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Sibylle Schroll

Publications and source records attributed to Sibylle Schroll.

At least 37 records · Page 2Linked to original sources
arXiv2022

Stability spaces of string and band modules

The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear spans. We show how stability spaces of thin modules are related to order polytopes. In the case of non-thin modules, we show how the stability spaces of string and band modules are related to the stability spaces of the thin modules corresponding to the abstract string and band. We use this to analyse the way in which the stability space of a band module is the limit of stability spaces of string modules. Namely, the stability space of the band module is a union of cones, each of which is the limit of the stability spaces of a family of string modules.

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arXiv2022

Characterisations of trivial extensions

In this paper we give a characterisation of trivial extension algebras in terms of quivers with relations. This result is based on a explicit description of the ideal of relations of the trivial extension of an algebra, given by the first author in the appendix. We also give a new proof of Wakamatsu's theorem in terms of their quiver and relations, which determines when two given algebras have isomorphic trivial extensions.

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arXiv2022

On higher torsion classes

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes.

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arXiv2022

Exceptional sequences in the derived category of a gentle algebra

In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $\mathsf{K^b(A)}$ of a gentle algebra $\mathsf{A}$. We show that full exceptional sequences in $\mathsf{K^b(A)}$ exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterise when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we show that the action of the braid group together with the grading shift on full exceptional sequences in $\mathsf{K^b(A)}$ is transitive. For the case of surfaces of higher genus, we reduce the problem of transitivity to the problem of the existence of certain sequences of pairs of exceptional objects. Finally, we interpret the duality of a full exceptional sequence induced by the longest element in the associated symmetric group using Koszul duality.

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arXiv2022

Cluster structures for the $A_{\infty}$ singularity

We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_\infty$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type $A$ cluster categories of Holm-Jorgensen, Fisher and Paquette-Yildirim. As a consequence, $\mathcal{C}_2$ has cluster tilting subcategories modelled by certain triangulations of the (completed) $\infty$-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed $\infty$-gon.

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arXiv2022

The role of gentle algebras in higher homological algebra

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $Λ$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $Λ$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(Λ)$ contains a $d$-cluster tilting subcategory that is closed under $[d]$, then $Λ$ is derived equivalent to an algebra of Dynkin type $A$. Furthermore, our approach gives a geometric characterization of all $d$-cluster tilting subcategories of $\mathcal{D}^b(Λ)$ that are closed under $[d]$.

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arXiv2021

Derived categories of skew-gentle algebras and orbifolds

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

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arXiv2021

A geometric realization of silting theory for gentle algebras

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff-Kapranov in the context of triangulated surfaces. We also show that silting reduction corresponds to the cutting of the underlying surface as is the case for Calabi-Yau reduction of surface cluster categories as shown by Marsh-Palu and Qiu-Zhou.

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arXiv2020

On band modules and $τ$-tilting finiteness

In this paper, motivated by a $τ$-tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $τ$-tilting finite if and only if no band module is a brick. We also recover a criterion for the $τ$-tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

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arXiv2020

A $τ$-tilting approach to the first Brauer-Thrall conjecture

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded.

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arXiv2020

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.

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arXiv2020

Skew-gentle algebras and orbifolds: Oberwolfach talk, January 2020

This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (19-24 January 2020). It introduces a geometric orbifold model for the bounded derived category of a skew-gentle algebra. This is joint work with Daniel Labardini-Fragoso and Yadira Valdivieso-Díaz.

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arXiv2020

On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in earlier work by the second author. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.

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arXiv2020

On extensions for gentle algebras

We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

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arXiv2020

Addendum and Erratum: Mapping cones for morphisms involving a band complex in the bounded derived category of a gentle algebra

In this note we correct two oversights in [Mapping cones in the bounded derived category of a gentle algebra, J. Algebra 530 (2019), 163--194, also arXiv:1609.09688] which only occur when a band complex is involved. As a consequence we see that the mapping cone of a morphism between two band complexes can decompose into arbitrarily many indecomposable direct summands.

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arXiv2019

Algebras and varieties

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The case of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra.

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arXiv2019

On quasi-hereditary algebras

Establishing whether an algebra is quasi-hereditary or not is, in general, a difficult problem. In this paper we introduce a sufficient criterion to determine whether a general finite dimensional algebra is quasi-hereditary by showing that the question can be reduced to showing that a closely associated monomial algebra is quasi-hereditary. For monomial algebras, we give an explicit, easily verifiable, necessary and sufficient criterion to determine whether it is quasi-hereditary.

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