Orbit Category Research Articles

Q-stability conditions on Calabi–Yau-𝕏 categories

We introduce $q$ -stability conditions $(\sigma,s)$ on Calabi–Yau- $\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$ , where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$ , where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$ . When $s=N$ is an integer, we show that $q$ -stability conditions can be identified with the stability conditions on $\mathcal {D}_N$ , provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$ -baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$ -stability conditions on $\mathcal {D}_\mathbb {X}$ . As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$ -stability conditions is a complex manifold of dimension $n+1$ . Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau- $\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$ -equivariant coherent sheaves on canonical bundles provide the Calabi–Yau- $\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster- $\mathbb {X}$ categories for acyclic quivers.

English

We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao's result when the ring is a finite dimensional algebra with finite global dimension over a field. As the first application, if $A,B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.

Serre Functors and Graded Categories

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 graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

On sheaves in finite group representations

Given a general finite group G , we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their quotients, equipped with atomic topology, we explicitly compute their sheaf categories via sheafification. This enables us to identify G -representations with various fixed-point sheaves. As a consequence, it provides an intrinsic new proof to the equivalence of M. Artin between the category of sheaves on the orbit category and that of group representations.

Higher limits over the fusion orbit category

The fusion orbit category F‾C(G) of a discrete group G over a collection C is the category whose objects are the subgroups H in C, and whose morphisms H→K are given by the G-maps G/H→G/K modulo the action of the centralizer group CG(H). We show that the higher limits over F‾C(G) can be computed using the hypercohomology spectral sequences coming from the Dwyer G-spaces for centralizer and normalizer decompositions for G.If G is the discrete group realizing a saturated fusion system F, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category Oc(F). This allows us to apply our results to the sharpness problem for the subgroup decomposition of a p-local finite group. We prove that the subgroup decomposition for every p-local finite group is sharp (over F-centric subgroups) if it is sharp for every p-local finite group with nontrivial center. We also show that for every p-local finite group (S,F,L), the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.

Endotrivial modules for finite groups via homotopy theory

Classifying endotrivialkGkG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite groupGG, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial modulekkdirect sum a projective module when restricted to a Sylowpp-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivialpp-subgroups with values in the unitsk×k^\times, viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers inGG, in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of thepp-subgroup complex andpp-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.

Sufficient and necessary conditions for hereditarity of infinite category algebras

We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects. As applications, we completely classify orbit categories, Quillen categories and transporter categories with left hereditary category algebras over a field.

Cluster categories of formal DG algebras and singularity categories

AbstractGiven a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau and realise its cluster category as a triangulated hull of an orbit category of a derived category and as the singularity category of a finite-dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results that stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by the Calabi-Yau dimension and thea-invariant of the algebra. Second, we prove that two DG orbit categories obtained from a DG endofunctor and its homotopy inverse are quasi-equivalent. As an application, we show that the higher cluster category of a higher representation infinite algebra is triangle equivalent to the singularity category of an Iwanaga-Gorenstein algebra, which is explicitly described. Also, we demonstrate that our results generalise the context of Keller–Murfet–Van den Bergh on the derived orbit category involving a square root of the AR translation.

Morita theorem for hereditary Calabi-Yau categories

We give a structure theorem for Calabi-Yau triangulated category with a hereditary cluster tilting object. We prove that an algebraic d-Calabi-Yau triangulated category with a d-cluster tilting object T such that its shifted sum T⊕⋯⊕T[−(d−2)] has hereditary endomorphism algebra H is triangle equivalent to the orbit category Db(modH)/τ−1/(d−1)[1] of the derived category of H for a naturally defined (d−1)-st root τ1/(d−1) of the AR translation, provided H is of non-Dynkin type. We also show that heredity of H follows from that of T when d=3, that of T⊕T[−1] when d=4, and similarly from a smaller endomorphism algebra for higher dimensions under vanishing of some negative self-extensions of T. Our result therefore generalizes the established theorems by Keller–Reiten and Keller–Murfet–Van den Bergh. Furthermore, we show that enhancements of such triangulated categories are unique. Finally we apply our results to Calabi-Yau reductions of a higher cluster category of a finite dimensional algebra and of the singularity category of an invariant subring.

Repetitive cluster categories of type $$D_n$$

In this paper, we show that the repetitive cluster category of type $$D_n$$ , defined as the orbit category $$\mathcal {D}^b(\mathrm {mod} {\textsf {k}}D_n)/(\tau ^{-1}[1])^p$$ , is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, [17], which we recover when $$p=1$$ .

The differential graded stable category of a self-injective algebra

Let A be a finite-dimensional, self-injective algebra, graded in non-positive degree. We define A -dgstab, the differential graded stable category of A , to be the Verdier quotient of the bounded derived category of dg-modules by the thick subcategory of perfect dg-modules. We express A -dgstab as the triangulated hull of the orbit category A -grstab / Ω ( 1 ) , reducing computations in the dg-stable category to those in the graded stable category. We provide a sufficient condition for the orbit category to be equivalent to A -dgstab and show this condition is satisfied by Nakayama algebras and Brauer tree algebras. We also provide a detailed description of the dg-stable category of the Brauer tree algebra corresponding to the star with n edges.

On weight complexes, pure functors, and detecting weights

This paper is dedicated to the study of weight complex functors (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that “ignore all non-zero weights”; these have a nice description in terms of weight complexes.An important example is the weight structure wG generated by the orbit category in the G-equivariant stable homotopy category SH(G); the corresponding pure cohomological functors into abelian groups are the Bredon cohomology associated to Mackey functors ones. Pure functors related to “motivic” weight structures are also quite useful.Our results also give some (more) new weight structures. Moreover, we prove that certain exact functors are conservative and “detect weights”.

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

Science orbits in the Saturn–Enceladus circular restricted three-body problem with oblate primaries

This contribution investigates the properties of a category of orbits around Enceladus. The motivation is the interest in the in situ exploration of this moon following Cassini’s detection of plumes of water and organic compounds close to its south pole. In a previous investigation, a set of heteroclinic connections were designed between halo orbits around the equilibrium points L1 and L2 of the circular restricted three-body problem with Saturn and Enceladus as primaries. The kinematical and geometrical characteristics of those trajectories makes them good candidates as science orbits for the extended observation of the surface of Enceladus: they are highly inclined, they approach the moon and they are maneuver-free. However, the low heights above the surface and the strong perturbing effect of Saturn impose a more careful look at their dynamics, in particular regarding the influence of the polar flattening of the primaries. Therefore, those solutions are here reconsidered by employing a dynamical model that includes the effect of the oblateness of Saturn and Enceladus, individually and in combination. The dynamical equivalents of the halo orbits around the equilibrium points L1 and L2 and their stable and unstable hyperbolic invariant manifolds are obtained in the perturbed models, and maneuver-free heteroclinic connections are identified in the new framework. A systematic comparison with the corresponding solutions of the unperturbed problem shows that qualitative and quantitative features are not significantly altered when the oblateness of the primaries is taken into account. Furthermore, it is found that the J2 coefficient of Saturn plays a larger role than that of Enceladus. From a mission perspective, the results confirm the scientific value of the solutions obtained in the classical circular restricted three-body problem and suggests that this simpler model can be used in a preliminary feasibility analysis.

Singularity categories of Gorenstein monomial algebras

In this paper, we consider the singularity category D s g ( mod A ) and the Z -graded singularity category D s g ( mod Z A ) for a Gorenstein monomial algebra A . Firstly, for a positively graded 1-Gorenstein algebra, we prove that its Z -graded singularity category admits silting objects. Secondly, for A = K Q / I being a Gorenstein monomial algebra, we prove that D s g ( mod Z A ) has tilting objects. As a consequence, D s g ( mod Z A ) is triangulated equivalent to the derived category of a hereditary algebra H which is of finite representation type. Finally, we give a characterization of 1-Gorenstein monomial algebras , and describe their singularity categories clearly by using the triangulated orbit categories of type A .

On isomorphisms of generalized multifold extensions of algebras without nonzero oriented cycles

Assume that a basic algebra A over an algebraically closed field with a basic set A 0 of primitive idempotents has the property that for all Let n be a nonzero integer, and and two automorphisms of the repetitive category of A with jump n (namely, they send to where is the i-th copy of A in for all ). If and coincide on the objects and if there exists a map such that for all morphisms then the orbit categories and are isomorphic as -graded categories.

Relative group homology theories with coefficients and the comparison homomorphism

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu’s theory to Adamson’s theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.

Classifying spaces and Bredon (co)homology for transitive groupoids

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.

Obstructions for gluing biset functors

We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the subquotients of a finite group G. The obstruction groups for this theory are the reduced cohomology groups of a category DG⁎ whose objects are the sections (U,V) of G, where 1≠V⊴U≤G, and whose morphisms are defined as a generalization of morphisms in the orbit category. Using this obstruction theory, we calculate the obstruction group for some well-known p-biset functors, such as the Dade group functor defined on p-groups with p odd.

Real ADE-Equivariant (co)Homotopy and Super M-Branes

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf's theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant 4-sphere-valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. We find that on each singular stratum the black brane's instanton contribution, namely its super Nambu-Goto/Green-Schwarz action, appears as the homotopy datum associated to the morphisms in the orbit category.