We consider $Λ$ an artin algebra and $n \geq 2$. We study how to compute the left and right degrees of irreducible morphisms between complexes in a generalized standard Auslander-Reiten component of ${\mathbf{C_n}({\rm proj}\, Λ)}$ with length. We give conditions under which the kernel and the cokernel of irreducible morphisms between complexes in $\mathbf{C_n}({\rm proj}\, Λ)$ belong to such a category. For a finite dimensional hereditary algebra $H$ over an algebraically closed field, we determine when an irreducible morphism has finite left (or right) degree and we give a characterization, depending on the degrees of certain irreducible morphisms, under which $\mathbf{C_n}({\rm proj} \,H)$ is of finite type.
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 explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.
In representation theory, the double centraliser property is an important property for a module (bimodule). It plays a fundamental role in many theories. In this paper, we extend this property to complexes in derived categories of finite dimensional algebras, under the name derived double centraliser property. Characterizations for complexes with the derived double centraliser property and (two-sided) tilting complexes in derived categories of hereditary algebras are given. In particular, all complexes with this property in the derived categories of lower triangular matrix algebras are classif…
We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with finitely-generated homogeneous components are shown to be isomorphic to direct sums of indecomposable string and band complexes. The corresponding isoclasses are described, and the Krull-Remak-Schmidt-Azumaya property is verified. This classification problem is solved using the idea of functorial filtrations.
We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many non-isomorphic and nontrivial bounded complexes of projective objects, and with small homology. We also prove a version for finite dimensional algebras with finitely generated cohomology.
We study the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of an additive category $\mathcal{B}$ and the derived category $\mathsf{D}_{N}(\mathcal{A})$ of an abelian category $\mathcal{A}$. First we show that both $\mathsf{K}_N(\mathcal{B})$ and $\mathsf{D}_N(\mathcal{A})$ have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category $\mathcal{A}$, we show that $\mathsf{D}_{N}(\mathcal{A})$ is triangle equivalent to the ordinary derived ca…
We study varieties of complexes of projective modules with fixed ranks, and relate these varieties to the varieties of their homologies. We show that for an algebra of global dimension at most two, these two varieties are related by a pair of morphisms which are smooth with irreducible fibres.
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…
We study the category of $\mathbb{Z}$-indexed sequences over an abelian category and certain generalized homology functors for this category of sequences which are indexed by positive integers $a$ and $b$. By looking at the corresponding derived category, we show that there is an "optimal" subcategory of sequences for every choice of our generalized homology functors, namely, the category of $N$-complexes (sequences for which the differential $d$ satisfies $d^N = 0$) where $N = a + b$. In this optimal case we show that our homology functors reduce to Kapranov's homology functors $\operatorname…
We study generic objects in triangulated categories and characterize the finite dimensional algebras $A$ such that the derived categories $D(\Mod A)$ are generically trivial. This is an analogue of a result of Crawley-Boevey for module categories. As a consequence, we show that $D(\Mod A)$ is generically trivial if and only if the category of perfect complexes $K^b(\proj A)$ is locally finite.
Let $\mathscr{A}$ be a small abelian category. For a closed subbifunctor $F$ of $\Ext_{\mathscr{A}}^{1}(-,-)$, Buan has generalized the construction of the Verdier's quotient category to get a relative derived category, where he localized with respect to $F$-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relatively derived categories, we give a relative version on derived equivalences induced by $F$-tilting complexes. We discuss the relationships between rela…
We study Verdier quotients of diverse homotopy categories of a full additive subcategory $\mathcal E$ of an abelian category. In particular, we consider the categories $K^{x,y}({\mathcal E})$ for $x\in\{\infty, +,-,b\}$, and $y\in\{\emptyset,b,+,-,\infty\}$ the homotopy categories of left, right, unbounded complexes with homology being $0$, bounded, left or right bounded, or unbounded. Inclusion of these categories give a partially ordered set, and we study localisation sequences or recollement diagrams between the Verdier quotients, and prove that many quotients lead to equivalent categories.
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a $d-$compatibility degree $(-||-)$ on any pair of ``colored'' almost positive real Schur roots which generalizes previous definitions on the non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $Φ$ corresponding to the va…
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The firs…
Gillespie's Theorem gives a systematic way to construct model category structures on $\mathscr{C}( \mathscr{M} )$, the category of chain complexes over an abelian category $\mathscr{M}$. We can view $\mathscr{C}( \mathscr{M} )$ as the category of representations of the quiver $\cdots \rightarrow 2 \rightarrow 1 \rightarrow 0 \rightarrow -1 \rightarrow -2 \rightarrow \cdots$ with the relations that two consecutive arrows compose to $0$. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, an…
We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay $\opn{T}_2(R)$-modules.
Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. Secondly, we characterize the Hall algebra of complexes of fixed size by generators and relations, and relate it to the derived Hall algebra of $\A$. Finally, we give the integration map on the Hall algebra of $2$-term complexes over projectives.
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
Chain complexes of finitely generated free modules over orbit categories provide natural algebraic models for finite G-CW complexes with prescribed isotropy. We prove a p-hypoelementary Dress induction theorem for K-theory over the orbit category of a finite group, and use it to re-interpret some results of Oliver and Kropholler-Wall on acyclic complexes.
For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's prin…
In this paper, we prove Faltings' annihilator theorem for complexes over a CM-excellent ring. As an application, we give a complete classification of the t-structures of the bounded derived category of finitely generated modules over a CM-excellent ring of finite Krull dimension.
Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with coefficients in $M_g$ carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus $g+1$. The ordinary Hochschild complex of $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor $\mathfra…
For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic c…
We study liftings of abelian model structures to categories of chain complexes and construct a realization functor from the derived category of a Grothendieck abelian category equipped with a cofibrantly generated, hereditary abelian model structure to the homotopy category of that model structure.
In the terms of an `$n$-periodic derived category', we describe explicitly how the orbit category of the bounded derived category of an algebra with respect to powers of the shift functor embeds in its triangulated hull. We obtain a large class of algebras whose orbit categories are strictly smaller than their triangulated hulls and a realization of the phenomenon that an automorphism need not induce the identity functor on the associated orbit category.
