Covering Theory of Categories without Free Actions and Derived Equivalences

Let G be a group of automorphisms of a category C. We give a definition for a functor F : C → C to be a G-covering and three constructions of the orbit category C/G, which generalizes the notion of a Galois covering of locally finitedimensional categories with group G whose action on C is free and locally bonded. Here C/G is defined for any category C and does not require that the action of G is free or locally bounded. We show that a G-covering is a universal “right Ginvariant” functor and is essentially given by the canonical functor C → C/G. By using this we improve a covering technique for derived equivalence. Also we prove theorems describing the relationships between smash product construction and the orbit category construction as in Cibils and Marcos [4] without an assumption that the G-action is free. In addition, we give a presentation of a skew monoid category by a quiver with relations, which enables us to calculate many examples. Introduction Throughout this paper k is a commutative ring, and all categories and functors are assumed to be k-linear. Further F : C → C is a functor between categories C and C, and G is a group acting on C. We always assume that G-actions are faithful, i.e., G-actions are given by monomorphisms G ֌ Aut(C), which we usually regard as the inclusion, where Aut(C) is the group of automorphisms of C (not the group of auto-equivalences of C modulo natural isomorphisms). The classical setting of covering technique required the following conditions: (1) C is basic (i.e., x 6= y ⇒ x 6∼= y); (2) C is semiperfect (i.e., C(x, x) is a local algebra, ∀x ∈ C); (3) G-action is free (i.e.,1 6= ∀α ∈ G, ∀x ∈ C, αx 6= x); and (4) G-action is locally bounded (i.e., ∀x, y ∈ C, {α ∈ G | C(αx, y) 6= 0} is finite). But these assumptions made it very inconvenient to apply the covering technique to usual additive categories such as the bounded homotopy category K(prjR) of finitely generated projective modules over a ring R or even the module category ModR of R because these categories do not satisfy the condition (2) and hence we have to construct the full subcategory of indecomposable objects, which destroys additional structures like a structure of a triangulated category; and to satisfy the condition (1) we have to choose a complete set of representatives of isoclasses of objects that should be stable under the G-action, which is not so easy in practice; and also the condition (3) is difficult to check in many cases, e.g., when we use G-actions on the two categories above induced from that on R. These made the proof of the main theorem of a covering technique for derived equivalences in [1] unnecessarily complicated and prevented wider 1 2 HIDETO ASASHIBA applications. In this paper we generalize the covering technique to remove all these assumptions. Cibils and Marcos [4] and Keller [9] gave a similar generalization. However, they usually assume that the G-action is free. Here we do not assume this condition. We will show that all results given in [4, Sections 3 and 4] hold without this condition. In this paper we give three constructions of the orbit category of C by G: C/G, C/1G and C/2G. The construction of C/1G is the same as the skew category construction in [4], and the construction of C/ 2 G is the same as the orbit category in [9]. Our construction of C/G is a modification of Gabriel’s in [5]. As in Proposition 2.8 there are explicit isomorphisms between these three orbit categories. The main difference of this paper from [4] and [9] is in the use of an admissible family of natural isomorphisms φ := (φα : F → Fα)α∈G to define for F to be a right G-invariant functor that enables us to define a general concept of a G-covering functor; and in the use of a slightly weaker concept of a precovering functor that is useful to induce covering functors by restricting the target category C. Our characterization (Theorem 2.6) of G-covering functors F : C → C combines the universality among right G-invariant functors and an explicit form of F as the canonical functor C → C/G up to equivalences. We will show that the pushdown (defined as in [5]) of a G-covering functor induces a precovering functors between categories of finitely generated modules (Theorem 4.3) and between homotopy categories of bounded complexes of finitely generated projective modules (Theorem 4.4). This property will be used to show derived equivalences. The paper is organized as follows. In section 1 by generalizing the classical covering functors we give a definition of G-covering functors as right G-invariant functors with some isomorphism conditions. In section 2 we construct orbit categories and canonical functors. Using their universality we prove Theorem 2.6, which will be used to prove the fundamental theorem of a covering technique for derived equivalence (Theorem 4.7) in section 4. In section 3 we introduce skew group categories in a general setting as done by Reiten and Riedtmann [10] in the finite group case. In section 4 we develop a covering technique for derived equivalence in our general setting. In section 5 we prove results in [4, Section 3] without the assumption that the G-action is free, which gives a relationship between smash products and orbit categories. In particular this gives us a way to make G-actions free up to “weakly G-equivariant equivalences”. In section 6 we prove the results in [4] (Theorems 4.3 and 4.5) without the assumption that the G-action is free. More precisely, we will show that the pullup functor π : Mod C/G→ Mod C (see section 4 for definition) induces an equivalence between Mod C/G and the full subcategory Mod C of Mod C consisting of “G-invariant modules” (see Definition 6.1), and the pushdown functor π : Mod C → Mod C/G (see section 4 for definition) induces an equivalence between Mod C and the subcategory ModG C/G of Mod C/G consisting of G-graded modules and homogeneous morphisms (see Definition 6.3). In section 7 we give a way to compute the first orbit category C/1G using a quiver with relations to apply theorems in section 4. We generalized it to the monoid case to include a computation of preprojective algebras, with a hope to have wider applications. In section 8 we give some examples to illustrate the contents in previous sections. COVERING THEORY WITHOUT FREE ACTIONS AND DERIVED EQUIVALENCES 3 In the sequel, the notation δα,β stands for the Kronecker delta, namely it has the value 1 if α = β, and the value 0 otherwise. 1. Covering functors Definition 1.1. A family φ := (φα)α∈G of natural isomorphisms φα : F → Fα (α ∈ G) is said to be admissible if (1) φ1,x = 1lFx for each x ∈ C; (in fact, this is superfluous, see Remark 1.2) and (2) The following diagram is commutative for each α, β ∈ G and each x ∈ C: Fx φα,x // φβα,x ## H H H H H H H H H Fαx φβ,αx Fβαx. A right G-invariant functor is a pair (F, φ) of a functor F and an admissible family φ := (φα)α∈G of natural isomorphisms φα : F → Fα (α ∈ G). For right G-invariant functors (F, φ) : C → C and (F , φ) : C → C, a morphism (F, φ) → (F , φ) is a natural transformation η : F → F ′ such that for each α ∈ G the following diagram commutes: F φα −−−→ Fα η 