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Rosanna Laking

Publications and source records attributed to Rosanna Laking.

12 recordsLinked to original sources
arXiv2026

Limits and colimits in silting theory with applications to the wall and chamber structure of an algebra

In this paper we consider a family of nested t-structures given by silting objects and construct a silting object corresponding to the intersection of aisles of these t-structures as a homotopy colimit. The dual construction for the cosilting case is given as a homotopy limit. The results are applied to construct two-term large silting objects corresponding to the numerical torsion pairs and the limiting walls in the wall and chamber structure of the real Grothendieck group of a finite dimensional algebra. In particular, in case the algebra is tame we can describe any numerical torsion pair in this way by combining our results with results of Plamondon and Yurikusa.

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arXiv2026

Mutation of torsion pairs for finite-dimensional algebras

We study the lattice $\mathbf{tors}(A)$ of torsion pairs in the category $\mathrm{mod}(A)$ of finitely generated modules over an artinian ring $A$. It was shown by the authors in previous work that $\mathbf{tors}(A)$ is isomorphic to a lattice formed by certain closed sets, called maximal rigid, in the Ziegler spectrum of the unbounded derived category $\mathrm{D}(A)$ of $A$. Moreover, the structure of this lattice is described by an operation on maximal rigid sets which encompasses (the dual of) silting mutation. In this paper we provide an explicit description of this operation and we discuss how it is reflected in the lattice $\mathbf{tors}(A)$. We establish a bijection between the wide intervals in $\mathbf{tors}(A)$ and the closed rigid sets in the Ziegler spectrum of $\mathrm{D}(A)$. Moreover, we show that the arrows in the Hasse quiver of $\mathbf{tors}(A)$ correspond to the closed rigid sets that are almost complete, or equivalently, that can be completed to a maximal rigid set in exactly two ways. Our results are most interesting in the case when $A$ is a finite dimensional algebra. In fact, we generalise results by Adachi, Iyama and Reiten, with an important difference: not every point in a maximal rigid set is mutable. We use the topology on the Ziegler spectrum to determine the mutable points. In the last section of the paper we illustrate our results by the example of a finite dimensional algebra arising from a triangulation of an annulus.

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arXiv2024

Simples in a cotilting heart

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.

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arXiv2024

Torsion pairs via the Ziegler spectrum

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $τ$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring.

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arXiv2022

Mutation and torsion pairs

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 triangulated categories. We show that mutation of pure-injective cosilting objects encompasses the classical concept of mutation for compact silting complexes. As an application we prove that any minimal inclusion of torsion classes in the category of finitely generated modules over an artinian ring corresponds to an irreducible mutation. This generalises a well-known result for functorially finite torsion classes.

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arXiv2019

Cotilting sheaves over weighted noncommutative regular projective curves

We consider the category $\operatorname{Qcoh}\mathbb{X}$ of quasicoherent sheaves where $\mathbb{X}$ is a weighted noncommutative regular projective curve over a field $k$. This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope $\infty$. In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in $\operatorname{Qcoh}\mathbb{X}$.

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arXiv2019

Classification of cosilting modules in type $\tilde{A}$

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.

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arXiv2018

Definability and approximations in triangulated categories

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.

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arXiv2018

Purity in compactly generated derivators and t-structures with Grothendieck hearts

We study t-structures with Grothendieck hearts on compactly generated triangulated categories $\mathcal{T}$ that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of $\mathcal{T}$ in terms of directed homotopy colimits. For a left nondegenerate t-structure ${\bf t}=(\mathcal{U},\mathcal{V})$ on $\mathcal{T}$, we show that $\mathcal{V}$ is definable if and only if ${\bf t}$ is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to ${\bf t}$ being homotopically smashing and to ${\bf t}$ being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of ${\bf t}$ are determined by purity conditions on the associated partial cosilting object.

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arXiv2016

The Ziegler spectrum for derived-discrete algebras

Let $Λ$ be a derived-discrete algebra. We show that the Krull-Gabriel dimension of the homotopy category of projective $Λ$-modules, and therefore the Cantor-Bendixson rank of its Ziegler spectrum, is $2$, thus extending a result of Bobiński and Krause. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective $Λ$-modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple.

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arXiv2015

Krull-Gabriel dimension of domestic string algebras

We calculate, confirming a conjecture of Schröer, the Krull-Gabriel dimension of the category of modules over any domestic string algebra, as well as the Cantor-Bendixson rank of each point of its Ziegler spectrum. We also determine the topology on this space.

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