扩张、Morita环及形式三角矩阵环中构造左（右）n-余挠对，研究其遗传性质，改进并提升了相关已知结果。
Salce \cite{MR565595} introduced the notion of a \emph{cotorsion pair} of classes of abelian groups, and asked whether every such pair is \emph{complete} (i.e., has enough injectives and projectives). We prove that it is consistent, relative to the consistency of Vopěnka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof-Shelah \cite{MR2031314}, this shows that Salce's Problem is independent of the ZFC axioms (modulo the consistency of VP).
$n$-cotorsion pairs over formal triangular matrix rings
针对形式三角矩阵环$Λ$，通过研究其上特殊类，从环$A$和$B$的左/右$n$-余挠对构造出$Λ$的左/右$n$-余挠对，并举例验证结果。
A parity obstruction to completeness of object cotorsion pairs
本文构造了一个完备的理想余挠对，其对象余挠对既非特殊预覆盖也非特殊预包络，通过奇偶性障碍否定了问题 29。
Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity
研究Grothendieck范畴中余挠对相关的余导出与反导出范畴，通过构造自然等价、证明嵌套余挠对的伴随等，探讨平坦型余挠对，证明特定余挠对相关范畴相同的条件，还讨论了相关示例及周期性猜想。
We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$ resolution is exactly the class of complexes with finite Gorenstein $\A$ dimension. This lets us give general techniques for computing Tate-Vogel cohomoloies of complexes with finite Gorenstein $\A$ dimension. As a consequence, relative cohomology groups for complexes with finite Gorenstein $\A$ dimension are investigated. Finally, the relationships between Gorenste…
We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(Φ(\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$ and $(\mathrm{Rep}(Q,\mathcal{A}),Ψ(\mathcal{B}))$ in $\mathrm{Rep}(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\widetilde{\mathcal{A}}, \ma…
Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion modules. In this note we extend this result to the class of all rings $R$ satisfying that each flat left $R$-module is filtered by totally ordered limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these ri…
An explicit self-dual construction of complete cotorsion pairs in the relative context
本文在环同态诱导的模范畴中，利用余诱导函子和有限分解维数条件，显式构造了完备余挠对，并推广到可数乘积、序数指标滤过及倾斜余挠对情形。
A higher dimensional Auslander-Iyama-Solberg correspondence
本文将 Auslander–Iyama–Solberg 对应推广到高维情形，用任意具有有限投射与内射维数的自正交模替代极小忠实投射内射模，建立相对 Auslander–Gorenstein 对与推广的预簇倾斜模之间的新双射，并应用于一般线性群的模表示论。
In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $Σ$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $Σ$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a…
We study the interplay between the notions of $n$-coherent rings and finitely $n$-presented modules, and also study the relative homological algebra associated to them. We show that the $n$-coherency of a ring is equivalent to the thickness of the class of finitely $n$-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to ${\rm Ext}^1_R(F,-)$ and ${\rm Tor}_1^R(F,-)$. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of $n$-coherent ring…
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
Serre duality, Mukai pairing and universal Auslander--Reiten triangle
本文研究了光滑正则dg代数中Serre双曲性与Mukai配对的关系，并构造了通用Auslander-Reiten三角形，提供了函数构造方法。
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。
Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ (resp. $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$) is finite, then the bounded homotopy category of $\mathscr{Y}$ (resp. $\mathscr{X}$) is contained in that of $\mathscr{X}$ (resp. $\mathscr{Y}$). As a consequence, we get that the right $\m…
Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm and}\ (\mathcal{B}_C(R),\mathcal{A}_C(R^{op}))$$ are coproduct-closed and product-closed duality pairs and both $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if $\mathcal{B}_C(R)$ is enveloping in $\Mod R$, then $\mathcal{A}_C(S)$ is enveloping in $\Mod S$. Then some…
L. Salce introduced the notion of a cotorsion pair (F,C) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (F,C) of classes of R-modules is cogenerated by a set, then it is complete. Recently Herzog, Fu, Asensio and Torrecillas developed the ideal approximation theory. In this article we look at a result motivated by the Eklof-Trlifaj argument for an ideal I when it is generated by a set of h…
Model structures on the category of Q-shaped modules
该研究开发了从AMod的余挠对构造Q-形模范畴上阿贝尔模型结构的方法，在Q无环时可得到投射模型结构，并将其应用于微分左A-模范畴，获得了相容的诱导余挠对及阿贝尔模型结构。
Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.
We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modu…
We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement $(\mathcal{A}$, $\mathcal{B}$, $\mathcal{C})$ of extriangulated categories, we show that cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$ induce cotorsion pairs in $\mathcal{B}$ under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomena when it is applied to abelian categories.
By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a modification of this conjecture with the word 'finite' replaced by 'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in …
Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of strongly right exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprim…
We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.
Let $R$ be a right and left Ore ring, $S$ its set of regular elements and $Q = R[S^{-1}] = [S^{-1}] R$ the classical ring of quotients of $R$. We prove that if F.dim$(Q_Q) = 0$, then the following conditions are equivalent: $(i)$ Flat right $R$-modules are strongly flat. $ (ii)$ Matlis-cotorsion right $R$-modules are Enochs-cotorsion. $(iii) $ $h$-divisible right $R$-modules are weak-injective. $(iv)$ Homomorphic images of weak-injective right $R$-modules are weak-injective. $(v)$ Homomorphic images of injective right $R$-modules are weak-injective. $(vi)$ Right $R$-modules of weak dimension $…
Auslander and Kleiner proved in 1994 an abstract version of Green correspondence for pairs of adjoint functors between three categories. They produce additive quotients of certain subcategories giving the classical Green correspondence in the special setting of modular representation theory. Carlson, Peng and Wheeler showed in 1998 that Green correspondence in the classical setting of modular representation theory is actually an equivalence between triangulated categories with respect to a non standard triangulated structure. In the present note we first define and study a version of relative …
In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $Φ$ and any generator $X$ of the $A$-module category, the $Φ$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idemp…
Gluing and lifting exact model structures for the recollement of exact categories
该研究提出正合范畴recollement下遗传正合模型结构的显式粘合方法，借助余挠对理论完成相关构造，并实现模型结构recollement到同伦范畴的提升，为三角范畴recollement的构造提供新途径，且在上三角矩阵环相关范畴中给出应用。
We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Au…
Acyclic complexes of FP-injective modules over Ding-Chen rings
本文提出了一种结合cotorsion对的方法，用于构建新的模型结构，并研究丁-陈环的稳定模块范畴，通过引入Gorenstein FP-pro-注入模块。
Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [14], and it does not provide for approximations when $R$ has cardinality $\leq \aleph_0$, [6]. We remove the cardinality restriction on $R$ in the latter result. We also prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair $(\mathcal A,\mathcal B)$ is of countable type when…
Generalized periodicity theorems
本文在由强有限表现模类生成的余挠对框架下证明广义周期性定理，统一推广了平坦/投射、fp-投射、fp-内射/内射及余挠周期性定理，并适用于任意环上的模和 Grothendieck 范畴。
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