We introduce a notion of generalized Serre duality on a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$. This duality induces the generalized Serre functor on $\mathcal{T}$, which is a linear triangle equivalence between two thick triangulated subcategories of $\mathcal{T}$. Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the third term of an Auslander-Reiten triangle in $\mathcal{T}$; dually, the range of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the first term of an Auslander-Reiten triangle in $\mathcal{T}$.
We compute explicitly the generalized Serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of Artin and Zhang. We obtain a characterization of Gorenstein algebras: an artin algebra $A$ is Gorenstein if and only if the bounded homotopy category of finitely generated projective $A$-modules has Serre duality in the sense of Bondal and Kapranov.
For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{\rm{b}}({\rm mod}{\mbox{-}}A^{\rm op})$ and ${\mathbb{K}}^{\rm{b}}({\rm mod}{\mbox{-}}A)$. If $A=Λ$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ^{\rm op})$ and ${\mathbb{K}}^{\rm{b}}_{\rm ac}({\rm mod}{\mbox{-}}Λ).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{\rm{b}}({\rm mod}{\mbox{-}}Λ)$ admits a Serre f…
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we prove that $\mathscr{C}$ has Auslander-Reiten-Serre duality if and only if $\mathscr{C}$ has Auslander-Reiten $n$-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander-Reiten-Serre duality). Finally, assume further that $\mathscr{C}$ has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection …
In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.
We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver.
For an abelian category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. Special cases and examples are discussed. In particular, an abelian category with a periodic Serre duality and its equivariantization are studied.
An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.
We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of Kálmán.
In an ongoing project to classify all hereditary abelian categories, we provide a classification of Ext-finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence. In order to prove the classification, we will study the shapes of the Auslander-Reiten components extensively and use appropriate generalisations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.
Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.
This is a report on recent work of Chalupnik and Touze. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.
A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in arXiv:1801.07995, motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities. We introduce "partial Serre duality", which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an …
We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality.
Let G be a connected complex reductive group and let K be a symmetric subgroup of G. We prove a formula for the Drinfeld-Gaitsgory functor for the dg-category of K-equivariant sheaves on the flag manifold of G in terms of the Matsuki duality functor. As byproducts, we obtain a description of the Serre functor and the Deligne-Lusztig duality for (g,K)-modules.
Drinfeld suggested the definition of a certain endo-functor, called the pseudo-identity functor, on the category of D-modules on an algebraic stack. We extend this definition to an arbitrary DG category, and show that if certain finiteness conditions are satisfied, this functor is the inverse of the Serre functor. We apply this to the category of (g,K)-modules, and we stipulate that the pseudo-identity functor is the analog of the Deligne-Lusztig functor. In order to support this, we show that the pseudo-identity functor for (g,K)-modules is isomorphic to the composition of cohomological and c…
In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…
We present a new characterization of Lusztig's canonical quotient group. We also define a duality map: to a pair consisting of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra.
We characterize the generalized Auslander--Reiten duality on the category of finitely presented modules over some certain Hom-finite category. Examples include the category FI of finite sets with injections, and the one VI of finite dimensional vector spaces with linear injections over a finite field.
We introduce a notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category $\mathcal{C}$. This duality induces the generalized Auslander-Reiten translation functors $τ$ and $τ^-$. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories $\mathcal{C}_r$ and $\mathcal{C}_l$ of $\mathcal{C}$. A non-projective indecomposable object lies in the domain of $τ$ if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of $τ^-$ if and only if it a…
We study the effect of Alvis-Curtis duality on the unipotent representations of $\mathrm{GL}_n(q)$ in non-defining characteristic $\ell$. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both $\ell$ and the order of $q$ modulo $\ell$.
