Lidia Angeleri Hügel

• University of Verona

We consider endomorphism algebras of $n$-term silting complexes in derived categories of hereditary algebras, and we show that the module category of such an endomorphism algebra has a separated $n$-section. For $n=3$ we obtain a trisection in the sense of [2].

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod-R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category $ R\operatorname{-}\operatorname{mod}$ of finitely presented modules over a left artinian ring $R$, we assign two wide subcategories in the category $ R\operatorname{-}\operatorname{Mod}$ of all $R$-modules and describe them e...

Mutation of compact silting objects is a fundamental operation in the representation theory of finite-dimensional algebras due to its connections to cluster theory and to the lattice of torsion pairs in module or derived categories. In this paper we develop a theory of mutation in the broader framework of silting or cosilting t-structures in triang...

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic mod...

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A...

We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory...

We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory is generated by a partial silting object. In particular, every such smashing subcategory admits...

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic mod...

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod-A) of a ring A. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A, which is a natural extension of the construction...

We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t‐structures and co‐t‐structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the...

We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory is generated by a partial silting object. In particular, every such smashing subcategory admits...

We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and co-t-structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the...

We study different types of localisations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localisation in the sense of Cohn and Schofield; and (b) when such universal localisations are classical rings of fractions. In order to find such criteria, we use the theory...

A classic result by Bass says that the class of all projective modules is covering if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules C, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and...

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or of a finite dimensional algebra of finite representation type, can be parametrised by silting modules. In thes...

We prove that a finite dimensional algebra is $\tau$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $\tau$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $\tau$-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epim...

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 noncom...

Let $\mathbb{X}$ be a weighted noncommutative regular projective curve over a field $k$. The category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves is a hereditary, locally noetherian Grothendieck category. We classify all tilting sheaves which have a non-coherent torsion subsheaf. In case of nonnegative orbifold Euler characteristic we...

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we...

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we...

Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal 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 al...

Tilting modules over commutative rings were recently classified in [12]: they correspond bijectively to faithful Gabriel topologies of finite type. In this note we extend this classification by dropping faithfulness. The counterpart of an arbitrary Gabriel topology of finite type is obtained by replacing tilting with the more general notion of a si...

There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For he...

For a commutative noetherian ring \(R\), we establish a bijection between the resolving subcategories consisting of finitely generated \(R\)-modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R[1]\) in their heart. Under this bijection, the t-structure...

We introduce the new concept of silting modules. These modules generalize tilting modules over an arbitrary ring, as well as support $\tau $-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 partia...

The workshop brought together experts on localisation theory and tilting theory from different parts of mathematics with the aim of fully exploiting the power of some recent developments in so far rather independent contexts. The intensive exchange during the workshop is expected to lead to new and strengthened synergies and to new applications.

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochster's conjecture on the...

The concept of recollement is used to obtain a stratification of the derived module category of a ring which may be regarded as an analogue of a composition series for groups or modules. This analogy raises the problem whether a ‘derived’ Jordan Hölder theorem holds true; that is, are such stratifications unique up to ordering and equivalence? This...

A Jordan H\"older theorem is established for derived module categories of piecewise hereditary algebras. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.

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. By Nicolás...

Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied by L. Angeleri Hügel [in J. Algebra 307, No. 1, 361-376 (2007; Zbl 1151.16016)]. We then apply the tilting and the cotilt...

We characterize the hereditary torsion pairs of nite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples. The notion of torsion theory (or torsion pair) was introduced by S. E. Dickson (21) in the sixties in t...

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 dire...

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation typ...

Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic objects. We discuss the question of uniqueness of such a stratification, up to ordering and derived equivalence,...

Let R = ⊕ Reλ = ⊕ eλ R be an associative ring with enough idempotents indexed over a possibly infinite set Λ. Assume that {eλ : λ ∈ Λ} is a set of pairwise orthogonal primitive idempotents, and that R is locally bounded, that is, the projective modules eλR and Reλ are of finite length for each λ ∈ Λ. We prove the existence of almost split sequences...

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 relationshi...

We show that a tilting module T over a ring R admits an exact sequence 0 → R → T 0 → T 1 → 0 such that \(T_0,T_1\in\text{Add}(T)\) and HomR (T 1,T 0) = 0 if and only if T has the form S ⊕ S/R for some injective ring epimorphism λ : R → S with the property that \(\text{Tor}_1^R(S,S)=0\) and pdS R ≤ 1. We then study the case where λ is a universal lo...

In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631–650], the Telescope Conjecture was formulated for the module category of an artin algebra R as follows: “If C=(A,B) is a complete hereditary cotorsion pair in with A and B closed under direct limits, then ”. We extend this conjecture to arbitrary rings R,...

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions....

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext1R(M, T)=0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map MÄR Õi Î IQi® Õi Î I(MÄRQi){M\oti...

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions....

Let R be a hereditary, indecomposable, left pure semisimple ring. Inspired by [I. Reiten, C.M. Ringel, Infinite dimensional representations of canonical algebras, Canad. J. Math. 58 (2006) 180–224], we investigate the perfect cotorsion pair (C,D) in R-Mod generated by the preinjective component q. We show that there is a finitely generated product-...

Let R be a commutative domain. We prove that an R-module B is projective if and only if Ext^1(B,T)=0 for any torsion module T. This answers in the affirmative a question raised by Kaplansky in 1962.

It is well known that a module M over an arbitrary ring admits an indecomposable decomposition whenever it has the property that every local direct summand of M is a direct summand [28]. Recently, J. L. Gomez Pardo and P. Guil Asensio [18] have shown that requiring this property not only for M but for any direct sum M (ℵ) of copies of M even yields...

We study coherence properties of a module M over its en- domorphism ring. Hereby we extend to modules known char- acterizations of coherent and π-coherent rings. Moreover, we discuss the case that the category addM is covariantly, re- spectively contravariantly, finite in modR. Finally, we give a new characterization of endofinite modules. A left m...

Let R be a ring and MR be an R-module. We characterize the existence of Add M-covers and Add M-envelopes in terms of finiteness conditions of M over its endomorphism ring. We then present some applications related to the existence of well-behaved direct sum decompositions for direct products of copies of M. Our results can be viewed as natural exte...

R is Iwanaga-Gorenstein if and only if the class of Gorenstein projective right modules is induced by a finitely generated cotilting module (Corollary 3.8). This work was started while the first author was a Ramon y Cajal Fellow at the Universitat Aut`onoma de Barcelona. The research of the second author was funded by the DGI (Spain) and FEDER thro...

We investigate the indecomposable direct summands of a product of modules M_i \prod_{i \in I} M_i by studying categorical properties of M = Mi | i Î I {\cal M} = {M_i | i \in I} as well as module-theoretical properties of M = \coprodi Î I Mi M = \coprod_{i \in I} M_i viewed as module over its endomorphism ring.

We study the existence of complements to a partial tilting module over an arbitrary ring. As a consequence, we show that a finitely generated partial tilting module over an artin algebra has a (possibly infinitely generated) complement.

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new noti...

We investigate the Auslander-Reiten quiver of a P-1-hereditary artin algebra Lambda by relating it to the Auslander-Reiten quiver of a suitable tilted algebra. We then obtain that the Auslander-Reiten components of Lambda are postprojective, or preinjective, or quasi-serial, or obtained from a quasi-serial translation quiver by coray insertions, or...

Over an arbitrary ring, we consider cotilting modules endowed with some finiteness conditions. We show that they correspond to pairs of dualities between certain cat-egories consisting of finitely presented modules. This extends the Cotilting Theorem proved by Colby for the noetherian case. The relationship with the cotilting modules studied by Col...

Let L {\mit\Lambda} be a connected artin algebra with the postprojective partition P0 , ¼, Pn , ¼, P¥ {\cal P}_0 , \cdots , {\cal P}_n , \cdots , {\cal P}\!_{\infty } , and assume that P0 ÈP1 {\cal P}_0 \cup {\cal P}_1 is closed under indecomposable submodules. Then either L {\mit\Lambda} is the radical square zero algebra given by the quiver [(A)\...

We discuss the existence of left and right almost split morphisms for a skeletal small category ℳ of modules over an arbitrary ring. To this end, we associate to ℳ a certain R-module M and investigate finiteness conditions on M viewed as a module over its endomorphism ring.