We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.
We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.
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…
We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for c…
We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate…
The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's $J$-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is …
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 subc…
The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach -- based on projective stabilization -- gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality -- the new concept is defined for any module over any ring. Un…
This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively …
We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a gr…
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.
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.
We prove that, in the category F of functors between F\_2-vector spaces, the tensor product between the second non constant standard injective functor and an exterior power functor is artinian. The only cas known to date was the artinian character of this injective ; our result is a step in the study of the third non constant standard injective. We use the division functor by the identity functor and facts from modular representation theory of the symmetric groups to obtain this theorem by detecing suitable composition factors.
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 this paper we consider the locally self-injective property of the product FI$^m$ of the category FI of finite sets and injections. Explicitly, we prove that the external tensor product commutes with the coinduction functor, and hence preserves injective modules. As corollaries, every projective FI$^m$-modules over a field of characteristic 0 is injective, and the Serre quotient of the category of finitely generated FI$^m$-modules by the category of finitely generated torsion FI$^m$-modules is equivalent to the category of finite dimensional FI$^m$-modules.
We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gor…
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…
Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. Denote by $D(G)$ the locally analytic distribution algebra of $G$. Orlik-Strauch and Agrawal-Strauch have studied certain exact functors defined on various categories of $\mathfrak{g}$-representations with image in the category of locally analytic $G$-representations or $D(G)$-modules. In this paper we prove that for suitably defined categories of $D(G)$-modules, this functor gives rise to injective homomorphisms at the level of Grothendieck groups. We also explain how this functor interacts with translation functors at…
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expec…
Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI$_G$ and VI$_q$, and answer positively an open question of Nagpal on representation stability theory.
In this paper we study adjunction relations between some natural functors on the representation category of the category of finite linearly ordered sets and order-preserving injections. We also prove that the Nakayama functor induces an equivalence from the Serre quotient of the category of finitely generated modules by the category of finitely generated torsion modules to the category of finite dimensional modules.
We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the Serre stable moduli stack. We explain why the method works by showing that the Serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lin…
We consider the homotopy category of complexes of projective modules over any gentle algebra. We prove that indecomposable $Σ$-pure-injective objects in s must be shifts of string or band complexes. We begin with a survey of purity in compactly generated triangulated categories, recalling some characterisations of $Σ$-pure-injective objects that mimic classical results from the model theory of modules. We then specify to the aforementioned homotopy category, and describe the compact objects in this case. Our proof uses a recent adaptation of a classification technique known as the functorial f…
In the previous article "Hearts of twin cotorsion pairs on exact categories", we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent.
Let $\mathscr{C}$ be a $(d+2)$-angulated category with $d$-suspension functor $Σ^d$. Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$-angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander--Reiten $(d+2)$-angles. Moreover, $τ_d=\mathbb{S}Σ^{-d}$ where $τ_d$ is $d$-Auslander-Reiten translation. These results generalize work by Bondal-Kapranov and Reiten-Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$, the quotient categ…
In this paper, we initiate the study of higher-dimensional Auslander-Reiten theory of self-injective algebras. We give a systematic construction of (weakly) $d$-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most $d$. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adap…
We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.
Let K 0 (Fp GLn(Fp)-proj) denote the Grothendieck group of finitely generated pro-jective Fp GLn(Fp)-modules. We show that the algebra C $\otimes$ n$\ge$0 K 0 (Fp GLn(Fp)-proj) with multiplication given by induction functors, is a polynomial algebra. We explicit generators and their relation with Deligne-Lusztig characters.
Let $\C$ be an $(n+2)$-angulated category with shift functor $Σ$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$ is equivalent to an $n$-cluster tilting subcategory of an abelian category $\textrm{mod}(Σ^{-1}\X)$. Moreover, we also prove that $\textrm{mod}(Σ^{-1}\X)$ is Gorenstein of Gorenstein dimension at most $n$. As an application, we generalize recent results of Jacobsen-Jørgensen and Koenig-Zhu.
Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In this paper, we first study some basic properties of pseudo-kernel of morphisms in $C$. When $C$ has pseudo-kernels, mod($C$) is abelian and then, in this case, we study radical functors, half exact functors, left exact functors and injective objects in mod($C$). At last, we extend the results for Mod($C$).
For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2}. We discuss the aspects of faithfulness, projective and injective objects, Ext-groups and some kind of adjunction formulas.
We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories.
For a noetherian ring $Λ$, the stabilization functor in the sense of Krause yields an embedding of the singularity category of $Λ$ into the homotopy category of acyclic complexes of injective $Λ$-modules. When $Λ$ contains a semisimple artinian subring $E$, we give an explicit description of the stabilization functor using the Hom complexes in the $E$-relative singular Yoneda dg category of $Λ$.
We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the …
A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ is fully faithful on the full subcategory of projective $A$-modules, for a given projective $A$-module $P$. By definition, Morita algebras are the covers of self-injective algebras and then $P$ is a faithful projective-injective module. Conversely, we show that $A$ is a Morita algebra and $\operatorname{End}_A(P)^{op}$ is self-injective whenever $(A, P)$ is a cover of $\operatorname{End}_A(P)^{op}$ for a faithful projective-injective module $P$.
For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.
We study the role of the Serre functor in the theory of derived equivalences. Let $\mathcal{A}$ be an abelian category and let $(\mathcal{U}, \mathcal{V})$ be a $t$-structure on the bounded derived category $D^b \mathcal{A}$ with heart $\mathcal{H}$. We investigate when the natural embedding $\mathcal{H} \to D^b \mathcal{A}$ can be extended to a triangle equivalence $D^b \mathcal{H} \to D^b \mathcal{A}$. Our focus of study is the case where $\mathcal{A}$ is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exi…
Auslander's formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of C. The same approach shows for an arbitrary Grothendieck abelian category that its derived category and…
