We show that every tilting module of projective dimension one over a ring R is associated in a natural way to the universal localization (in the sense of Schofield) of R at a set of finitely presented modules of projective dimension one. We then investigate tilting modules arising from universal localization. Furthermore, we discuss the relationship between universal localization and the localization given by a perfect Gabriel topology. Finally, we give some applications to Artin algebras and to Pruefer domains.
Given a ring R, we investigate tilting modules of the form S \oplus S/R for some injective ring epimorphism R \to S. In particular, we are interested in tilting modules arising from Schofield's universal localization. For some rings, in this way one obtains a classification of all tilting modules.
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.
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…
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.
We study universal localisations, in the sense of Cohn and Schofield, for finite dimensional algebras and classify them by certain subcategories of our initial module category. A complete classification is presented in the hereditary case as well as for Nakayama algebras and local algebras. Furthermore, for hereditary algebras, we establish a correspondence between finite dimensional universal localisations and finitely generated support tilting modules. In the Nakayama case, we get a similar result using $τ$-tilting modules, which were recently introduced by Adachi, Iyama and Reiten.
We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categ…
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…
Given a non-unit, non-zero-divisor, central element $x$ of a ring $Λ$, it is well known that many properties or invariants of $Λ$ determine, and are determined by, those of $Λ/ x Λ$ and $Λ_x$. In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over $Λ$ gives rise to tilting modules over $Λ_x$ and $Λ/ x Λ$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over $Λ$ is tilting if its corresponding localization and quotient are tilti…
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…
We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.
We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of (-2)-curves, we obtain an equivalence with modules over a well known algebra.
This is an appendix to the Handbook of Tilting Theory, edited by Angeleri-Huegel, Happel and Krause, to be published soon. Part 1 of the appendix provides an outline of the core of tilting theory. Part 2 is devoted to topics where tilting modules and tilted algebras have shown to be relevant. Both Parts 1 and 2 contain historical annodations and reminiscences. The final Part 3 is a short report on some striking recent developments which are motivated by the cluster theory of Fomin and Zelevinsky. In particular, we will guide the reader to the basic properties of cluster tilted algebras, to the…
This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. For some pairs of blocks $\overline{\mathcal{O}}[λ]$ and $\overline{\mathcal{O}}[μ]$, the subcategories whose objects have finite length are equivalence via functors obtained by the direct limits of translation functors. Tilting objects can also…
We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard g…
We show that silting modules are closely related with localisations of rings. More precisely, every partial silting module gives rise to a localisation at a set of maps between countably generated projective modules and, conversely, every universal localisation, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.
We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the Serre stable moduli stack. We explain why the method works by showing that the Serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lin…
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…
Let $k$ be an algebraically closed field and $Φ$ a finite dimensional $k$-algebra. The universal localization $Φ\rightarrow Φ_\mathcal{S}$ of $Φ$ with respect to a set of morphisms between finitely generated projective $Φ$-modules $\mathcal{S}$ always exists. Moreover, when $Φ$ is hereditary, Krause and Šťovíček proved that the universal localizations of $Φ$ are in bijective correspondence with various natural structures. Taking inspiration from an alternative definition of universal localizations involving a triangulated subcategory of $\mathcal{D}^{\text{perf}}(Φ)$, we introduce a higher ana…
Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which f…
We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories $\mathcal{T}$ of $\mathcal{A}$ closed under extensions and direct summands are precisely those for which $(_{}^{\perp_1}\mathcal{T},\mathcal{T})$ is a right complete cotorsion pair, where $_{}^{\perp_1}\mathca…
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）
