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Weighted projective lines of tubular type and equivariantization

We prove that the categories of coherent sheaves over weighted projective lines of tubular type are explicitly related to each other via the equivariantization with respect to certain cyclic group actions.


Introduction
The notion of weighted projective lines is introduced in [7,8], which gives a geometric treatment to the representation theory of the canonical algebras in the sense of [15]. We are interested in weighted projective lines of tubular type. Recall that the category of coherent sheaves over such a weighted projective line is derived equivalent to the module category over a tubular algebra of the same type.
It is known due to [7,11,2] that the category of coherent sheaves over a weighted projective line of tubular type is equivalent to the category of equivariant coherent sheaves over an elliptic curve with respect to a certain cyclic group action; compare [13]. This result explains well that the classification of indecomposable modules over a tubular algebra in [15,12] has similar features as the classification of indecomposable coherent sheaves over an elliptic curve in [1].
In this paper, we show that the categories of coherent sheaves over weighted projective lines of different tubular types are related to each other, via the equivariantization with respect to certain cyclic group actions. Indeed, these cyclic groups are of order two or three, and the actions are the degree-shift actions, which are induced from the grading on the homogeneous coordinate algebras. Here, the equivariantization means forming the category of equivariant objects for a given finite group action on a category; compare [14,4,5].
Let us describe the main results of this paper. Let k be an algebraically closed field, whose characteristic is different from two or three. According to the types, weighted projective lines X of tubular type are denoted by X(2, 2, 2, 2; λ), X(3, 3, 3), X(4, 4, 2) and X(6, 3, 2), respectively. Here, λ ∈ k is not 0 or 1. The Auslander-Reiten translation on the category coh-X of coherent sheaves over X is induced from the degree-shift automorphism by the dualizing element ω, which is an element in the grading group of the homogeneous coordinate algebra of X.
In the tubular types, the dualizing element ω has order 2, 3, 4 and 6, according to their types. By the degree-shift automorphisms, we have a strict action on coh-X by the cyclic group Z ω and also by its subgroup. For a finite group G and a (strict) G-action on a category A, we denote by A G the category of equivariant objects. In particular, we have the category (coh-X) G for any subgroup G ⊆ Z ω.
The following theorem combines Propositions 3.2, 3.4 and 3.6. Here, we fix ǫ ∈ k satisfying ǫ 2 − ǫ + 1 = 0. Theorem. Keep the notation and assumptions as above. Denote by ω the dualizing element in the grading group of the homogeneous coordinate algebra of X. Then we have the following equivalences of categories.
The acting groups in (1) and (3) are of order two, and the one in (2) is of order three. For the proof, we construct explicit algebra homomorphisms between the corresponding homogeneous coordinate algebras. In all these three cases, the algebra homomorphisms are injective, whose images equal certain restriction subalgebras [2] of the homogeneous coordinate algebras. We mention that the idea using the restriction subalgebras goes back to [7,8,10].
The paper is structured as follows. In Section 2, we recall basic facts on the homogeneous coordinate algebras of weighted projective lines, and their restriction subalgebras. We introduce admissible homomorphisms between the string groups, which play a role in Proposition 2.4. Indeed, Proposition 2.4 claims that under certain conditions, the categories of coherent sheaves over different weighted projective lines are related via the equivariantization with respect to a degree-shift action.
In Section 3, we prove Theorem. For the proof, we construct explicit admissible homomorphisms between the string groups and algebra homomorphisms between the homogeneous coordinate algebras, which are verified to satisfy the conditions in Proposition 2.4. In Proposition 3.8, we show that the categories of coherent sheaves over weighted projective lines of the same weight type (2, 2, 2, 2) but with different parameters, might be related to each other via the equivariantization with respect to a certain degree-shift action.

The restriction subalgebras and equivariantization
In this section, we recall from [7,8] basic facts on the homogeneous coordinate algebra of a weighted projective line. We study admissible homomorphisms between the string groups and algebra homomorphisms between the homogeneous coordinate algebras, whose images equal the restriction subalgebras with respect to an effective subgroup of the string group. We formulate Proposition 2.4, which gives sufficient conditions on when two categories of coherent sheaves over different weighted projective lines are related by the equivariantization with respect to a certain degree-shift action.
2.1. Let t ≥ 1 be an integer. A weight sequence p = (p 1 , p 2 , · · · , p t ) of length t consists of positive integers satisfying p i ≥ 2. We might assume that the weight sequence p satisfies p 1 ≥ p 2 ≥ · · · ≥ p t .
The string group L(p) is an abelian group with generators x 1 , x 2 , · · · , x t subject to the relations p 1 x 1 = p 2 x 2 = · · · = p t x t , where this common element is denoted by c and called the canonical element. Here, the abelian group is written additively.
The string group L(p) is of rank one, where c is of infinite order. There is an isomorphism of abelian groups sending x i + Z c to the vector (0, · · · , 0,1, 0, · · · , 0) with1 on the i-th component. From this isomorphism, we deduce that each element x in L(p) is uniquely written in its normal form For each 1 ≤ i ≤ t, we have a surjective group homomorphism π i : L(p) → Z/p i Z with π i ( x j ) = δ i,j1 . Here, δ is the Kronecker symbol. Following [2, Definition 6.5], an infinite subgroup H ⊆ L(p) is effective provided that π i (H) = Z/p i Z, or equivalently,1 lies in π i (H) for each 1 ≤ i ≤ t.
Let p = lcm(p) = lcm(p 1 , p 2 , · · · , p t ) be the least common multiple of p. There is a unique surjective group homomorphism δ : L(p) → Z, called the degree map, given by δ( x i ) = p pi . We observe that the kernel of δ equals the torsion subgroup of L(p).
We mention that if the weight sequence p is of non-tubular type, or equivalently, the dualizing element ω is of infinite order, then the subgroup Z ω ⊆ L(p) is effective. For Let q = (q 1 , q 2 , · · · , q s ) be another weight sequence and L(q) be the corresponding string group. Then we have the map mult : L(q) → N as above.
We observe that by (AH1) the fiber π −1 ( x) for each x ∈ Imπ is a finite set.
2.2. Let k be an arbitrary field. A parameter sequence λ = (λ 1 , λ 2 , · · · , λ t ) of length t consists of a collection of pairwise distinct rational points on the projective line P 1 k . The parameter sequence is normalized if λ 1 = ∞, λ 2 = 0 and λ 3 = 1. A weighted projective line X = X(p, λ) of weight type p and parameter sequence λ is by definition the projective line P 1 k such that each point λ i has weight p i . We will assume that the parameter sequence λ is normalized. The homogeneous coordinate algebra S = S(p, λ) of the weighted projective line X is defined to be For an infinite subgroup H ⊆ L(p), we have the restriction subalgebra S H = x∈H S x of S, which is viewed as an H-graded algebra. We recall from [2, Lemma 6.2] that this algebra S H is a finitely generated k-algebra.
In general, the structure of these restriction subalgebras S H is not known. We mention that if p is of non-tubular type and H = Z ω, the corresponding restriction subalgebras are related to Kleinian singularities and Fuchsian singularities; see [8,Proposition 8.4] and [10,Sections 5 and 6]. In what follows, we explain the importance of effective subgroups H in L(p) = L.
We recall that the weighted projective line X is endowed with a structure sheaf of L-graded commutative noetherian algebras. Then the abelian category coh-X of (L-graded) coherent sheaves on X is defined. In what follows, we will recall a more convenient description of coh-X via graded S-modules.
We denote by mod L -S the abelian category of finitely generated L-graded Smodules, and by mod L 0 -S its Serre subcategory formed by finite dimensional modules. We denote by qmod L -S = mod L -S/mod L 0 -S the quotient abelian category. By [7, Theorem 1.8] the sheafification functor yields an equivalence We will identify these two categories.
We suppose that π : L(q) → L(p) is a group homomorphism with Imπ ⊆ L(p) an infinite subgroup, and that there is an algebra homomorphism φ : S(q, µ) → S(p, λ) satisfying φ(S(q, µ) y ) ⊆ S(p, λ) π( y) for each y ∈ L(q). We observe that φ induces an Imπ-graded algebra homomorphism Here, S(p, λ) Imπ is the restriction subalgebra with respect to the subgroup Imπ ⊆ L(p). For the Imπ-graded algebra π * S(q, µ), we apply the notation in the following paragraph to the surjective homomorphism L(q) → Imπ.
Given a surjective group homomorphism θ : G → K and a G-graded algebra A = g∈G A G , we define a K-graded algebra θ * A as follows: as an ungraded algebra θ * A = A, while its homogeneous component (θ * A) h = g∈θ −1 (h) A g for each h ∈ K. In other words, θ * A equals A, but with a coarser grading. Lemma 2.3. Keep the notation as above. Assume that the Imπ-graded algebra homomorphismφ is surjective and that π : L(q) → L(p) satisfies (AH2) in Definition 2.1. Thenφ is an isomorphism.
Proof. Let x ∈ Imπ. We apply (2.3) and (AH2) to infer that dim k (π * S(q, µ)) x = dim k S(p, λ) x , both of which are finite. By the surjectivity assumption, we infer thatφ is bijective on each degree. Then we are done.
We recall from [5,4,2] the equivariantization briefly. Let G be a group with unit e and A be a category. A strict G-action on A is a group homomorphism from G to the automorphism group of A, which assigns for each g ∈ G an automorphism F g on A. Hence, we have F e = Id A and F g F h = F gh . Temporarily, we write G mutliplicatively.
A G-equivariant object in A is a pair (X, α), where X is an object in A and α assigns to each g ∈ G an isomorphism α g : X → F g (X) subject to the relations This gives rise to the category A G of equivariant objects.
We observe if A is abelian, so is A G . Indeed, a sequence of equivariant objects is exact in A G if and only if so is the sequence of underlying objects in A.
For each subgroup N ⊆ L(q), we have a strict N -action on mod L(q) -S(q, µ) by setting F x = (− x) for each x ∈ N . Here, (− x) is the degree-shift functor by the element − x. This N -action induces strict N -actions on both qmod L(q) -S(q, µ) and coh-X(q, µ). These resulted N -actions are called the degree-shift actions. In particular, we will consider the categories (qmod L(q) -S(q, µ)) N and (coh-X(q, µ)) N of N -equivariant objects.

The proof of Theorem
In this section, we study the homogeneous coordinate algebras of weighted projective lines of tubular type. We prove Theorem in Propositions 3.2, 3.4 and 3.6. We construct explicit admissible homomorphisms between the string groups and algebra homomorphisms between the homogeneous coordinate algebras, which satisfy the conditions in Proposition 2.4.
We list these homogeneous coordinate algebras explicitly as follows.
). Here, we use different letters for the generators to avoid confusion. Moreover, we will use letters in the lower case to represent their images in the quotient algebras. For example, y i will represent the image of Y i in S (3, 3, 3).
For an element z = r c + r 1 z 1 + r 2 z 2 + r 3 z 3 ∈ L(4, 4, 2) in its normal form, we have π( z) = 2r c + r 1 x 1 + r 2 x 2 + r 3 x 3 + r 3 x 4 . This expression might not be a normal form. Indeed, it depends on whether r 1 and r 2 are larger than two or not.
We omit the remaining details.