Abelian Category Research Articles

Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

Monoidal Categories, 2-Traces, and Cyclic Cohomology

AbstractIn this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ($\mathscr{C},\otimes$) endowed with a symmetric 2-trace,i.e., an$F\in \text{Fun}(\mathscr{C},\text{Vec})$satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in$F$”. Furthermore, we observe that if$\mathscr{M}$is a$\mathscr{C}$-bimodule category and$(F,M)$is a stable central pair,i.e.,$F\in \text{Fun}(\mathscr{M},\text{Vec})$and$M\in \mathscr{M}$satisfy certain conditions, then$\mathscr{C}$acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

K-theoretic obstructions to bounded t-structures

Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $$-1$$ . The main results of this paper are that $$\mathrm {K}_{-1}(E)$$ vanishes when E is a small stable $$\infty $$ -category with a bounded t-structure and that $$\mathrm {K}_{-n}(E)$$ vanishes for all $$n\geqslant 1$$ when additionally the heart of E is noetherian. It follows that Barwick’s theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.

GORENSTEIN PROJECTIVE OBJECTS IN FUNCTOR CATEGORIES

Let $k$ be a commutative ring, let ${\mathcal{C}}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let ${\mathcal{B}}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$ of the functor category ${\mathcal{B}}^{{\mathcal{C}}}$, and we show that it is a subcategory of the Gorenstein projective objects ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ in ${\mathcal{B}}^{{\mathcal{C}}}$. Furthermore, we obtain criteria for when ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$. We show in examples that this can be used to compute ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ explicitly.

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.

Hereditary triangulated categories

We call a triangulated category hereditary provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the triangulated category is algebraical, we may replace the equivalence by a triangle equivalence. We give two intrinsic characterizations of hereditary triangulated categories using a certain full subcategory and the non-existence of certain paths. We apply them to piecewise hereditary algebras.

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.

Partial Traces on Additive Categories

In this paper, we study partial traces on additive categories. Haghverdi and Scott introduced partially traced symmetric monoidal categories generalizing traced symmetric monoidal categories given by Joyal, Street and Verity. The original example of a partial trace is given in terms of the execution formula on the category of vector spaces and linear functions. Malherbe, Scott and Selinger gave another example of a partial trace on the category of vector spaces, and they observed that we can define these two partial traces on arbitrary additive categories. A natural question is: what kind of partial traces does the category of vector spaces have? We give a (partial) answer to this question. Our main result is: every abelian category has a largest partial trace. Here, “largest” means that every partial trace on the abelian category is obtained by restricting the domain of the largest partial trace. As a corollary, we show that the partial trace given by Malherbe, Scott and Selinger is the largest partial trace on the category of vector spaces.

D-abelian quotients of (d + 2)-angulated categories

Let T be a triangulated category. If T is a cluster tilting object and I=[addT] is the ideal of morphisms factoring through an object of add T, then the quotient category T/I is abelian. This is an important result of cluster theory, due to Keller–Reiten and König–Zhu. More general conditions which imply that T/I is abelian were determined by Grimeland and the first author.Now let T be a suitable (d+2)-angulated category for an integer d⩾1. If T is a cluster tilting object in the sense of Oppermann–Thomas and I=[addT] is the ideal of morphisms factoring through an object of add T, then we show that T/I is d-abelian.The notions of (d+2)-angulated and d-abelian categories are due to Geiss–Keller–Oppermann and Jasso. They are higher homological generalisations of triangulated and abelian categories, which are recovered in the special case d=1. We actually show that if Γ=EndTT is the endomorphism algebra of T, then T/I is equivalent to a d-cluster tilting subcategory of mod Γ in the sense of Iyama; this implies that T/I is d-abelian. Moreover, we show that Γ is a d-Gorenstein algebra.More general conditions which imply that T/I is d-abelian will also be determined, generalising the triangulated results of Grimeland and the first author.

Computations of atom spectra

AbstractThis is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of non‐commutative rings. We illustrate our method and results by several examples.

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.

Auslander–Reiten (d + 2)-angles in subcategories and a (d + 2)-angulated generalisation of a theorem by Brüning

Let Φ be a finite dimensional algebra over an algebraically closed field k and assume gldimΦ≤d, for some fixed positive integer d. For d=1, Brüning proved that there is a bijection between the wide subcategories of the abelian category mod Φ and those of the triangulated category Db(mod Φ). Moreover, for a suitable triangulated category M, Jørgensen gave a description of Auslander–Reiten triangles in the extension closed subcategories of M.In this paper, we generalise these results for d-abelian and (d+2)-angulated categories, where kernels and cokernels are replaced by complexes of d+1 objects and triangles are replaced by complexes of d+2 objects. The categories are obtained as follows: if F⊆modΦ is a d-cluster tilting subcategory, consider F‾:=add{ΣidF|i∈Z}⊆Db(mod Φ). Then F is d-abelian and plays the role of a higher mod Φ having for higher derived category the (d+2)-angulated category F‾.

AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

Standard derived equivalences and $~\bf{D}$-standard abelian categories

In 1991, Rickard raised the following conjecture: Any derived equivalence between modulecategories is standard, i.e., isomorphic to the derived tensor functor by a two-sided tilting complex.We introduce the notion of a $~\bf{D}$-standard Abelian category, and conjecture that any module categoryis $~\bf{D}$-standard. We survey recent progress on these conjectures.

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.

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of Q-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.

Generalized Springer theory for D-modules on a reductive Lie algebra

Given a reductive group $G$, we give a description of the abelian category of $G$-equivariant $D$-modules on $\mathfrak{g}=\mathrm{Lie}(G)$, which specializes to Lusztig's generalized Springer correspondence upon restriction to the nilpotent cone. More precisely, the category has an orthogonal decomposition in to blocks indexed by cuspidal data $(L,\mathcal{E})$, consisting of a Levi subgroup $L$, and a cuspidal local system $\mathcal{E}$ on a nilpotent $L$-orbit. Each block is equivalent to the category of $D$-modules on the center $\mathfrak{z}(\mathfrak{l})$ of $\mathfrak{l}$ which are equivariant for the action of the relative Weyl group $N_G(L)/L$. The proof involves developing a theory of parabolic induction and restriction functors, and studying the corresponding monads acting on categories of cuspidal objects. It is hoped that the same techniques will be fruitful in understanding similar questions in the group, elliptic, mirabolic, quantum, and modular settings.

Auslander-Reiten duality for Grothendieck abelian categories

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.

Frobenius–Perron theory of modified ADE bound quiver algebras

The Frobenius–Perron dimension for an abelian category was recently introduced in [5]. We apply this theory to the category of representations of the finite-dimensional radical square zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius–Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius–Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius–Perron dimensions.

Injectivity and Projectivity Properties of The Category of Representation Modules of Rings

Let R, S be two rings with unity, M an S-module, and f: R → S a ring homomorphism. If the map M → M, m ↦ f(r)m is S-linear for any r ∈ R, then M is a representation module of ring R. This condition will be true if sf (r) − f (r)s ∈ Ann(M) for all r ∈ R and s ∈ S. The class of S-modules M, where sf(r) − f(r)s ∈ Ann(M) for all r ∈ R and s ∈ S, forms a category with its morphisms are all module homomorphisms. This class is denoted by \U0001d50d. The purpose of this paper is to prove that the category \U0001d50d is an abelian category which is under sufficient conditions enabling the category \U0001d50d has enough injective objects and enough projective objects. First, we prove the category \U0001d50d is stable under kernel and image of module homomorphisms, and a finite direct sum of objects of \U0001d50d is also the object of \U0001d50d. By using this two properties, we prove that \U0001d50d is the abelian category. Next, we determine the properties of the abelian category \U0001d50d, such that it has enough injective objects and enough projective objects. We obtain that, if S as R-module is an element of \U0001d50d, then the category \U0001d50d has enough projective objects and enough injective objects.