Let $R$ be a right notherian ring. We introduce the concept of relative singularity category $Δ_{\mathcal{X}}(R)$ of $R$ with respect to a contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along with some finiteness conditions on $\mathcal{X}$, we prove that $Δ_{\mathcal{X}}(R)$ is triangle equivalent to a subcategory of the homotopy category $\mathbb{K}_{\rm{ac}}(\mathcal{X})$ of exact complexes over $\mathcal{X}$. As an application, a new description of the classical singularity category $\mathbb{D}_{\rm{sg}}(R)$ is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right notherian ring to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.
In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively.
We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.
We consider the relationship between the relative stable category of Benson, Iyengar, and Krause and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism…
In this paper, we consider the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits silting objects. Secondly, for $A=KQ/I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mod^{\mathbb Z} A)$ has tilting objects. As a consequence, $D_{sg}(\mod^{\mathbb Z}A)$ is triangulated equivalent to the derived category of a hereditary algebra $H$ which is of finite repre…
We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations of quivers with potentials (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algeb…
We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We give some applications to proving that certain blocks have equivalent derived categories.
Let $K$ be an algebraically closed field. Let $(Q,Sp,I)$ be a skewed-gentle triple, $(Q^{sg},I^{sg})$ and $(Q^g,I^{g})$ be its corresponding skewed-gentle pair and associated gentle pair respectively. It proves that the skewed-gentle algebra $KQ^{sg}/< I^{sg}>$ is singularity equivalent to $KQ/< I>$. Moreover, we use $(Q,Sp,I)$ to describe the singularity category of $KQ^g/< I^g>$. As a corollary, we get that $\mathrm{gldim} KQ^{sg}/< I^{sg}><\infty$ if and only if $\mathrm{gldim} KQ/< I><\infty$ if and only if $\mathrm{gldim} KQ^{g}/< I^{g}><\infty$.
In this paper we show that the (un)bounded derived categories$\colon$(i) of the monomorphism category, (ii) of the morphism category and (iii) of the double morphism category, admit a periodic infinite ladder of recollements. These results are based on a characterization that we provide for a recollement of (compactly generated) triangulated categories to admit a ladder of some height going either upwards or downwards. Moreover, we introduce and study the singularity category of the monomorphism category over an Artin algebra $Λ$ and show that there is a periodic infinite ladder that connects …
The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence o…
We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed.
We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which are known or generalise known results, on singularity categories of algebras with radical square zero, of non-commutative deformations of Kleinian singularities, of $SL_3(\mathbb{C})$-quotient singularities and of Gorenstein toric threefolds.
Let $\Mod \CS$ denote the category of $\CS$-modules, where $\CS$ is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for $\Mod \CS$. This will have several interesting applications. In the second part, we apply our techniques to get some interesting recollements of derived categories in different levels. We specialize our results to path rings as well as graded rings.
In this paper, we consider $n$-perforated Yoneda algebras for $n$-angulated categories, and show that, under some conditions, $n$-angles induce derived equivalences between the quotient algebras of $n$-perforated Yoneda algebras. This result generalizes some results of Hu, König and Xi. And it also establishes a connection between higher cluster theory and derived equivalences. Namely, in a cluster tilting subcategory of a triangulated category, an Auslander-Reiten $n$-angle implies a derived equivalence between two quotient algebras. This result can be compared with the fact that an Auslander…
Let $\X$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the stable category $\underline\X$ of $\X$. We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we characterize simple hypersurface singularities of type $(\a_1)$ as comple…
Let $Λ= \left[\begin{array}{cc} A & 0 \\ M & B \end{array}\right] $ be an Artin algebra and $_BM_A$ a $B$-$A$-bimodule. We prove that there is a triangle equivalence $D_{sg}(Λ) \cong D_{sg}(A)\coprod D_{sg}(B)$ between the corresponding singularity categories if $_BM$ is semi-simple and $M_A$ is projective. As a result, we obtain a new method for describing the singularity categories of certain bounded quiver algebras.
We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for a subgroup $H$, often the normalizer of a $p$-subgroup of $G$. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.
We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.
We determine the singularity category of an arbitrary finite dimensional gentle algebra $Λ$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $Λ$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.
We construct a differential graded algebra (DGA) modelling certain $A_\infty$ algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely $H^*BG$ and $H_*ΩBG^{^\wedge}_p$. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander--Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.
We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint triple between the module categories. In particular, we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of Gorenstein projective modules. Moreover, we study this problem at the level of `big singularity categories' in the sense of Krause. Along the way we establish an explicit const…
We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $ω$ of an abelian category. We introduce the Frobenius category of $ω$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $ω$-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable category of (unnecessarily finitely-generated) Gorenstein-projective modules with singularity categories of rings. We prove that for a Gorenstein ring, the stable category of Gorenstein-projective mod…
In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krause's recollement for the stable derived category. In the special case of curved mixed Z-graded complexes, we show that one of our singu…
Let $Λ$ be a finite-dimensional algebra with finite global dimension, $R_k=K[X]/(X^k)$ be the $\mathcal{Z}$-graded local ring with $k\geq1$, and $Λ_k=Λ\otimes_K R_k$. We consider the singularity category $\mathcal{D}_{sg}(\mathrm{mod}^\mathcal{Z}(Λ_k))$ of the graded modules over $Λ_k$. It is showed that there is a tilting object in $\mathcal{D}_{sg}(\mathrm{mod}^\mathcal{Z}(Λ_k))$ such that its endomorphism algebra is isomorphic to the triangular matrix algebra $T_{k-1}(Λ)$ with coefficients in $Λ$ and there is a triangulated equivalence between $\mathcal{D}_{sg}(\mathrm{mod}^{\mathcal{Z}/k\m…
In this note we use results of Minamoto and Amiot, Iyama, Reiten to construct an embedding of the graded singularity category of certain graded Gorenstein algebras into the derived categories of coherent sheaves over its projective scheme. These graded algebras are constructed using the preprojective algebras of $d$-representation infinite algebras as defined by Herschend, Iyama and Oppermann. We relate this embedding to the construction of a semi-orthogonal decomposition of the derived category of coherent sheaves over the projective scheme of a Gorenstein algebra of parameter 1 described by …
This paper gives upper bounds for the dimension of the singularity category of a Cohen-Macaulay local ring with an isolated singularity. One of them recovers an upper bound given by Ballard, Favero and Katzarkov in the case of a hypersurface.
For an exact category $\mathcal{E}$ with enough projectives and with a $d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $\mathcal{E}$ and $\underline{\mathcal{E}}$. We also deduce that the Gorenstein projectives of $\mathcal{E}$ admit a $d\mathbb{Z}$-cluster tilting subcategory under some assumptions. Finally, we compute the $…
We construct a triangle equivalence between the singularity categories of two isolated cyclic quotient singularities of Krull dimensions two and three, respectively. This is the first example of a singular equivalence involving connected commutative algebras of odd and even Krull dimension. In combination with Orlov's localization result, this gives further singular equivalences between certain quasi-projective varieties of dimensions two and three, respectively.
We study the following generalization of singularity categories. Let X be a quasi-projective Gorenstein scheme with isolated singularities and A a non-commutative resolution of singularities of X in the sense of Van den Bergh. We introduce the relative singularity category as the Verdier quotient of the bounded derived category of coherent sheaves on A modulo the category of perfect complexes on X. We view it as a measure for the difference between X and A. The main results of this thesis are the following. (i) We prove an analogue of Orlov's localization result in our setup. If X has isolated…
We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.
We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let $ω$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $ω$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_ω}$, where $\mathcal{G}_ω$ is the subcategory of all $ω$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent …
We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.
We associate a t-structure to a family of objects in D(A), the derived category of a Grothendieck category A. Using general results on t-structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Beilinson's equivalences.
For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces…
In this article, we investigate the condition for the hearts of twin cotorsion pairs to be equivalent, compatibly with the associated functors. This is related to the vanishing of components of pairs through the associated functors.
We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.
We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.
The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $\mathsf{D_{sg}}(R)$ of a hypersurface $R$ of countable representation type. For a thick subcategory $\mathcal{T}$ of $\mathsf{D_{sg}}(R)$ and a full subcategory $\mathcal{X}$ of $\mathcal{T}$, we calculate the Rouquier dimension of $\mathcal{T}$ with respect to $\mathcal{X}$. Furthermore, we prove that the level in $\mathsf{D_{sg}}(R)$ of the residue field of $R$ with respect to each …
We study two constructions related to the intervals of finite posets. The first one is a poset. The second one is more complicated. Loosely speaking it can be seen as a poset with some extra zero-relations. As main result, we show that these two constructions are equivalent at the level of derived categories.
We study the properties of the relative derived category $D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$ relative to a full and additive subcategory $\mathscr{C}$. In particular, when $\mathscr{A}=A{\text -}\mod$ for a finite-dimensional algebra $A$ over a field and $\mathscr{C}$ is a contravariantly finite subcategory of $A$-$\mod$ which is admissible and closed under direct summands, the $\mathscr{C}$-singularity category $D_{\mathscr{C}{\text sg}}$($\mathscr{A}$)=$D_{\mathscr{C}}^{b}$($\mathscr{A}$)/$K^{b}(\mathscr{C})$ is studied. We give a sufficient condition wh…
Given 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 realize 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 which 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 …
We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued gro…
