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Gustavo Jasso

Publications and source records attributed to Gustavo Jasso.

At least 19 recordsLinked to original sources
arXiv2026

Hopfological algebra, revisited

We propose an $\infty$-categorical approach to Khovanov--Qi's Hopfological algebra that, in particular, refines several foundational aspects of the theory by recasting the previous constructions in terms of $\infty$-categories of modules in monoidal $\infty$-categories. This perspective leads to a more general variant of Hopfological algebra that takes place over an arbitrary rigidly-compactly generated symmetric monoidal stable $\infty$-category, which we also outline in the article. In the appendix, we compare the construction of Hopfological derived categories to that of Holm--J{\o}rgensen's $Q$-shaped derived categories.

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arXiv2025

The Derived Auslander--Iyama Correspondence II: Bimodule Calabi--Yau Structures

Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras (dg) with finite-dimensional $0$-th cohomology such that the canonical generator of their perfect derived category is a basic $d\ZZ$-cluster tilting object, and basic Frobenius algebras that are twisted $(d+2)$-periodic as bimodules. In this article, we prove a variant of our general correspondence for bimodule right Calabi--Yau dg algebras. A novel ingredient is a new cohomology theory which contains obstructions to the existence and uniqueness of minimal $A_\infty$-bimodule structures on a graded bimodule. As an application of our results, we obtain, to our knowledge, the first example of an algebraic triangulated category with a triangulated Calabi--Yau structure that cannot be lifted to a bimodule right Calabi--Yau structure on any of its dg enhancements.

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arXiv2025

Rickard's Derived Morita Theory: Review and Outlook

We survey the main results in Jeremy Rickard's seminal papers `Morita theory for derived categories' and `Derived equivalences and derived functors'. These papers catalysed the later development of the Morita theory of (enhanced) compactly generated triangulated categories by Keller in the algebraic setting and by Schwede and Shipley in the topological setting. We also discuss the role of Rickard's notion of splendid equivalence in the context of Brou\'e's abelian defect group conjecture, and indicate an alternative proof of parts of Rickard's Derived Morita Theorem that leverages the notion of completion of a triangulated category.

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arXiv2025

Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.

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arXiv2025

Obstruction theory for $A$-infinity bimodules

We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of Bousfield--Kan type. The second page of this spectral sequence is mostly given by a new cohomology theory associated to a pair consisting of a graded algebra and a graded bimodule over it. This new cohomology theory fits in a long exact sequence involving the Hochschild cohomology of the algebra and the self-extensions of the bimodule. We show that the second differential of this spectral sequence is given by the Gerstenhaber bracket with a bimodule analogue of the universal Massey product of a minimal $A$-infinity algebra. We also develop a closely-related obstruction theory for truncated minimal $A$-infinity bimodule structures over (the truncation of) a fixed minimal $A$-infinity algebra; the second page of the corresponding spectral sequence is now mostly given by the vector spaces of self-extensions of the underlying graded bimodule and the second differential is described analogously to the previous one. We also establish variants of the above for graded algebras and graded bimodules that are $d$-sparse, that is they are concentrated in degrees that are multiples of a fixed integer $d\geq1$. These obstruction theories are used to establish intrinsic formality and almost formality theorems for differential graded bimodules over differential graded algebras. Our results hold, more generally, in the context of graded operads with multiplication equipped with an associative operadic ideal, examples of which are the endomorphism operad of a graded algebra and the linear endomorphism operad of a pair consisting of a graded algebra and a graded bimodule over it.

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arXiv2025

$Q$-shaped derived categories as derived categories of differential graded bimodules

We prove that, under mild assumptions, the $Q$-shaped derived categories introduced by Holm and J{\o}rgensen are equivalent to derived categories of differential graded bimodules over differential graded categories. This yields new derived invariance results for $Q$-shaped derived categories that allow us to extend known descriptions of such categories as derived categories of differential graded bimodules over (possibly graded) algebras.

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arXiv2024

Nakayama-type phenomena in higher Auslander--Reiten theory

This paper surveys recent contructions in higher Auslander--Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type $\mathbb{A}$ quivers. These include higher dimensional analogues of Nakayama algebras, of the mesh category of type $\mathbb{Z}\mathbb{A}_\infty$ and the tubes, and of the triangulated category generated by an $m$-spherical object. For $m=2$, the latter category can be regarded as the higher cluster category of type $\mathbb{A}_\infty$ whose cluster-tilting combinatorics are controlled by the triangulations of the cylic apeirotope.

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arXiv2024

On a theorem of B. Keller on Yoneda algebras of simple modules

A theorem of Keller states that the Yoneda algebra of the simple modules over a finite-dimensional algebra is generated in cohomological degrees $0$ and $1$ as a minimal $A_\infty$-algebra. We provide a proof of an extension of Keller's theorem to abelian length categories by reducing the problem to a particular class of Nakayama algebras, where the claim can be shown by direct computation.

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arXiv2023

The Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence

We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Derived Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors.

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arXiv2022

The Derived Auslander-Iyama Correspondence

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.

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arXiv2019

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

We show that the perfect derived categories of Iyama's $d$-dimensional Auslander algebras of type $\mathbb{A}$ are equivalent to the partially wrapped Fukaya categories of the $d$-fold symmetric product of the $2$-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk and those of its $(n-d)$-fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type $\mathbb{A}$. As a byproduct of our results, we deduce that the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk organise into a paracyclic object equivalent to the $d$-dimensional Waldhausen $\operatorname{S}$-construction, a simplicial space whose geometric realisation provides the $d$-fold delooping of the connective algebraic $K$-theory space of the ring of coefficients.

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arXiv2019

Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $\operatorname{S}$-constructions

These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type $\mathbb{A}$ to Eilenberg--Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen $\operatorname{S}$-construction from algebraic $K$-theory.

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arXiv2019

Higher Nakayama algebras I: Construction

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{ZA}_\infty$ and the tubes.

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arXiv2018

An introduction to higher Auslander-Reiten theory

This article consists of an introduction to Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander-Reiten theory, including the existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories, following the approach to classical Auslander-Reiten theory due to Auslander, Reiten, and Smalø. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for $d$-exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in $d$-cluster-tilting subcategories.

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arXiv2018

Simplicial structures in higher Auslander-Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, a family of algebras arising in the context of Iyama's higher Auslander-Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg-MacLane spaces and higher-dimensional versions of Waldhausen's $\operatorname{S}_\bullet$-construction in algebraic $K$-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable $\infty$-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type $\mathbb{A}$ quivers developed by Groth and Šťov\'ıček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable $\infty$-category.

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