arXiv subjects

Jenny August

Publications and source records attributed to Jenny August.

8 recordsLinked to original sources

arXiv2026

Higher torsion classes, $\tau_d$-tilting theory and silting complexes

Initiated in work by Adachi, Iyama and Reiten, the area known as $\tau$-tilting theory plays a fundamental role in contemporary representation theory. In this paper we explore a higher-dimensional analogue of this theory, formulated with respect to the higher Auslander-Reiten translation $\tau_d$. In particular, we associate to any functorially finite $d$-torsion class a maximal $\tau_d$-rigid pair and a $(d+1)$-term silting complex. In the case $d=1$, the notions of maximal $\tau_d$-rigid and support $\tau$-tilting pairs coincide, and our theory recovers the classical bijections. However, the proof strategies for $d>1$ differ significantly. As an intermediate step, we prove that a $d$-cluster tilting subcategory of a module category induces a $d$-cluster tilting subcategory of the category of $(d+1)$-term complexes, producing novel examples of $d$-exact categories. We introduce the notion of a $d$-torsion class in the exact setup, and use this to obtain the aforementioned $(d+1)$-term silting complex. We moreover apply our theory to study $d$-APR tilting modules and slices. To illustrate our results, we provide explicit combinatorial descriptions of maximal $\tau_d$-rigid pairs and $(d+1)$-term silting complexes for higher Auslander and higher Nakayama algebras.

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arXiv2023

Categories for Grassmannian cluster algebras of infinite rank

We construct Grassmannian categories of infinite rank, providing an infinite analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. Each Grassmannian category of infinite rank is given as the category of graded maximal Cohen-Macaulay modules over a certain hypersurface singularity. We show that generically free modules of rank $1$ in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra of infinite rank. Moreover, this bijection is structure preserving, as it relates rigidity in the category to compatibility of Plücker coordinates. Along the way, we develop a combinatorial formula to compute the dimension of the $\mathrm{Ext}^1$-spaces between any two generically free modules of rank $1$ in the Grassmannian category of infinite rank.

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arXiv2023

A characterisation of higher torsion classes

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and $d$-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the $d$-torsion classes in $\mathcal{M}$ form a complete lattice. Moreover, we use the characterisation to classify the $d$-torsion classes associated to higher Auslander algebras of type $\mathbb{A}$, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.

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arXiv2022

Stability conditions for contraction algebras

This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a 3-fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial, and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms, (2) a classification of tilting complexes in the derived category of a contraction algebra, (3) contractibility of a certain stability space associated to the flop, and (4) a new proof of the $K(π,1)$-theorem in various finite settings, which includes ADE braid groups.

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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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arXiv2021

Silting and Tilting for Weakly Symmetric Algebras

If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.

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arXiv2020

The Tilting Theory of Contraction Algebras

To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these algebras and then use the structure of an associated hyperplane arrangement to control the compositions, obtaining a faithful group action on the bounded derived category. Further, we determine precisely those standard equivalences which are induced by two-term tilting complexes and show that any standard equivalence between contraction algebras (up to algebra isomorphism) can be viewed as the composition of our constructed functors. Thus, for a contraction algebra, we obtain a complete picture of its derived equivalence class and, in particular, of its derived autoequivalence group.

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arXiv2020

On the Finiteness of the Derived Equivalence Classes of some Stable Endomorphism Rings

We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $f \colon X \to \mathrm{Spec}\ R$, where $X$ has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $A_{\mathrm{con}}$ known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $A_{\mathrm{con}}$ and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from $f$. This provides evidence towards a key conjecture in the area.

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