Covering Theory of Categories without Free Action Assumption 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 we do not require that the action of G is free or locally bounded. We show that a G-covering is a universal “G-invariant” 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 by Cibils and Marcos (2006) without the assumption that the G-action is free. The orbit category construction by a cyclic group generated by an auto-equivalence modulo natural isomorphisms (e.g., the construction of cluster categories) is justified by a notion of the “colimit orbit category”. 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 G is a group (except for sections 8, 9) and k is a commutative ring, and all categories, functors and algebras are assumed to be k-linear. A pair (C, A) of a category C and a group homomorphism A : G→ Aut(C) is called a category with a G-action or a G-category, where Aut(C) is the group of automorphisms of C (not the group of auto-equivalences of C modulo natural isomorphisms). We set Aα := A(α) for all α ∈ G. If there is no confusion we always (except for sections 8, 9) denote G-actions by the same letter A, and simply write C = (C, A), and further we usually write αx := Aαx, αf := Aαf for all x ∈ C and all morphisms f in C. Classical covering technique. Let F : C → C be a functor with C a G-category. The classical setting of covering technique (see e.g., [8]) 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 2000 Mathematics Subject Classification. 18A32, 16B50, 16G20. This work is partially supported by Grants-in-Aid for Scientific Research (C) 17540036 from JSPS. 1 2 HIDETO ASASHIBA 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., even in the case when we use G-actions on the category K(prjR) or on ModR 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 applications. The first purpose of this paper is to generalize the covering technique to remove all these assumptions. Orbit categories and covering functors. Recall that to define a so-called “root category” D(modH)/[2] of a hereditary algebra H over a field in Happel [11] or in Peng-Xiao [16] we needed a generalization that removes at least conditions (1) and (2). It seems, however, even such a simple generalization was not found explicitly in the literature for a long time. The definition of root categories given in [16] works only for itself, and does not give a general definition of orbit categories. (Nevertheless, their definition was useful to show that the obtained orbit category is a triangulated category.) This gave us one of the motivations to start this work. Recently general definitions of orbit categories was given in [6] by Cibils and Marcos (let us denote it by C/1G) and in [14] by Keller (in the case that G is cyclic, let us denote it by C/2G). But we still did not understand the relationship between the notion of covering functors by Gabriel [8] and the orbit categories defined by them. We wanted to generalize Gabriel’s covering technique as much as possible. To this end it was necessary to generalize the definition of a covering functor. In the classical setting the first condition for a functor F to be a (Galois) covering functor (with group G) is that F = FAα for all α ∈ G. This leads us naturally to a definition of an invariance adjuster, a family of natural isomorphisms φ := (φα : F → FAα)α∈G (see Definition 1.1). The pair (F, φ) is called a (right) G-invariant functor, further which is called a G-covering functor if F is a dense functor such that both F (1) x,y : ⊕ α∈G C(αx, y) → C(Fx, Fy), (fα)α∈G 7→ ∑ α∈G F (fα) · φα,x, and F (2) x,y : ⊕ β∈G C(x, βy) → C(Fx, Fy), (fβ)β∈G 7→ ∑ β∈G φ β,y · F (fβ) are isomorphisms of k-modules for all x, y ∈ C. (In fact, it is enough to require that either F (1) x,y or F (2) x,y is an isomorphism for each x, y ∈ C.) Roughly speaking the definition of C := C/1G (resp. C ′ := C/2G) yields by setting all the F (1) x,y (resp. F (2) x,y ) to be the identities. In this paper we give a “left-right symmetric” construction of the orbit category C/G of C by G, which is a direct modification of Gabriel’s in [8], and give explicit isomorphisms between C/G, C/1G and C/2G (Proposition 2.10). If F has the same property but not necessarily a dense functor, then F is called a G-precovering functor, which is useful to induce G-covering functors by restricting the target category C. Our characterization (Theorem 2.8) of G-covering functors F : C → C combines the universality among G-invariant functors and an explicit form of F as the canonical COVERING THEORY WITHOUT FREE ACTION AND DERIVED EQUIVALENCES 3 functor P : C → C/G up to equivalences. We will show that the pushdown (defined as in [8]) of a G-covering functor induces a G-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. Free action assumption and a categorical generalization of CM-duality. Now, in [6] Cibils and Marcos gave two definitions of orbit categories. The first one (let us denote it by C/ f G) is defined only if the G-action is free, and the second one is the orbit category C/1G stated above, called the skew category, which is defined without the free action assumption. These two constructions coincide if the G-action is free. But they mainly used C/ f G and treated only the free action case in their main discussions in [6, sections 3, 4], where they recovered Cohen-Montgomery duality ([7]) in the categorical setting (section 3), and described the module category of C by that of C/G, which generalizes [10, Theorem 3.2] of Green, and conversely the module category of C/G by that of C (section 4). The second purpose of this paper is to show that all the corresponding statements in [6, sections 3, 4] hold without the free action assumption. Namely, (a) we show by elementary proofs that the orbit category construction and the smash product construction are mutual inverses. This gives us a full categorical generalization of Cohen-Montgomery duality, and is regarded as a categorical version of [5, Theorems 1.3, 2.2] of Beattie. In particular, this gives us a way to make G-actions free up to “(weakly) G-equivariant equivalences” (liberalization). Further (b) we will show again by elementary proofs that the pullup functor P : Mod(C/G) → Mod C (see section 4 for definition) induces an isomorphism from Mod(C/G) to the full subcategory Mod C of Mod C consisting of “G-invariant modules” (see Definition 6.1), and the pushdown functor P : Mod C → Mod(C/G) (see section 4 for definition) induces an equivalence from Mod C to the subcategory ModG(C/G) of Mod(C/G) consisting of G-graded modules and degree-preserving morphisms (see Definition 6.4). The latter gives a generalization of a categorical version of [4, Theorem 2.6] of Beattie. We note that the definition of smash products given in [6] is easy to handle and very useful, and that we can regard it as a categorical version of the definition of smash products by Quinn [17] (when the group is finite), and it enables us to formulate the covering construction by Green [10], and recovers the usual smash product of a k-algebra and the k-dual of a group algebra. Lax action of cyclic group. In [14] Keller defined the orbit category C/2G only when G is cyclic. This seems to be mainly because he only needed to construct an orbit category by a cyclic group generated by an auto-equivalence S of C modulo natural isomorphisms. As he remarked there, by replacing both C and S in a standard way by a category C and an automorphism S ′ of C, respectively, we can form the orbit category C′/2〈S 〉, which he denoted by C/S by abuse of notation and call it the orbit category of C by S. But after that some authors seem to forget this remark (e.g., when constructing cluster categories) and simply identified as C = C and S = S , and used the same formula for the definition of C/S as if S were an automorphism of C, which is not well-defined. The third purpose of this paper is to give a definition of the orbit 4 HIDETO ASASHIBA category C/S directly by replacing neither C nor S. More precisely, it is known that there are at least two standard ways of replacing the pair (C, S). One way is to replace C by a full subcategory consisting of a complete list of representatives of isoclasses of objects in C. Another way is to replace C by a category containing more objects as done in Keller and Vossieck [15]. We realized that the second construction has a form C/S#Z of the smash product of a Z-graded category C/S (called the “colimit orbit category” of C by S) and the group Z. Applying the generalization of Cohen-Montgomery duality above we see that the orbit category C/S is justified by using the colimit orbit category C/S. When S is an automorphism, of course we have C/S = C/〈S〉. Computation by quivers with relations. Finally, we give a way to compute the first orbit category C/1G using a quiver with relations to apply theorems in preceding sections. We generalized it to the monoid case to include a computation of preprojective algebras, with a hope to have wider applications.