Silvana Bazzoni
We generalize Brenner and Butler's Theorem as well as Happel's Theorem on the equivalences induced by a finitely generated tilting module over artin algebras, to the case of an infinitely generated tilting module over an arbitrary associative ring establishing the equivalences induced between subcategories of module categories and also at the level of derived categories.
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Let $T_R$ be a right $n$-tilting module over an arbitrary associative ring $R$. In this paper we prove that there exists a $n$-tilting module $T'_R$ equivalent to $T_R$ which induces a derived equivalence between the unbounded derived category $\D(R)$ and a triangulated subcategory $\mathcal E_{\perp}$ of $\D(\End(T'))$ equivalent to the quotient category of $\D(\End(T'))$ modulo the kernel of the total left derived functor $-\otimes^{\mathbb L}_{S'}T'$. In case $T_R$ is a classical $n$-tilting module, we get again the Cline-Parshall-Scott and Happel's results.
We relate the notions of BB-tilting and perverse derived equivalence at a vertex. Based on these notions, we define mutations of algebras, leading to derived equivalent ones. We present applications to endomorphism algebras of cluster-tilting objects in 2-Calabi-Yau categories and to algebras of global dimension at most 2.
Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S$ the endomorphism ring. We prove that $T$ induces an equivalence between stable categories of repetitive algebras of $R$ and $S$.
We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
We give a category theoretic approach to several known equivalences from (classic) tilting theory and commutative algebra. Furthermore, we apply our main results to establish a duality theory for relative Cohen-Macaulay modules in the sense of Hellus, Schenzel, and Zargar.
The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that…
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}_…
Derived equivalence for the simple flop of type $G_2^{\dagger}$ via tilting bundles
本文通过倾转包证明了类型$G_2^{\dagger}$简单翻转的局部模型的导出等价性，并给出了非交换crepant解析，该解析与翻转两侧导出等价。
Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be the Happel-Reiten-Smalø tilt of $\mathcal{A}$ with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between $\mathcal{B}$ and $\mathcal{A}$, which is compatible with the inclusion of $\mathcal{B}$ into the derived category of $\mathcal{A}$. In particular, any splitting torsion pair induces a derived equivalence. We prove that for the realization functor of any bounded $t$-structure, its denseness implies its fully-faithfulness.
In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.
We prove that a finite dimensional algebra is $τ$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $τ$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $τ$-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms $A\longrightarrow B$ with ${\rm Tor}_1^A(B,B)=0$. It follows that a finite dimensional algebra is $τ$-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.
We give a necessary and sufficient condition for that the support $τ$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite dimensional algebras whose support $τ$-tilting posets are isomorphic to the poset of symmetric group with weak order.
Tilting modules and tilting torsion pairs - filtrations induced by tilting modules
本文研究倾斜模如何通过诱导滤子揭示模块范畴的整体结构，探讨其在代数与表示论中的应用与贡献。
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforwa…
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a hereditary abelian category $\mathcal{H}$, we verify that the tilting functor Hom$_\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\mathcal{C(H)}$ to the cluster category $\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\mathc…
A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an S-module, one can extend T to a tilting complex over A inducing a derived equivalence between A and another triangular matrix ring specified by (S',R,M'), where the ring S' and the R-S'-bimodule M' depend only on T and M, and S' is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebr…
This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster) categories of finite dimensional algebras.
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.
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。
Relative tilting theory in abelian categories II: $n$-$\mathcal{X}$-tilting theory
本文在阿贝尔范畴中引入了$n$-$\mathcal{X}$-相对倾斜理论，统一了多种已有倾斜概念，并展示了其在正合范畴、extriangulated范畴及箭图表示范畴中构建新倾斜理论和余挠对的应用。
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".
Quasi-equivalence of relabeled positroid cluster structures
本文证明了Fraser--Sherman-Bennett猜想，通过构造倾斜模和Frobenius范畴化，建立了重标记正丛簇结构的拟等价性，并利用导出等价和幺模变换证明了坐标环上的恒等映射是拟簇同构。
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.
For a certain Wakamatsu-tilting bimodule over two artin algebras $A$ and $B$, Wakamatsu constructed an explicit equivalence between the stable module categories over the trivial extension algebra of $A$ and that of $B$. We prove that Wakamatsu's functor is a triangle functor, thus a triangle equivalence.
We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tiltin…
On $τ$-tilting finiteness of symmetric algebras of polynomial growth
本文研究代数闭域上多项式增长对称代数等有限维代数的$τ$-倾斜有限性，发现导出等价保持该性质，并证明了相关代数中表示有限性与$τ$-倾斜有限性的一致性。
We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is ju…
Characteristic tilting modules and Ringel duality in the Noetherian world
本文研究交换诺特环上分裂拟遗传代数的Ringel对偶，刻画了包含所有标准模的最小子解消子范畴，并证明其间的正合等价可提升为保持拟遗传结构的Morita等价。
In this paper, we study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-cluster tilting submodule of $X$ is a two-sided …
We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $\mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.
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.
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ Λ) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(Λ)$ for a finite dimensional algebra $Λ$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } Λ< \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (Λ)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(Λ)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(Λ)$ is replaced with a finitely graded IG algebra $A$…
Dualities of resolving subcategories of finitely generated modules over Artin algebras are characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of these dualities to resolving subcategories of finitely generated modules with finite projective or Gorenstein-projective dimensions, Miyashita's duality and Huisgen-Zimmermann's correspondence on tilting modules as well as their Gorenstein version are obtained. Applications include constructing triangle equivalences of derived categories of finitely generated Gorenstein-projective modules and showing the invariance …
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。
Support $τ$-tilting posets and Hochschild reconstruction for matrix centralizer algebras
该研究确定矩阵中心化子代数的支持τ-倾斜偏序集与Hochschild不变量，给出其Morita等价准则，实现对应代数类的Morita重构。
Singular equivalences and homological conjectures
通过中心化子矩阵代数的奇异范畴和奇异等价，验证了Auslander-Reiten和Cartan行列式猜想，进而证明所有经典同调猜想对域上中心化子矩阵代数成立。
We show the existence of tilting objects in the singularity category $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ and the derived category of a finite-dimensional algebra. In particular, we obtain a tilting object if the Beilinson algebra of $A$ is a levelled Koszul algebra. This generalises the existence of a tilting object in $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(S^G)$, where $S$ is a Koszul AS-regular algebra and $G$ i…
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。
In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal U}\oplus R_{\mathcal U}/R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ is a recollement of the derived module category $\D{R}$ of $R$ and the derived module category $\D{{\mathbb A}_{\mathcal{U}}}$ of the adèle ring ${\mathbb A}_{\mathcal{U…
Tilting objects in periodic triangulated categories
本文在 $m$-周期三角范畴中引入 $m$-周期倾斜对象概念，并证明具有该对象的范畴在同调假设下三角等价于代数的 $m$-周期导出范畴，同时构建了自内射代数稳定范畴与遗传代数导出范畴间的三角等价。
From $τ$-tilting to $τ_2$-tilting via splitting tilting triples
本文通过分裂倾斜三元组，将τ-倾斜模构造为τ_2-倾斜模，并推广了Peng的构造，确定了自同态代数及半正交分解。
Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly $n$-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry,and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals…
Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism $B\ra C$ and a recollement among the (unbounded) derived module categories $\D{C}$ of $C$, $\D{B}$ of $B$, and $\D{A}$ of $A$. In particular, the kernel of the total left derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived module category $\D{C}$. Conversely, if the functor $T\otimes_B^{\mathbb …
Transfer of Wakamatsu tilting modules along Frobenius extensions
本文研究Frobenius扩张下Wakamatsu倾斜模的转移，给出诱导模仍为Wakamatsu倾斜模的充分条件，刻画倾斜情形，证明相关自同态环映射为Frobenius扩张并讨论其应用。
Let $\mathcal{P}^{<\infty} (Λ$-mod$)$ be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra $Λ$. We develop an applicable criterion that reduces the test for contravariant finiteness of $\mathcal{P}^{<\infty} (Λ$ -mod$)$ in $Λ$-mod to corner algebras $e Λe$ for suitable idempotents $e \in Λ$. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of $\mathcal{P}^{<\infty} (Λ$-mod$)$. The consequences pursued hinge on the fact that this finiteness condition is known to be equ…
Tilting mutation of gentle algebras and its combinatorial description
本文通过广义 BB-倾斜模研究 gentle 代数的倾斜突变，给出基于箭向和关系的判据，并构造突变后自同态代数的组合描述，证明所得代数仍为 gentle 且导出等价。
We give a classification theorem for a relevant class of $t$-structures in triangulated categories, which includes in the case of the derived category of a Grothendieck category, the $t$-structures whose hearts have at most $n$ fixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the $t$-tree, a new technique which generalises the filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the $1$-tilting equivalence proved by Happel, Reiten and Smalø [HRS96]. The last section provides applic…
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…
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.
Let $R$ be a perfect ring and $T$ be an $R$-module. We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. It is proved that $T$ is sincere silting if and only if $T$ is presilting satisfing the vanishing condition $\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0$, and that $T$ is tilting if and only if $\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0$ and $\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant 2}_{R}(T,-)$. As an application, we prove that a sincere silting $R$-module $T$ of fin…
We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes.
A homotope, or a mutation, of a $k$-algebra is a new algebra with the same underlying space, but with the multiplication law dependent on the multiplication law of the original algebra. In this paper, we show that a generic finite-dimensional algebra of dimension greater than 3 has infinitely many non-isotopic homotopes, and that, more generally, a similar result is true for generic trilinear forms. We also study a particular class of homotopes called $Δ$-homotopes, where $Δ$ is an element of the algebra, and show that there are algebras with infinitely many non-isomorphic homotopes even under…
We construct two functors from the submodule category of a self-injective representation-finite algebra $Λ$ to the module category of the stable Auslander algebra of $Λ$. These functors factor through the module category of the Auslander algebra of $Λ$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a ch…
We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition …
Applications of Gorenstein projective $τ$-rigid modules
本文引入 $CM$-$\ au$-倾斜自由代数并研究其同调性质，通过等价关系建立 Gorenstein 投射 $\ au$-刚性模与某些模的双射，并部分解决了 Tachikawa 第一猜想。
Let $ψ: A \to A'$ be a cyclic contraction of dimer algebras, with $A$ non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a finitely generated module over its center. In contrast, $A$ is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of $A$ with that of $A'$. We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus of $A'$, and give an explicit classification of the simple $A$-modules parameterized by the Azumaya locus. Furthermore, we sho…
Tilting in $Q$-shaped derived categories
本文证明了在某些条件下，代数 $A$ 的 $Q$-形导出范畴与另一代数 $B$ 的经典导出范畴之间存在三角等价，并推广了 Iyama-Kato-Miyachi 关于 $N$-导出范畴的结果。
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