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Let
$\textsf{T}$
be a triangulated category with shift functor
$\Sigma \colon \textsf{T} \to \textsf{T}$
. Suppose
$(\textsf{A},\textsf{B})$
is a co-t-structure with coheart
$\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$
and extended coheart
$\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{S}* \Sigma \textsf{S}$
, which is an extriangulated category. We show that there is a bijection between co-t-structures
$(\textsf{A}^{\prime},\textsf{B}^{\prime})$
in
$\textsf{T}$
such that
$\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$
and complete cotorsion pairs in the extended coheart
$\textsf{C}$
. In the case that
$\textsf{T}$
is Hom-finite,
$\textbf{k}$
-linear and Krull–Schmidt, we show further that there is a bijection between complete cotorsion pairs in
$\textsf{C}$
and functorially finite torsion classes in
$\textsf{mod}\, \textsf{S}$
.
We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.
Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.
The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport, and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Additionally, we include some computations of cosupport and provide a comparison of support and cosupport for cohomologically finite objects. Finally, we assign to any object of the category a subset of
$\mathrm {Spec}R$
, called the big cosupport, and study some of its properties.
For a weight structure w on a triangulated category
$\underline {C}$
we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case
$w=w^{sph}$
(the spherical weight structure on
$SH$
), we deduce the following converse to the stable Hurewicz theorem:
$H^{sing}_{i}(M)=\{0\}$
for all
$i<0$
if and only if
$M\in SH$
is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.
The main idea is to study M that has no weights
$m,\dots ,n$
(‘in the middle’). For
$w=w^{sph}$
, this is the case if there exists a distinguished triangle
$LM\to M\to RM$
, where
$RM$
is an n-connected spectrum and
$LM$
is an
$m-1$
-skeleton (of M) in the sense of Margolis’s definition; this happens whenever
$H^{sing}_i(M)=\{0\}$
for
$m\le i\le n$
and
$H^{sing}_{m-1}(M)$
is a free abelian group. We also consider morphisms that kill weights
$m,\dots ,n$
; those ‘send n-w-skeleta into
$m-1$
-w-skeleta’.
We prove that the derived categories of abelian categories have unique enhancements—all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a Grothendieck abelian category are also shown to have unique enhancements. Finally, we show that the derived category of complexes with quasi-coherent cohomology and the category of perfect complexes have unique enhancements for quasi-compact and quasi-separated schemes.
We prove that a finite-dimensional algebra
$ \Lambda $
is
$ \tau $
-tilting finite if and only if all the bricks over
$ \Lambda $
are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for
$ \tau $
-tilting infinite algebras.
As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to
$\tau$
-tilting reduction and brick labelling.
We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally
$3$
-Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank
$1$
modules and Plücker coordinates.
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.
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 strong 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 surfaces with orbifold points, 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.
Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is introduced in this paper. We give a Bazzoni characterization of tilting pairs in this setting. We also obtain the Auslander–Reiten correspondence of tilting pairs which classifies finite $\mathcal {C}$-tilting subcategories for a certain self-orthogonal subcategory $\mathcal {C}$ with some assumptions. This generalizes the known results given by Wei and Xi for the categories of finitely generated modules over Artin algebras, thereby providing new insights in exact and triangulated categories.
In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system
$\textbf {A}$
with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that
$\lim ^n\textbf {A}$
(the nth derived limit of
$\textbf {A}$
) vanishes for every
$n>0$
. Since that time, the question of whether it is consistent with the
$\mathsf {ZFC}$
axioms that
$\lim ^n \textbf {A}=0$
for every
$n>0$
has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces.
We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the
$\mathsf {ZFC}$
axioms that
$\lim ^n \textbf {A}=0$
for all
$n>0$
. We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to
$\lim ^n\textbf {A}=0$
will hold for each
$n>0$
. This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions
$\mathbb {N}^2\to \mathbb {Z}$
which are indexed in turn by n-tuples of functions
$f:\mathbb {N}\to \mathbb {N}$
. The triviality and coherence in question here generalise the classical and well-studied case of
$n=1$
.
This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.
There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.
For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$, there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$. Examples related to inflation categories and weighted projective lines are discussed.
We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimal d-Auslander–Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.
In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.
We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).
We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence.