STABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS ^{1}

Kunio Yamagata ^{2}

Abstract. A finite dimensional self-injective algebra will be determined when it is stably equivalent to a positive self-injective algebra of Dynkin, tilted or quasi-tilted type.

1. Introduction

In a series of joint work with A. Skowroński, we have been studying self-injective algebras with Galois coverings by repetitive algebras [10] [11] [12] [13], and [4]. One of the aims of the work is to characterize self-injective algebras ring theoretically or module categorically which have Galois coverings by the repetitive algebras. A ring theoretical criterion theorem was found in [11] for a self-injective algebra to have a Galois covering by a repetitive algebra. On the other hand, study of the module category over a self-injective algebra depends on the type of a repetitive algebra which defines a Galois covering. In this survey paper, some of main theorems related to module categories in the joint work are arranged into two theorems (Theorems 2, 3). The proofs, however, are not unified (and refer the proofs in the references mentioned above), but a common idea of the proofs is to find an ideal so that a criterion theorem is applicable to conclude the existence of a Galois covering by a repetitive algebra.

Throughout this paper, K will be a fixed (commutative) field, and by an algebra we mean a basic and associative K-algebra which is not necessarily finite dimensional, but with a complete set of orthogonal primitive idempotents, that is, with a set of orthogonal primitive idempotents, say {e_{i}}_{i∈I} of an algebra Λ, such that Λ = ⨁_{i∈I} Λe_{i} = ⨁_{i∈I} e_{i}Λ. A K-category R is a category whose hom-sets are Kspaces and composition of morphisms are K-bilinear. Then, K-categories (finite Kcategories, respectively) are in one-to-one correspondence with algebras having fixed complete set of orthogonal primitive idempotents (finite dimensional algebras with identity, respectively). We freely identify K-categories with K-algebras. A will be a finite dimensional, connected self-injective algebra, and B and Λ will be finite dimensional algebras. Two finite dimensional self-injective algebras A_{1}, A_{2} are said to be socle equivalent if A_{1}/ soc A_{1} ≅ A_{2}/ soc A_{2} where soc is the socle of an algebra, and stably equivalent if their stable categories \underline{mod}A_{1} and \underline{mod}A_{2} of the categories of finite dimensional modules are isomorphic.

1The paper is in a final form and no version of it will be submitted for publication elsewhere.

2Supported by the Japan Society for the Promotion of Science, Grant in Aid for Science Research (C) (1) No. 155 400 12.

2. Repetitive categories

2.1. Let B be a K-category, and B_{n} = B, DB_{n} = DB the copies of B and DB, respectively. The repetitive category B̂ of B is the direct sum of K-spaces B̂ = ⨁_{n∈ℤ}(B_{n} ⊕ DB_{n}) with multiplication:

( ∑_{i}(b_{i}, f_{i})) · ( ∑_{j}(c_{j}, g_{j})) = ∑_{i}(b_{i}c_{i}, b_{i}g_{i} + f_{i}c_{i+1})

for b_{i}, c_{j} ∈ B_{i} and f_{i}, g_{j} ∈ DB_{i} for i ∈ ℤ. The category may be written as a matrix algebra without identity in the following way:

B̂ = ( \begin{matrix}... & ... & & & 0 \\ & B_{n−1} & DB_{n−1} \\ & & B_{n} & DB_{n} \\ & & & B_{n+1} & ... \\ 0 & & & & ... \end{matrix} )

where all B_{n} lie on the diagonal, and B̂ is the set of those matrices that all but finitely many entries are zero. Summation and multiplication are defined as ones of matrices with DB ⊗DB → 0 zero map. B̂ is a K-category whose object set is the disjoint union of copies of Obj(B) in each B_{n}.

2.2. A group G of automorphisms of a K-category B̂ is said to be admissible if any automorphism of G acts freely on Obj(B̂), and the G-orbit sets of Obj(B̂) is finite. The category B̂/G is then naturally defined by the G-orbits of B̂-objects and B̂- morphisms. See [1] for details. The orbit category B̂/G is clearly finite dimensional and self-injective.

An automorphism νB̂ of B̂ is called the Nakayama automorphism of B̂ if the restriction to any B_{n} ⊕ DB_{n} is identity onto B_{n+1} ⊕ DB_{n+1}. An automorphism φ of B̂ is said to be positive if φ(B_{n}) ⊆ ∑_{i≥n}(B_{i} ⊕ DB_{i}) (n ∈ ℤ) or, equivalently, φ(Obj(B_{n})) ⊆ ⋃_{i≥n} Obj(B_{i}) (n ∈ ℤ), and strictly positive if φ is positive and φ(B_{n}) ≠ B_{n} for all n.

Importance of the orbit categories by repetitive categories may be suggested by the classification theorem of representation-finite self-injective algebras over an algebraically closed field, which was proved by C. Riedtmann [8] [9] (and also by D. Hughes - J. Waschbüsch [3]). Also, D. Happel (1991) showed that the bounded category D^{b}(mod B) of a finite dimensional algebra B is embedded into the stable category \underline{mod}B̂ of the category of finite dimensional modules over B̂. Moreover, D^{b}(mod B) is isomorphic to \underline{mod}B̂ if and only if gldim B < ∞.

2.3. (1) A typical example of the orbit algebras by repetitive algebras is B̂/⟨νB̂⟩ which is isomorphic to the trivial extension algebra B ⋉ DB.

(2) Let A be 4-dimensional, local and self-injective algebras such that

A ≅ B̂/⟨φ⟩; B = k[x]/(x^{2}), φ^{2} = ψνB̂

for a positive automorphisms φ, ψ, and they are socle equivalent. Those algebras were first presented by T. Nakayama and C. Nesbitt in 1938 [5]. J. Rickard showed that such algebras are isomorphic if they are stably equivalent (see [15] [6]). Thus it should be noted that a socle equivalence does not imply a stable equivalence, in general.

3. Type of self-injective algebras

3.1. An algebra B is called quasi-tilted algebra if B ≅ End_{H}(T) where T is a tilting object in a hereditary abelian K-category H, that is, Hom_{H}(X, Y) and Ext_{H}^{1}(X, Y) are finite dimensional and Ext_{H}^{2}(X, Y) = 0 for all objects X, Y of H. This is equivalent to the statement that gldim(B) ≤ 2 and every finitely generated indecomposable Bmodule X has pd(X) ≤ 1 or id(X) ≤ 1. An algebra is said to be canonical [7] if its ordinary quiver has a unique source ω and a unique sink o, and consists of paths p_{1}, …, p_{n} (n ≥ 2) from ω to o, which meet each other only at ω and o. Moreover, in case n = 2 there is no non-trivial relation (i.e., the algebra is hereditary), and in case n ≥ 3 the length of each path is more than 1 and relations are

p_{1} + p_{2} + p_{3} = 0, p_{1} + λ_{i}p_{2} + p_{i} = 0 (i = 4, …, n)

for λ_{i} ≠ 0, 1 and λ_{i} ≠ λ_{j} for all i ≠ j .

Theorem 1. [2] An algebra B is quasi-tilted if and only if B ≅ End_{Λ}(T) where T is a tilting module over a hereditary algebra or a canonical algebra Λ.

3.2. Let A be a self-injective algebra. A is said to be positive (strictly positive, respectively) if A ≅ B̂/⟨φνB̂⟩ for some algebra B and positive (strictly positive, respectively) automorphism φ of B̂. (See [16].) A positive self-injective algebra A is said to be of

(i) (Δ-)tilted type if B is a tilted algebra of type Δ,

(ii) canonical type if B is the endomorphism algebra of a tilting module over a canonical algebra,

(iii) quasi-tilted type if B is a quasi-tilted algebra.

Thus, by Theorem 1, a positive self-injective algebra A is of quasi-tilted type if and only if A is of tilted type or canonical type.

4. Main Theorems

4.1. Some of main results on the module categories in [10] [12] [13] [4] are stated as follows.

Theorem 2. A self-injective algebra A is stably equivalent to a positive self-injective algebra of Δ-tilted type if and only if A is socle equivalent to a positive self-injective algebra of Δ-tilted type. Moreover, in case K is algebraically closed, those statements are equivalent to the statement that A is isomorphic to a positive self-injective algebra of Δ-tilted type.

Theorem 3. The followings are equivalent.

(1) A is stably equivalent to a strongly positive self-injective algebra of quasi-tilted type.

(2) A is isomorphic to a strongly positive self-injective algebra of quasi-tilted type.

4.2. Theorem 2 was first proved in [10] for algebras of non-Dynkin type, and later in [13] for algebras of Dynkin type (Δ = A_{n}, B_{n}, C_{n}, D_{n} (n ≥ 4), E_{6}, E_{7}, E_{8}, F_{4}, G_{2}). On the other hand, Theorem 3 was proved in [12] for non-Dynkin type, and recently in [4] for canonical type where a new characterization of a quasi-tilted algebra is given and a precise observation is required for the form of Auslander-Reiten components.

In any case, however, we have to find an ideal I satisfying the annihilator condition r_{A}(I) = eI for some e = e^{2}, and then, apply the following criterion theorem for the existence of a positive Galois covering.

Criterion Theorem 4 . [11] A self-injective algebra Λ is positive if there is an ideal I of Λ such that, for some idempotent e of Λ, the following conditions are satisfied:

(1) r_{Λ}(I) = eI,

(2) The canonical algebra epimorphism eΛe → eΛe/eIe splits. In this case, Λ is isomorphic to B̂/⟨φνB̂⟩ for B = Λ/I and a positive automorphism φ of B̂.

The idea of the application of the criterion theorem is to construct another selfinjective algebra Λ by making use of the annihilator condition (1) (see [11]), so that Λ has both conditions (1) and (2), and A and Λ are socle equivalent. Thus we can know the existence of a positive Galois covering of A up to socle. The converse of Theorem 4 is also true [14] and refer to the survey paper [16].

References

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[2] D. Happel, I. Reiten, Hereditary abelian categories with tilting objects over arbitrary basic fields, J. Algebra 256 (2002) 414–432.

[3] D. Hughes, J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. 47 (1983) 347–364.

[4] O. Kerner, A. Skowroński, K. Yamagata, Invariability of self-injective algebras of quasitilted type under stable equivalences, to appear

[5] T. Nakayama, C. Nesbitt, Note on symmetric algebras, Ann. of Math. 39 (1938) 659–668.

[6] Y. Ohnuki, K. Takeda, K. Yamagata, Automorphisms of repetitive algebras, J. Algebra 232 (2000) 708–724.

[7] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., 1099, Springer-Verlag, 1984.

[8] C. Riedtmann, Representation-finite self-injective algebras of class 𝔸_{n}, In: Representation Theory II, Lecture Notes in Math., 832, Springer-Verlag, 1980, pp. 449–520.

[9] C. Riedtmann, Representation-finite selfinjective algebras of class 𝔻_{n}, Compositio Math., 49 (1983), 231–282.

[10] A. Skowroński, K. Yamagata, Stable equivalence of selfinjective algebras of tilted type, Arch. Math. (Basel) 70 (1998), 341–350.

[11] A. Skowroński, K. Yamagata, Galois coverings of selfinjective algebras by repetitive algebras, Trans. Amer. Math. Soc. 351 (1999), 715–734.

[12] A. Skowroński, K. Yamagata, On invariability of selfinjective algebras of tilted type under stable equivalences, Proc. Amer. Math. Soc. 132 (2004), 659–667.

[13] A. Skowroński, K. Yamagata, Stable equivalence of selfinjective algebras of Dynkin type, Algebras and Representation Theory, to appear [14] A. Skowroński, K. Yamagata, Positive Galois coverings of selfinjective algebras, Adv in Math.,

in press

[15] K. Yamagata, Frobenius algebras, in: Handbook of Algebra 1, Vol.1, Elsevier, 1996, 841–887.

[16] K. Yamagata, Galois coverings of selfinjective algebras by twisted repetitive algebras, The 4th China-Japan-Korea International Symposium on Ring Theory, Nanjing (China), June 2004

Department of Mathematics,

Tokyo University of Agriculture and Technology Nakacho 2-24-16, Koganei, Tokyo 184-8488, Japan

  E-mail address: yamagata@cc.tuat.ac.jp