Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of module categories for artin algebras $Λ'$, $Λ$ and $Λ''$. We provide a sufficient condition such that a glued torsion pair in $\mbox{mod} Λ$ is tilting when the given two torsion pairs are tilting in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively. Using this result, we give a construction of gluing of tilting modules in $\mbox{mod} Λ$ with respect to tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively.
For any good tilting module $T$ over a ring $A$, there exists an $n$-symmetric subcategory $\mathscr{E}$ of a module category such that the derived category of the endomorphism ring of $T$ is a recollement of the derived categories of $\mathscr{E}$ and $A$ in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of $\mathscr{E}$.
We study connections between recollements of the derived category D(Mod-R) of a ring R and tilting theory. We first provide constructions of tilting objects from given recollements, recovering several different results from the literature. Secondly, we show how to construct a recollement from a tilting module of projective dimension one. Our results will be employed in a forthcoming paper in order to investigate stratifications of D(Mod-R).
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a 'nice' set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a…
Let $(\mbox{mod} Λ',\mbox{mod} Λ,\mbox{mod} Λ'')$ be a recollement of abelian categories for artin algebras $Λ'$, $Λ$ and $Λ''$. Under certain conditions, we present an explicit construction of gluing of (support) $τ$-tilting modules in $\mbox{mod} Λ$ with respect to (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ respectively; conversely, we study the construction of (support) $τ$-tilting modules in $\mbox{mod} Λ'$ and $\mbox{mod} Λ''$ obtained from (support) $τ$-tilting modules in $\mbox{mod} Λ$.
Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions.
Let $\ca$ be a small dg category over a field $k$ and let $\cu$ be a small full subcategory of the derived category $\cd\ca$ which generate all free dg $\ca$-modules. Let $(\cb,X)$ be a standard lift of $\cu$. We show that there is a recollement such that its middle term is $\cd\cb$, its right term is $\cd\ca$, and the three functors on its right side are constructed from $X$. This applies to the pair $(A,T)$, where $A$ is a $k$-algebra and $T$ is a good $n$-tilting module, and we obtain a result of Bazzoni--Mantese--Tonolo. This also applies to the pair $(\ca,\cu)$, where $\ca$ is an augmente…
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 …
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.
In the paper, we investigate the lifting of recollements with respect to Gorenstein-projective modules. Specifically, a homological ring epimorphism can induce a lifting of the recollement of the stable category of finitely generated Gorenstein-projective modules; the recollement of the bounded Gorenstein derived categories of some upper triangular matrix algebras can be lifted to the homotopy category of Gorenstein-projective modules. As a byproduct, we give a sufficient and necessary condition on the upper triangular matrix algebra T_{n}(A) to be of finite CM-type for an algebra A of finite …
In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring A and its idempotent ideal J, we show that an A/J-module T is a silting A-module if and only if T is a silting A/J-module. For the finite dimensional k-algebras, with k a field, we show that the tensor products of silting modules are still silting. We also show that the (partial) Gorenstein silting properties can be glued by the recollements of module categ…
In this article, we prove that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then torsion pairs in $\mathcal A$ and $\mathcal C$ can induce torsion pairs in $\mathcal B$, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.
Surjective homological epimorphisms with stratifying kernel can be used to construct recollements of derived module categories. These `stratifying' recollements are derived from recollements of module categories. Can every recollement be put in this form, up to equivalence? A negative answer will be given after providing a characterisation of recollements equivalent to stratifying ones. Moreover, criteria for a ring epimorphism to be `stratifying' will be presented as well as constructions of such epimorphisms.
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (noncompact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Ric…
Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct resolving subcategories in $(\mathcal{A},\mathcal{B},\mathcal{C})$. As applications of the Auslander-Reiten correspondence, we get the gluing of cotilting modules in a recollement of module categories for artin algebras. We also give some bounds of resolution dimensions of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$ with respect to re…
Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms…
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）
We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay $\opn{T}_2(R)$-modules.
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…
The relation between the $n$-recollements of stable categories of Gorenstein projective modules and the virtual Gorensteinness of algebras are investigated. Let $A,B$, and $C$ be finite dimensional algebras. We prove that if the stable category of Gorenstein projective $A$-modules admits a $n$-recollement relative to that of $B$ and $C$ with $n\geq 2$, then $A$ is virtually Gorenstein if and only if so are $B$ and $C$.
Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we…
The higher APR tilting modules and higher BB tilting modules were introduced and studied in higher Auslander-Reiten theory. Our objective is to consider these tilting modules by the corresponding simple modules, and show that the tensor product of higher APR (BB) tilting modules is a higher APR (BB) tilting module.
Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $τ$-tilting $B$-module is a support $τ$-tilting $A$-module; and, conversely, the restriction of a given support $τ$-tilting $A$-module is a support $τ$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $τ$-tilting modules.
In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $τ$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.
The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing "Gorenstein tilting approximations" and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of compleme…
Mutation of τ-tilting modules is a basic operation to construct a new support τ-tilting module from a given one by replacing a direct summand. The aim of this paper is to give a positive answer to the question posed in [AIR, Question 2.31] about mutation of τ-tilting modules.
Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological objects. The purpose of this series of papers is to study recollements in terms of derived module categories and homological ring epimorphisms, and then to apply our results to both representation theory and algebraic K-theory. In this paper we present a new and systematic method to construct recollements of derived module categories. For this aim, we intro…
In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey-Sauter, Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give characterizations of almost $n$-Auslander-Gorenstein algebras and almost $n$-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost $1$-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.
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.
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
Let $Λ$ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $τ$-tilting $Λ$-modules. In this paper, we will show that it can be restricted to $τ$-tilting modules.
The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposabl…
We introduce the notions of Gorenstein projective $τ$-rigid modules, Gorenstein projective support $τ$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support $τ$-tilting modules. More precisely, for an algebra $Λ$, We prove that there is a bijection between the set of Gorenstein projective support $τ$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$τ$-tilting finite algebras and show that $Λ$ is CM-$τ$-tilting finite if and only if…
Support $τ$-tilting modules correspond to some classes of categorical objects bijectively, such as two-term tilting complexes for any finite dimensional symmetric algebra. This fact motivates us to classify support $τ$-tilting modules over blocks of finite groups. Therefore we classify support $τ$-tilting modules over particular blocks of finite groups by using the modular representation theoretical approaches including Clifford theory, Green's indecomposability theorem and so on.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting $A$-modules and the tilting $B$-modules, and we prove that there is a one-one correspondence between the basic $T$-tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$. Moreover, we show that there is a one-one correspondence between the $T$-contravariantly finite $T$-resolving subcategories of $T^{\perp}$ and the basic $T$-t…
We study (support) $τ$-tilting modules over the trivial extensions of finite dimensional algebras. More precisely, we construct two classes of (support)$τ$-tilting modules in terms of the adjoint functors which extend and generalize the results on (support) $τ$-tilting modules over triangular matrix rings given by Gao-Huang.
We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.
Following the previous work, we determine exactly the highest weights for which a tensor product of two induced modules is a tilting module, for the algebraic group $SL_2$ over an algebraically closed field of positive characteristic.
We give an example of a finite-dimensional algebra with a 2-cluster tilting module and a simple module which has infinite complexity. This answers a question of Erdmann and Holm.
In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.
The class of support $τ$-tilting modules was introduced recently by Adachi, Iyama and Reiten. These modules complete the class of tilting modules from the point of view of mutations. Given a finite dimensional algebra $A$, we study all basic support $τ$-tilting $A$-modules which have given basic $τ$-rigid $A$-module as a direct summand. We show that there exist an algebra $C$ such that there exists an order-preserving bijection between these modules and all basic support $τ$-tilting $C$-modules; we call this process $τ$-tilting reduction. An important step in this process is the formation of $…
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent.
Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $τ$-tilting $A$-module can be extended to be a support $τ$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$
We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form T=R_U\oplus R_U /R where U is a union of tubes, and R_U denotes the universal localization of R at U in the sense of Schofield and Crawley-Boevey. Here R_U/R is a direct sum of the Prüfer modules corresponding to the tubes in U. Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class Gen L consists of all modules wit…
Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $τ$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $τ$-tilting modules and support $τ$-tilting modules over linearly Dynkin type algebras whose square radical are zero.
Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. We establish an isomorphism of $G$-modules between a direct sum of modules $\text{St} \otimes \text{St}$ and a direct sum of tensor products of simple modules of restricted highest weight with tilting modules that are projective over the Frobenius kernel of $G$. This isomorphism holds precisely when Donkin's Tilting Module Conjecture does, and thus can be seen as providing a $G$-module theoretic characterization of this conjecture.
Let G be a semisimple group over an algebraically closed field of characteristic p>0. We give a (partly conjectural) simple, closed formula for the character of many indecomposable tilting rational G-modules, assuming that p is large.
We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.
We give a complete picture of when the tensor product of an induced module and a Weyl module is a tilting module for the algebraic group $SL_2$ over an algebraically closed field of characteristic $p$. Whilst the result is recursive by nature, we give an explicit statement in terms of the $p$-adic expansions of the highest weight of each module.
We construct families of representations for quantum groups over $\mathbb{Z}[v,v^{-1}]$-algebras that interpolate between Weyl modules and tilting modules. These families might be candidates for objects with characters satisfying the {\em generations of characters} philosophy of Lusztig and Lusztig-Williamson.
In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. These results are well-known by several specialists, but seem difficult to find in the existing literature.
