Quotient Category Research Articles

Triangulated Quotient Categories

A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jørgensen, respectively.

Factorial algebraic group actions and categorical quotients

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space.

Some mixed Hodge structures on $$l^{2}$$ -cohomology groups of coverings of Kähler manifolds

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.

Framed moduli and Grassmannians of submodules

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$.

On monads of exact reflective localizations of Abelian categories

In this paper we define Gabriel monads as the idempotent monads associated to exact reflective localizations in Abelian categories and characterize them by a simple set of properties. The coimage of a Gabriel monad is a Serre quotient category. The Gabriel monad induces an equivalence between its coimage and its image, the localizing subcategory of local objects.

Vector fields and Luna strata

Let V be a G-module, where G is a complex reductive group. Let Z≔V//G denote the categorical quotient. One can ask if the Luna stratification of Z is intrinsic. That is, if φ:Z→Z is any automorphism, does φ send strata to strata? In Kuttler and Reichstein (2008) [1], the answer was shown to be yes for V a direct sum of sufficiently many copies of a G-module W. We show that the answer is yes for almost all V. The key is to consider the vector fields on Z. Our methods also show that complex analytic automorphisms preserve the stratification.

Category equivalences involving graded modules over path algebras of quivers

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and kQ its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(kQ)≔Gr(kQ)/Fdim(kQ) of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(kQ)≡ModS(Q)≡GrL(Q∘)≡ModL(Q∘)0≡QGr(kQ(n)). Here S(Q)=lim⟶EndkI(kQ1⊗n) is a direct limit of finite dimensional semisimple algebras; Q∘ is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from Q; L(Q∘) is the Leavitt path algebra of Q∘; L(Q∘)0 is its degree zero component; and Q(n) is the quiver whose incidence matrix is the nth power of that for Q. It is also shown that all short exact sequences in qgr(kQ), the full subcategory of finitely presented objects in QGr(kQ), split. Consequently qgr(kQ) can be given the structure of a triangulated category with suspension functor the Serre degree twist (−1); this triangulated category is equivalent to the “singularity category” Db(Λ)/Dperf(Λ) where Λ is the radical square zero algebra kQ/kQ≥2, and Db(Λ) is the bounded derived category of finite dimensional left Λ-modules.

An equivalence of categories for graded modules over monomial algebras and path algebras of quivers

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations, or, more generally, the path algebra of a finite quiver modulo a finite number of relations of the form “ path = 0 ”. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which there is an equivalence of categories QGr A ≡ QGr ( k Q ) . Here QGr A is the quotient category Gr A / Fdim of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules. The proof makes use of an algebra homomorphism A → k Q that may be of independent interest.

Cohomology theories based on flats

Let A be an associative ring with identity, K ( Flat A ) the homotopy category of flat modules and K p ( Flat A ) the full subcategory of pure complexes. The quotient category K ( Flat A ) / K p ( Flat A ) , called here the pure derived category of flats, was introduced by Neeman. In this category flat resolutions are unique up to homotopy and so can be used to compute cohomology. We develop theories of Tate and complete cohomology in the pure derived category of flats. These theories extend naturally to sheaves over semi-separated noetherian schemes, where there are not always enough projectives, but we do have enough flats. As applications we characterize rings with finite sfli and schemes which are locally Gorenstein.

On local categories of finite groups

Let G be a finite group. Over any finite G-poset \({\mathcal{P}}\) we may define a transporter category as the corresponding Grothendieck construction. Transporter categories are generalizations of subgroups of G, and we shall demonstrate the finite generation of their cohomology. We record a generalized Frobenius reciprocity and use it to examine some important quotient categories of transporter categories, customarily called local categories.

Cartier modules: Finiteness results

On a locally Noetherian scheme X over a field of positive characteristic p, we study the category of coherent X-modules M equipped with a pe-linear map, i.e. an additive map C : XX satisfying rC(m) C(rpe m) for all mM, rX. The notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main result in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F-finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of HartshorneSpeiser Ann. Math. 105: 4579, 1977, Lyubeznik J. reine angew. Math. 491: 65130, 1997, Sharp Trans. Amer. Math. Soc. 359: 42374258, 2007, EnescuHochster Alg. Num. Th. 2: 721754, 2008, and Hochster Contemp. Math. 448: 119127, 2007 about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F-finite scheme X Lyubeznik's F-finite modules J. reine angew. Math. 491: 65130, 1997 have finite length.

Recollement of homotopy categories and Cohen-Macaulay modules

AbstractWe study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.

Locally Stable Grothendieck Categories: Applications

We introduce for any Grothendieck category the notion of stable localizing subcategory, as a localizing subcategory that can be written as an intersection of localizing subcategories defined by indecomposable injectives. A Grothendieck category in which every localizing subcategory is stable is called a locally stable category. As a main result we give a characterization of these categories in terms of the local stability of a localizing subcategory and its quotient category. The locally coirreducible categories (in particular, the categories with Gabriel dimension) and the locally noetherian categories are examples of locally stable categories. We also present some applications to the category of modules over a left fully bounded noetherian ring, to the category of comodules over a coalgebra and to the category of modules over graded rings.

Coring stabilizers for a Hopf algebra coaction

The coring stabilizer Stab ( P ) is introduced for any prime ideal P of a right H-comodule algebra A such that the factor ring A / P is either right or left Goldie. This notion is used to obtain Hopf algebraic analogs of two category equivalences associated with a homogeneous space. The category of linearized quasicoherent sheaves on a noncommutative homogeneous space is interpreted as a suitable quotient category of the category of Hopf modules. Birational invariance of such quotient categories is proved. It is shown that for a birational H-coequivariant extension B of A properly defined subsets of prime ideals of A and B correspond to each other bijectively.

Lifting automorphisms of generalized adjoint quotients

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let \( \pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G \) be the categorical quotient where \( \mathfrak{g} \) is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism φ of X lifts to a map \( \Phi :{\mathfrak{g}^r} \to {\mathfrak{g}^r} \) commuting with π. As an application we consider the action of φ on the Luna stratification of X.

SEMI-DIVISORIALITY OF HOM-MODULES AND INJECTIVE COGENERATOR OF A QUOTIENT CATEGORY

In this paper, we study w-ity and (co-)semi-divisoriality of Hom-modules and the semi-divisorial envelope of <TEX>$Hom_R$</TEX>(M,N) under suitable conditions on R, M, and N. We also investigate an injective cogenerator of a quotient category.

Geometric invariant theory and Einstein–Weyl geometry

In this article, we give a survey of geometric invariant theory for Toric Varieties, and present an application to the Einstein–Weyl geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space C P 1 , 1 , 2 . We also find and classify all possible quotients.

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C , or, more generally, the enveloping algebra of any locally free Lie algebroid on X . The category of finitely generated graded modules of the Rees algebra D ˜ has a natural quotient category P D which imitates the category of modules on P r o j of a graded commutative ring. We show that the derived category D b ( P D ) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X . This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D .

General Heart Construction on a Triangulated Category (II): Associated Homological Functor

In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., t-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call the heart. Two extremal cases of torsion pairs are t-structures and cluster tilting subcategories. If the torsion pair comes from a t-structure, then its heart is nothing other than the heart of this t-structure. In this case, as is well known, by composing certain adjoint functors, we obtain a homological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes homological. In this paper, we unify these two constructions, to obtain a homological functor from the triangulated category, to the heart of any torsion pair.

Direct decompositions of mixed abelian groups

We consider a sufficiently large subcategory of the category of mixed abelian groups of finite torsion-free rank and its quotient catego ry obtained by annihilating those homomorphisms which factor through the torsion. We prove that the second category provides a good approximation to the first category, but is much simpler: the groups of morphisms in the second category are finite rank torsion-free groups. This renders it possible to exam ine direct decompositions of mixed groups by the same methods as in the case of finite rank torsion-free groups.