Amnon Yekutieli, James J. Zhang
We prove that two-sided tilting complexes, and dualizing complexes, over simple Goldie rings (with some technical conditions) are always shifts of invertible bimodules. This allows us to describe the derived Picard groups of such rings, and to deduce these are Gorenstein (and sometimes even Auslander-Gorenstein Cohen-Macaulay) rings.
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A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing com…
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.
Dualizing complexes over $\mathbb{Z}$-algebras
本文定义ℤ-代数上的对偶复形与平衡对偶复形，证明诺特连通ℤ-代数存在平衡对偶复形的等价条件，并由此推出对应光滑非交换射影概型存在Serre函子。
Let $G,H$ be groups, $ϕ: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $ϕ$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.
Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra.
Rigid dualizing complexes of affine Hecke algebras
本文确定了约化根系关联的仿射Hecke代数$H_q$的刚性对偶复形，并推导出经典Hecke代数在特定条件下在其中心上为Frobenius的，且Nakayama自同构由显式对合给出。
A differential algebra of finite type over a field k is a filtered algebra A, such that the associated graded algebra is finite over its center, and the center is a finitely generated k-algebra. The prototypical example is the algebra of differential operators on a smooth affine variety, when char k = 0. We study homological and geometric properties of differential algebras of finite type. The main results concern the rigid dualizing complex over such an algebra A: its existence, structure and variance properties. We also define and study perverse A-modules, and show how they are related to th…
We give a bijection between the tilting complexes in the bounded homotopy category of the Auslander algebra of the truncated polynomial ring and ZxB where B is the Artin braid goup of type A with n-1 generators. The tilting complexes have mutation components parametrized by Z and each component has a natural faithful and transitive operation of B. This also implies that the derived Picard group of this algebra is isomorphic to the direct product of its outer isomorphism group and ZxB. This work is to be seen as a continuation of the work of Geuenich and an application of the work of Aihara and…
Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the derived tensor product with T is an equivalence. The complex T is called a tilting complex. When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. So we can compute D…
First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements $\{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)}$ and $\{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}$, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex $\varDelta^{\centerdot}_{T}$ which induces an equivalence between $\cat{D}_{A/AeA}(A)$ and $\cat{D}_…
Let $R$ be a commutative artinian ring. We extend higher Auslander correspondence from Artin $R$-algebras of finite representation type to dualizing $R$-varieties. More precisely, for a positive integer $d$, we show that a dualizing $R$-variety is $d$-abelian if and only if it is a $d$-Auslander dualizing $R$-variety if and only if it is equivalent to a $d$-cluster-tilting subcategory of the category of finitely presented modules over a dualizing $R$-variety.
Two-sided tilting complexes for generalized Brauer tree algebras
针对广义Brauer树代数，显式构造出与Membrillo-Hernández树到星倾斜复形相对应的双侧倾斜复形。
This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any such complex $\mathcal L^\bullet$, we construct a triangulated category endowed with a pair of (possibly degenerate) t-structures of the derived type, whose hearts are the abelian categories of left $\mathcal C$-comodules and left $\mathcal D$-contramodules. A weak version of pseudo-derived categories arising out of (co)resolving subcategories in abelian/exact…
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
Pseudo-dualizing complexes and pseudo-derived categories
本文通过去掉内射维数条件定义伪对偶化复形，并证明在一对结合环上它诱导伪余导出范畴与伪反导出范畴之间的三角等价，推广了先前导出函子构造技术。
We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $τ$-tilting modules, is a derived invariant. Moreover, we apply our result to the …
Let $p$ be a prime number, $k$ an algebraically closed field of characteristic $p$, $\tilde{G}$ a finite group, and $G$ a normal subgroup of $\tilde{G}$ having a $p$-power index in $\tilde{G}$. Moreover let $B$ be a block of $kG$ with a cyclic defect group and $\tilde{B}$ be the unique block of $k\tilde{G}$ covering $B$. We study tilting complexes over the block $\tilde{B}$ and show that the block $\tilde{B}$ is a tilting-discrete algebra. Moreover we show that the set of all tilting complexes over $\tilde{B}$ is isomorphic to that over $B$ as partially ordered sets.
Two-sided tilting complexes over symmetric algebras sending simple modules to shifts of modules
本文针对代数闭域上有限维对称代数的倾斜复形，其不同分次分量无公共不可分解投射直和项，从诱导Morita型稳定等价的双模出发，显式地（重新）构造了对应于该倾斜复形的双侧倾斜复形。
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。
Topological endomorphism rings of tilting complexes
本文在紧致生成三角范畴中引入正派倾斜对象，证明其倾斜心等价于自同态环上的反模范畴，推广了已有结果，并建立了正派倾斜复形与倾斜和余倾斜导出等价对之间的Morita理论。
We consider recollements of derived categories of dg-algebras induced by self orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results on triangle equivalences proved in [B2] and [BMT]. In the end we focus on the connection between recollements of derived categories of rings, bireflective subcategories and generalized universal localizations".
In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the $τ$-tilting theo…
For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notio…
Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.
Two-term tilting complexes of biserial fractional Brauer graph algebras
本文研究双序列分数Brauer图代数的两项倾斜复形和Kauer移动，证明其斜群代数是斜Brauer图代数，并给出倾斜离散性的充要条件及导出等价封闭性。
We introduce Wakamatsu-silting complexes (resp., Wakamatsu-tilting complexes) as a common generalization of both silting complexes (resp., tilting complexes) and Wakamatsu-tilting modules. Characterizations of Wakamatsu-silting complexes are given. In particular, we show that a complex $T$ is Wakamatsu-silting if and only if its dual $DT$ is Wakamatsu-silting. It is conjectured that all compact Wakamatsu-silting complexes are just silting complexes. We prove that the conjecture lies under the finitistic dimension conjecture.
By Rickard's work, two rings are derived equivalent if there is a tilting complex, constructed from projective modules over the first ring such that the second ring is the endomorphism ring of this tilting complex. In this work I describe, under some conditions, the homomorphism space in the derived category between two complexes as iterated pull-back of homomorphism spaces between the homologies of the complexes.
Higher torsion classes, $τ_d$-tilting theory and silting complexes
本文研究了高维 $τ_d$-tilting 理论，通过 $τ_d$-rigid 对和 $(d+1)$-项 silting 复合体揭示了 $d$-torsion 类与 $d$-exact 类别的关系。
Non-kissing complexes and tau-tilting for gentle algebras
本文为温和代数建立了支持τ-倾斜复形与非亲吻复形的对应，并证明了有限情形下递增翻转图构成同余均匀格，其g-向量扇为非亲吻关联多面体的法扇。
The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly 2 cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exa…
For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.
It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for comple…
Gorenstein acyclic complexes and finitistic dimensions
本文研究带对偶复形的双边Noether环的大有限维数有限性问题，给出其与Gorenstein投射-无环上链复形可缩性的等价刻画，并对Artin代数证明Rickard定理的Gorenstein版本。
We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our method is the notion of a strong homotopy action in the sense of Stasheff.
Recollements of derived categories from $n$-term big tilting complexes
针对环A上的n项大倾斜复形T，构造B-Mod的扩张闭正合子范畴ℰ，证明导出范畴DB可由Dℰ和DA重排，还得到ℰ的d-对称性、阿贝尔性条件，将两项经典情形推广到任意有限项，构造了n≥2时的非紧大倾斜复形。
We prove a theorem which gives a bijection between the support $τ$-tilting modules over a given finite-dimensional algebra $A$ and the support $τ$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $τ$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $τ$-tilting modules for all algeb…
We introduce the new concept of silting modules. These modules generalise tilting modules over an arbitrary ring, as well as support $τ$-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. We show that silting modules generate torsion classes that provide left approximations, and that every partial silting module admits an analogue of the Bongartz complement. Furthermore, we prove that silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category. We also see how som…
For a ring $A$, we consider the question whether every bounded above cochain complex of injective $A$-modules which is acyclic is null-homotopic. We show that if $A$ is left and right noetherian and has a dualizing complex, then this implies that the finitistic dimension of $A$ is finite. In the appendix, Nakamura and Thompson show that the opposite holds over any ring. Our results give several new necessary and sufficient conditions for a ring to have finite finitistic dimension in a very general setting. Applications include a generalization of a recent result of Rickard about relations betw…
We consider a finite dimensional strongly $G$-graded algebra $A$ with { self-injective} $1$-component $B$, and in our main result we prove that the induction from $B$ to $A$ of a basic support $τ$-tilting pair of $B$-modules is a support $τ$-tilting pair $(M,P)$ of $A$-modules if and only if $M$ is $G$-invariant. A similar statement holds for the restriction from $A$ to $B$, so our results may be viewed as Clifford and Maschke type theorems for $2$-term silting complexes. We also give applications to semibricks and the associated wide subcategories.
A description of tilting complexes is given for a class of PI algebras whose prime spectrum is canonically homeomorphic to the prime spectrum of its center. Some Sklyanin algebras are the kind of algebras considered. As an application, it is proved that any algebra derived equivalent to such kind of algebra, is Morita equivalent to it.
Maximal Cohen-Macaulay DG-complexes
本文提出极小Cohen-Macaulay DG复形的定义，扩展了Iyengar等人关于局部环的极小Cohen-Macaulay复形的研究，并建立了具有恒定振幅的DG环的导出改进新交集定理。
Calabi-Yau completions for roots of dualizing dg bimodules
本文针对 dg 范畴上移位逆对偶化双模的根，引入了 $a$ 次根对及其 Calabi-Yau 完备化，证明了其循环不变性条件，并建立了与 Adams 分次 Calabi-Yau dg 范畴的双射，进而推广了簇范畴及倾斜丛定理。
Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated sc…
Schur roots and tilting modules of acyclic quivers over commutative rings
本文将域上无环箭图支撑倾斜模与簇代数簇的双射推广至不可分解交换Noetherian环，建立2-term倾斜复形与簇的双射，并证明Dynkin箭图的挠类与Spec R到簇集的保序映射双射对应。
Dimensions of $τ$-tilting modules over path algebras and preprojective algebras of Dynkin type
本文引入了$d$-多项式这一新生成函数，用于计算有限维代数上$τ$-倾斜模的维数，揭示了其与$τ$-刚性对单纯复形$f$-多项式的关系，并给出了Dynkin型预投射代数和路代数的显式公式及闭式表达式。
We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are "componentwise" tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizing the ADE-chain related to singularity theory, incidence algebras of posets and certain Auslander algebras or more generally endomorphism algebras of initial preprojective modules over path algebras of …
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