Torsion Classes Research Articles

A categorification of biclosed sets of strings

We consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick.Finite congruence-uniform lattices admit an alternate partial order known as the core label order. In many cases, the core label order of a congruence-uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra. Analogous to torsion shadows, we introduce wide shadows, and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.

The Cohomological Brauer Group of a Torsion 𝔾m-Gerbe

Abstract Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a ${\mathbb{G}}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${\operatorname{Br}}^{\prime}(S)$ of $S$. We show that the cohomological Brauer group ${\operatorname{Br}}^{\prime}(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\operatorname{Br}}^{\prime}(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.

Relations for Grothendieck groups and representation-finiteness

For an exact category E, we study the Butler's condition “AR=Ex”: the relation of the Grothendieck group of E is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that E has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker assumption, AR=Ex implies that the category of syzygies in E has finitely many indecomposables.

On the Combinatorics of Gentle Algebras

AbstractFor $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

Rowmotion in slow motion

Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and mod A has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

On positivity in Sasaki geometry

It is well known that if the dimension of the Sasaki cone $${\mathfrak {t}}^+$$ is greater than one, then all Sasakian structures in $${\mathfrak {t}}^+$$ are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that $$\dim {\mathfrak {t}}^+>1$$ there are three possibilities, either all elements of $${\mathfrak {t}}^+$$ are positive, all are indefinite, or both positive and indefinite Sasakian structures occur in $${\mathfrak {t}}^+$$. We illustrate by examples how the type can change as we move in $${\mathfrak {t}}^+$$. If there exists a Sasakian structure in $${\mathfrak {t}}^+$$ whose total transverse scalar curvature is non-positive, then all elements of $${\mathfrak {t}}^+$$ are indefinite. Furthermore, we prove that if the first Chern class is a torsion class or represented by a positive definite (1, 1) form, then all elements of $${\mathfrak {t}}^+$$ are positive.

Minimal inclusions of torsion classes

Let Λ be a finite-dimensional associative algebra. The torsion classes of modΛ form a lattice under containment, denoted by torsΛ. In this paper, we characterize the cover relations in torsΛ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of torsΛ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice torsΛ. In particular, we study the torsion theory of a quotient of the preprojective algebra of type A n . We show that its torsion class lattice is isomorphic to the weak order on A n .

Independence posets

Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph $G$. We characterize when an independence poset is a lattice with a graph-theoretic condition on $G$. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-simpleminded collections arising in the representation theory of certain acyclic finite-dimensional algebras.

A torsion Jacquet-Langlands correspondence

We prove a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds.

T-structures on hereditary categories

We study aisles, equivalently t-structures, in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t-structures in the derived category. In some cases, including the case of finite-dimensional modules over a finite-dimensional hereditary algebra, we refine our results and show that the t-structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring.

A classification of torsion classes in abelian categories

We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s classification of Serre subcategories.

Non-Perturbative Superpotentials and Discrete Torsion

We discuss the non-perturbative superpotential in $${{E}_{8}} \times {{E}_{8}}$$ heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space $$\tilde {X}.$$ The superpotential is induced by the string wrapping holomorphic, isolated, genus zero curves. We show, in a specific example, that the superpotential is non-zero both on $$\tilde {X}$$ and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.

Horizontal and vertical mutation fans

We introduce diagrams for m-cluster categories which we call “horizontal” and “vertical” mutation fans. These are analogous to the mutation fans (also known as “semi-invariant pictures” or “scattering diagrams”) for the standard (m = 1) cluster case, which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the m-cluster category with special emphasis on the c-vectors (the “brick” labels).

Weakly Stable Torsion Classes

Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring $A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by studying torsion and completion with respect to a single regular and normal element.

Weak proregularity, weak stability, and the noncommutative MGM equivalence

Let A be a commutative ring, and let a be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived a-torsion and a-adic completion functors to be nicely behaved is the weak proregularity of a. In particular, the MGM Equivalence holds.Because weak proregularity is defined in terms of elements of the ring (it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory.In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem says that in the commutative case, the ideal a is weakly proregular if and only if the corresponding torsion class T is weakly stable.We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: ▷ For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. ▷ The Noncommutative MGM Equivalence, that holds under the same assumptions on T. ▷ A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli & Zhang from 2003.

English

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules $$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$ such that F0,..., Fn−1 are finitely generated free and Kn−1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

Torsion classes in the cohomology of KHT Shimura varieties

Torsion classes in the cohomology of KHT Shimura varieties

Pseudocomplements and strong pseudocomplements in lattices of module classes

In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices [Formula: see text]-tors (of hereditary torsion classes), [Formula: see text]-nat (the lattice of natural classes) and [Formula: see text]-conat (the lattice of conatural classes) coincide.

Lattice Properties of Oriented Exchange Graphs and Torsion Classes

The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Br\"ustle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graphs with finitely many elements are semidistributive lattices. Furthermore, if the quiver is mutation-equivalent to a type A Dynkin quiver or is an oriented cycle, then the oriented exchange graph is a lattice quotient of a lattice of biclosed subcategories of modules over the cluster-tilted algebra, generalizing Reading's Cambrian lattices in type A. We also apply our results to address a conjecture of Br\"ustle, Dupont, and P\'erotin on the lengths of maximal green sequences.

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in $${\mathrm {Spin}}(7)$$ starting with a cocalibrated $$\mathrm {G}_2$$ -structure on a seven-dimensional manifold. As $${\mathrm {Sp}}(2)\subseteq {\mathrm {SU}}(4)\subseteq {\mathrm {Spin}}(7)$$ , one may also obtain Calabi–Yau fourfolds or hyperkahler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics g with $$Hol(g)\subseteq {\mathrm {SU}}(4)$$ but $$Hol(g)\ne {\mathrm {Sp}}(2)$$ . We investigate when we actually get $$Hol(g)={\mathrm {SU}}(4)$$ and obtain many new explicit examples of Calabi–Yau fourfolds. The results rely on the connection between cocalibrated $$\mathrm {G}_2$$ -structures and hypo $${\mathrm {SU}}(3)$$ -structures and between the Hitchin and the hypo flow and on a systematic study of hypo $${\mathrm {SU}}(3)$$ -structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo $${\mathrm {SU}}(3)$$ -structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or $${\mathrm {Sp}}(2)$$ -holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to $${\mathrm {SU}}(4)$$ .