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Eric J. Hanson

Publications and source records attributed to Eric J. Hanson.

At least 19 recordsLinked to original sources
arXiv2025

Para-exceptional sequences for tame hereditary algebras and McCammond-Sulway lattices

Noncrossing partition posets in a Coxeter group $W$ can fail to be lattices when $W$ is not finite. When the lattice property fails for $W$ of affine type, McCammond and Sulway's construction provides a larger lattice that contains the noncrossing partition poset and that furthermore is a combinatorial Garside structure. We construct a lattice, isomorphic to McCammond and Sulway's lattice, using the representation theory of a corresponding connected tame hereditary algebra and give a representation-theoretic proof that it is a combinatorial Garside structure. To construct the lattice, we introduce para-exceptional sequences and para-exceptional subcategories in the module categories of tame hereditary algebras. Para-exceptional sequences are generalizations of exceptional sequences obtained by enlarging the set of allowed entries to include all non-homogeneous bricks. A para-exceptional subcategory is a subcategory obtained by applying a certain closure-like operator to the wide subcategory generated by a para-exceptional sequence.

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arXiv2025

Bricks and $τ$-tilting theory under base field extensions

Let $K:k$ be a field extension and let $Λ$ be a finite-dimensional $k$-algebra. We investigate the relationship between $Λ$ and $Λ_K = Λ\otimes_k K$ with particular emphasis on various aspects of $τ$-tilting theory and bricks. We show that many types of objects for $Λ$ lift injectively to the same type of object for $Λ_K$, and many common constructions in $τ$-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the $τ$-cluster morphism category $\mathfrak{W}(Λ)$ of $Λ$ to the $τ$-cluster morphism category $\mathfrak{W}(Λ_K)$ of $Λ_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(Λ)$ to a group whenever $k$ is of characteristic zero which has many important consequences. In the appendix, E. J. Hanson shows the analogous result whenever $k$ is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of $τ$-tilting finiteness under base field extension.

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arXiv2025

Stabilization of the Spread-Global Dimension

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A prominent example is the spread exact structure on the category of representations of a fixed poset, in which the indecomposable projectives are the spread representations (that is, the indicator representations of convex and connected subsets). The spread-global dimension is known to be finite for finite posets and not uniformly bounded on the collection of all Cartesian products between two arbitrary finite total orders. It was conjectured in [AENY23] that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite total order and an arbitrary finite total order. We provide a positive answer to this conjecture and, more generally, prove that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite poset and an arbitrary finite total order. In doing so, we also establish the existence of finite spread-resolutions for finitely presented representations of arbitrary grid posets.

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arXiv2025

Exact structures for persistence modules

We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets, classical arguments about the relative projective modules of an exact structure make use of Auslander-Reiten theory. One of our results establishes a new adjunction which allows us to ``lift'' these arguments to certain infinite posets over which Auslander-Reiten sequences do not always exist. We give several examples of this lifting, in particular highlighting the non-existence and existence of resolutions by upsets when working with finitely presentable representations of the plane and of the closure of the positive quadrant, respectively. We then restrict our attention to finite posets. In this setting, we discuss the relationship between the global dimension of an exact structure and the representation dimension of the incidence algebra of the poset. We conclude with our second novel contribution. This is an explicit description of the irreducible morphisms between relative projective modules for several exact structures which have appeared previously in the literature.

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arXiv2024

A uniqueness property of τ exceptional sequences

Recently, Buan and Marsh showed that if two complete $τ$-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is $τ$-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a $τ$-exceptional sequence are linearly independent.

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arXiv2024

An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$

Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid $A$-modules are in bijection with the triangulations of the polygon $S.$ In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal{E}_\diamond$ on $\text{mod}\,A$ such that the maximal almost rigid $A$-modules in the usual exact structure are exactly the maximal rigid $A$-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal{E}_\diamond$ translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure $\mathcal{E}_\diamond$, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type $\mathbb{D}$ and gentle algebras.

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arXiv2024

Sequences of ICE-closed subcategories via preordered $τ^{-1}$-rigid modules

Let $Λ$ be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated $Λ$-modules to classify certain (generalized) intermediate $t$-structures in the bounded derived category. We classifying these "contravariantly finite ICE-sequences" using concepts from $τ$-tilting theory. More precisely, we introduce "cogen-preordered $τ^{-1}$-rigid modules" as a generalization of (the dual of) the "TF-ordered $τ$-rigid modules" of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered $τ^{-1}$-rigid modules and certain sequences of intervals of torsion-free classes. Combined with the results of Sakai, this yields a bijection with the set of contravariantly finite ICE-sequences (of finite length), and thus also with the set of $(m+1)$-intermediate $t$-structures whose aisles are homology-determined.

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arXiv2024

Homological approximations in persistence theory

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement.

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arXiv2024

Morphisms and extensions between bricks over preprojective algebras of type A

The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called "arcs". In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the "weak exceptional sequences" over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A.

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arXiv2024

$τ$-perpendicular wide subcategories

Let $Λ$ be a finite-dimensional algebra. A wide subcategory of $\mathsf{mod}Λ$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $\mathsf{mod}Λ$ arising from $τ$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $τ$-tilting reduction. This leads to a natural way to extend the definition of the "$τ$-cluster morphism category" of $Λ$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan-Marsh in the $τ$-tilting finite case and by Igusa-Todorov in the hereditary case.

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arXiv2024

Infinitesimal semi-invariant pictures and co-amalgamation

The purpose of this paper is to study the local structure of the semi-invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co-amalgamation, we show that this local structure is completely described by the semi-invariant pictures of a collection of self-injective Nakayama algebras. We then describe the cones of this local structure using cluster-like structures that we call support regular clusters. Finally, we show that the local structure is (piecewise linearly) invariant under cluster tilting.

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arXiv2024

Mutation of τ-exceptional pairs and sequences

We introduce a notion of mutation for $τ$-exceptional sequences of modules over arbitrary finite dimensional algebras. For hereditary algebras, we show that this coincides with the classical mutation of exceptional sequences. For rank two algebras, we show that mutation of $τ$-exceptional sequences is transitive if and only if mutation of support $τ$-tilting modules in the sense of Adachi-Iyama-Reiten is transitive.

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arXiv2024

Invariants of persistence modules defined by order-embeddings

One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these non-totally ordered posets $P$ can be obtained from restricting a given module to a subposet $X$ of $P$ that is totally ordered (or more generally, of finite representation type), and then computing the barcode (or the general direct sum decomposition) over $X$. We consider in this paper general order-preserving embeddings of representation-finite subposets $X$ into $P$ and study systematically the invariants obtained by decomposing the restriction of a given $P$-module $M$ to $X$ into its indecomposable summands. The restriction functor from $\mathrm{mod}\ P$ to $\mathrm{mod}\ X$ is well-studied, and it is known to be exact and admits both left and right adjoint functors, known as induction and co-induction functors. This allows us to obtain new homological insights, and also to re-interpret previous results. We use this approach also to determine bases of the image of these invariants, thus generalizing the concept of signed barcodes which is considered in the literature in relation to stability results. It turns out that considering only order-embeddings of one fixed poset $X$ into the poset $P$, and studying the set of all indecomposables obtained from $X$ introduces a lot of redundancy. We therefore also study iterated embeddings of several posets of increasing sizes, while limiting attention to only some indecomposables (that have not been obtained from embedding of smaller posets previously).

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arXiv2023

Pop-Stack Operators for Torsion Classes and Cambrian Lattices

The pop-stack operator of a finite lattice $L$ is the map $\mathrm{pop}^{\downarrow}_L\colon L\to L$ that sends each element $x\in L$ to the meet of $\{x\}\cup\text{cov}_L(x)$, where $\text{cov}_L(x)$ is the set of elements covered by $x$ in $L$. We study several properties of the pop-stack operator of $\mathrm{tors}Λ$, the lattice of torsion classes of a $τ$-tilting finite algebra $Λ$ over a field $K$. We describe the pop-stack operator in terms of certain mutations of 2-term simple-minded collections. This allows us to describe preimages of a given torsion class under the pop-stack operator. We then specialize our attention to Cambrian lattices of a finite irreducible Coxeter group $W$. Using tools from representation theory, we provide simple Coxeter-theoretic and lattice-theoretic descriptions of the image of the pop-stack operator of a Cambrian lattice (which can be stated without representation theory). When specialized to a bipartite Cambrian lattice of type A, this result settles a conjecture of Choi and Sun. We also settle a related enumerative conjecture of Defant and Williams. When $L$ is an arbitrary lattice quotient of the weak order on $W$, we prove that the maximum size of a forward orbit under the pop-stack operator of $L$ is at most the Coxeter number of $W$; when $L$ is a Cambrian lattice, we provide an explicit construction to show that this maximum forward orbit size is actually equal to the Coxeter number.

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arXiv2023

Composition series of arbitrary cardinality in modular lattices and abelian categories

For a certain family of complete modular lattices, we prove a Jordan--Hölder--Scheier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices which are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5*). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.

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arXiv2023

A facial order for torsion classes

We generalize the "facial weak order" of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting "facial semistable order": one using cover relations, and one using Bongartz completions of 2-term presilting objects. For $τ$-tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.

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arXiv2023

Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank

Let $Λ$ be a finite-dimensional associative algebra over a field. A semibrick pair is a finite set of $Λ$-modules for which certain Hom- and Ext-sets vanish. A semibrick pair is completable if it can be enlarged so that a generating condition is satisfied. We prove that if $Λ$ is $τ$-tilting finite with at most 3 simple modules, then the completability of a semibrick pair can be characterized using conditions on pairs of modules. We then use the weak order to construct a combinatorial model for the semibrick pairs of preprojective algebras of type $A_n$. From this model, we deduce that any semibrick pair of size $n$ satisfies the generating condition, and that the dimension vectors of any semibrick pair form a subset of the column vectors of some $c$-matrix. Finally, we show that no "pairwise" criteria for completability exists for preprojective algebras of Dynkin diagrams with more than 3 vertices.

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