We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim}\nolimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim}\nolimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim}\nolimits \mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ infinite.
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.
We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.
Let $A$ be a finite dimensional algebra over a field $k$ and $\textbf{P}$ be a 2-term silting complex in $K^{b}(\text{proj}A)$. In this paper, we investigate the representation dimension of $\text{End}_{D^{b}(A)} (\textbf{P})$ by using the silting theory. We show that if $\textbf{P}$ is a separating silting complex with certain homological restriction, then rep.dim $A=$rep.dim $\text{End}_{D^{b}(A)}(\textbf{P})$. This gives a proper generalization of the classical compare theorem of representation dimensions showed by Chen and Hu. It is well-known that $\text{H}^{0}(\textbf{P})$ is a tilting $…
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…
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.
(Partial) Gorenstein silting modules are introduced and investigated. It is shown that for finite dimensional algebras of finite CM-type, partial Gorenstein silting modules are in bijection with τ_G-rigid modules; Gorenstein silting modules are the module-theoretic counterpart of 2-term Gorenstein silting complexes; and the relation between 2-term Gorenstein silting complexes, t-structures and torsion pairs in module categories. Furthermore, the corresponding version of the classical Brenner-Butler theorem in this setting are characterised; and the upper bound of the global dimension of endomo…
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
Higher torsion classes, $τ_d$-tilting theory and silting complexes
University of Glasgow（格拉斯哥大学）；Department of Mathematical Sciences NTNU（挪威科技大学数学科学系）；Aarhus Universitet（奥胡斯大学）；LAMFA, Université de Picardie Jules Verne（皮卡第儒勒·凡尔纳大学 LAMFA）；Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales（布宜诺斯艾利斯大学理学院）；CONICET - Universidad de Buenos Aires. Instituto de Investigaciones Matemáticas “Luis A. Santaló” (IMAS)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学路易斯·桑塔洛数学研究所）
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, $\mathcal {D}^b(A)$ be the bounded derived category of $A$-mod and $A^{(m)}$ be the $m$-replicated algebra of $A$. In this paper, we investigate the structure properties of endomorphism algebras arising from silting mutation in $\mathcal {D}^b(A)$ and tilting mutation in $A^{(m)}$-mod.
We develop silting theory of a noetherian algebra $Λ$ over a commutative noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $Λ$, and as a consequence, we see that mutation exists. As in the case of finite dimensional algebras, functorially finite torsion classes of $Λ$ bijectively correspond to silting $Λ$-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $Λ$, and by using this theorem, we study torsion classes of the module category of $Λ$. When $R$ has Krull dimension one, we describe the set of torsion classes of $Λ$ …
Let $Λ$ be an Artin algebra and $K^b(proj(Λ))$ be the triangulated category of bounded co-chain complexes in $proj(Λ).$ It is well known that two-terms silting complexes in $K^b(proj(Λ))$ are described by the $τ$-tilting theory. The aim of this paper is to give a characterization of certain $n$-term silting complexes in $K^b(proj(Λ))$ which are induced by $Λ$-modules. In order to do that, we introduce the notions of $τ_n$-rigid, $τ_n$-tilting and $τ_{n,m}$-tilting $Λ$-modules. The latter is both a generalization of $τ$-tilting and tilting in $mod(Λ).$ It is also stated and proved some variant,…
We introduce a method that produces a bijection between the posets ${\rm silt-}{A}$ and ${\rm silt-}{B}$ formed by the isomorphism classes of basic silting complexes over finite-dimensional $k$-algebras $A$ and $B$, by lifting $A$ and $B$ to two $k[[X]]$-orders which are isomorphic as rings. We apply this to a class of algebras generalising Brauer graph and weighted surface algebras, showing that their silting posets are multiplicity-independent in most cases. Under stronger hypotheses we also prove the existence of large multiplicity-independent subgroups in their derived Picard groups as wel…
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p \ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent $G$-orbit $C$ and each indecomposable tilting vector bundle $T$ on $C$ a certain complex $S(C,T)$ of $G \times \mathbb{G}_m$-equivariant coherent sheaves on $\mathcal{N}$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-$t$-structure. We then show that if $p$ is larger than the Coxeter number, then the hypercoho…
We study silting mutations (Okuyama-Rickard complexes) for selfinjective algebras given by quivers with potential (QPs). We show that silting mutation is compatible with QP mutation. As an application, we get a family of derived equivalences of Jacobian algebras.
This paper studies silted algebras, namely, endomorphism algebras of 2-term silting complexes, over path algebras of Dynkin quivers. We will describe an algorithm to produce all basic 2-term silting complexes over the path algebra of a Dynkin quiver, and use this algorithm to compute some examples.
The main goal of this paper is to compare the silting theory of an $R$-algebra $Λ$ over a Noetherian ring $R$ with that of its tensor product $Λ\otimes Γ$ with another $R$-algebra $Γ$. In the case that the $R$-algebra $Λ$ is Noetherian, $R$ a complete local ring and $\mathfrak{a}$ a certain ideal of the ring $R$, we obtain an isomorphism between the silting poset of $Λ$ and that of its quotient $Λ\otimes R/ \mathfrak{a}$. Furthermore, we study the restrictions of such a bijection to tilting complexes and deduce silting embedding and descent results for the algebra $Λ$ and a certain family of a…
We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex $\mathbf{P}$ in the bounded homotopy category $K^b(\mathop{\rm proj}\nolimits A)$ of finitely generated projective modules of a finite dimensional algebra $A$, the algebra $B = \mathop{\rm End}\nolimits_{K^b(\mathop{\rm proj}\nolimits A)}(\mathbf{P})$ admits a 2-term silting complex $\mathbf{Q}$ with the following properties: (i) The endomorphism algebra of $\mathbf{Q}$ in $K^b(\mathop{\rm proj}\nolimits B)$ is a factor algebra of $A$, and (ii) there are induced torsion pairs in $\mathop{\rm mo…
We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of $\mathop{\rm Ext}\nolimits$-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded $t$-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension a…
We show that indecomposable two-term presilting complexes over $Π_{n}$, the preprojective algebra of $A_{n}$, are in bijection with internal $n$-simplices in the prism $Δ_{n} \times Δ_{1}$, the product of an $n$-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of $Δ_{n} \times Δ_{1}$ and two-term silting complexes over $Π_{n}$ such that bistellar flips of triangulations correspond to mutations of two-term silting complexes. These bijections are shown to compatible with the known bijections involving the symmetric group.
In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras.
In this paper all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.
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 …
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
From $τ$-tilting to $τ_2$-tilting via splitting tilting triples
Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院教育部非线性科学重点实验室）；School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）
In the representation theory of finite-dimensional algebras $A$ over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group $K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_{\mathbb{Z}} {\mathbb{R}}$. The aim of this paper is to study the complement $\operatorname{\mathsf{NR}}$ of the union $\operatorname{\mathsf{Cone}}$ of all g-vector cones, which we call the non-rigid region. By the work of Iyam…
Recollements of derived categories from $n$-term big tilting complexes
School of Mathematics and Statistics, Beijing Jiaotong University（北京交通大学数学与统计学院）；Beijing Key Laboratory of Biological Big Data and Topological Statistics, Beijing Jiaotong University（北京交通大学生物大数据与拓扑统计北京市重点实验室）；School of Science, Beijing University of Civil Engineering and Architecture（北京建筑大学理学院）
Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated categories) for which the endomorphism rings of different terms are derived equivalent. We then study applications involving 0-Calabi-Yau triangulated categories. In particular, applying our results in the category of perfect complexes over a symmetric algebra gives a nice way of producing pairs of derived equivalent symmetric algebras. Included in the exampl…
