STABLE EQUIVALENCES INDUCED FROM GENERALIZED TILTING MODULES II ^{1}

Takayoshi Wakamatsu

1. Introduction

Let A and B be finite dimensional algebras over a field K. We suppose that _{B}T_{A} is a generalized tilting module and (φ, ψ) an admissible system for a symmetric algebra, where _{A}M ⊗_{A} M_{A} →^{φ} _{A}M_{A} and _{A}M ⊗_{A} M_{A} →^{ψ} _{A}DA_{A}. Then the transformed system (φ^{T} , ψ^{T}) is defined over the bimodule _{B}M_{B}^{T} = _{B}T ⊗_{A} Hom_{A}(T, M)_{B} and we have two symmetric algebras Λ(φ, ψ) = A ⊕ M ⊕ DA and Λ(φ^{T} , ψ^{T}) = B ⊕ M^{T} ⊕ DB under the assumption (1) the canonical map _{B}T ⊗_{A} Hom_{A}(T, M)_{B} →^{θ} _{B}Hom_{A}(T, T ⊗_{A} M)_{B} defined by θ(t⊗f)(t′) = t⊗f(t′) is bijective. In the previous note[2], we have shown the existence of a stable equivalence

S : \underline{mod} − Λ(φ, ψ) ≈ \underline{mod} − Λ(φ^{T} , ψ^{T})

by using the assumptions (2) the class C(T_{A}) = gen^{∗}(T_{A}) ∩ ⋂_{n>0} KerExt_{A}^{n}(T,?) is contravariantly finite in mod−A and, dually, D(DT_{B}) = cog^{∗}(DT_{B}) ∩ ⋂_{n>0} KerExt_{B}^{n}(?, DT) covariantly finite in mod−B, and (3) the modules M_{A} and T ⊗_{A} M_{A} are in the class C(T_{A}). Those assumptions (1) to (3) are satisfied if we suppose

(a) the module _{A}M_{A} is of the form ⨁_{(X,Y)} _{A}X ⊗_{K} Y_{A} with all Y_{A}’s are in the class C(T_{A}), and

(b) one of the algebras A and B is representation-finite. The purpose of the present note is to give an example of a couple of an admissible system

(φ, ψ) and a generalized tilting module _{B}T_{A} for which the symmetric algebras Λ(φ, ψ) and Λ(φ^{T} , ψ^{T}) are stably equivalent but not derived equivalent. Such an example means that our stable equivalence S is not induced from Morita theory of derived categories.

2. An Example

Define an algebra A by the quiver

Q(A) : \begin{aligned}& β \\ 1 & → & 2 \\ ⟲ & & ⟳ \\ α & & γ \end{aligned}

with the relations α^{2} = 0, γ^{2} = 0, β · α = 0 and γ · β = 0. It is checked that the algebra A is representation-finite with only eight non-isomorphic indecomposable modules. We also have

1The detailed version of this paper will be submitted for publication elsewhere.

A_{A} = ( \begin{matrix}1 \\ 1 \end{matrix} ) ⊕ \binom{2}{1  2} and DA_{A} = \binom{1  2}{1} ⊕ ( \begin{matrix}2 \\ 2 \end{matrix} ) .

Choose a generalized tilting module as T_{A} = ( \begin{matrix}1 & & 2 \\ & 1 & & 2 \end{matrix} ) ⊕ ( \begin{matrix}2 \\ 2 \end{matrix} ). Then, the quiver

Q(B) of B = End(T_{A}) is given by \begin{matrix}1 & ⇆ & 2 \\ ⟲ \end{matrix} and we have

B_{B} = ( \begin{matrix}& 1 \\ 1 & & 2 \\ & & & 1 \end{matrix} ) ⊕ ( \begin{matrix}2 \\ 1 \\ 2 \\ 1 \end{matrix} ) and DB_{B} = ( \begin{matrix}2 \\ & 1 \\ & & 2 & & 1 \\ & & & 1 \end{matrix} ) ⊕ ( \begin{matrix}2 \\ 1 \\ 2 \end{matrix} ) .

Now, we set _{A}M_{A} = _{A}Ae_{1} ⊗_{K} e_{1}DA_{A} = \binom{1}{1  2}⊕\binom{1  2}{1} and φ = 0 the zero

map from _{A}M ⊗_{A} M_{A} to _{A}M_{A}. Since our module _{A}M_{A} is canonically isomorphic to its dual _{A}DM_{A}, we have a map ψ : _{A}M ⊗_{A} M_{A} → _{A}DA_{A} and (φ, ψ) becomes an admissible system for a symmetric algebra. Then, the assumptions (a) and (b) are satisfied and, therefore, the symmetric algebras Λ = Λ(φ, ψ) and Γ = Λ(φ^{T} , ψ^{T}) are stably equivalent.

In order to prove that the algebras Λ and Γ are not derived equivalent, we use the following well-known result. The proof can be seen in the paper [1] by Usami.

Lemma 1. If the algebras Λ and Γ are derived equivalent, there exists a regular matrix P ∈ Mat_{n}(ℤ) and their Cartan matrices satisfy the equation ^{t}P · C_{Λ} · P = C_{Γ}.

We have C_{Λ} = ( \begin{matrix}8 & 3 \\ 3 & 5 \end{matrix} ) since

e_{1}Λ_{A} = ( \begin{matrix}1 \\ 1 \end{matrix} ) ⊕ \{ \binom{1  2}{1} ⊕ \binom{1  2}{1} \} ⊕ \binom{1  2}{1}

and

e_{2}Λ_{A} = \binom{2}{1  2} ⊕ \{ \binom{1  2}{1} \} ⊕ ( \begin{matrix}2 \\ 2 \end{matrix} ) .

Similarly, we have C_{Γ} = ( \begin{matrix}10 & 3 \\ 3 & 4 \end{matrix} ) from

f_{1}Γ_{B} = ( \begin{matrix}& 1 \\ 1 & & 2 \\ & & & 1 \end{matrix} ) ⊕ \{ ( \begin{matrix}1 \\ 1 \end{matrix} ) ⊕ ( \begin{matrix}1 \\ 1 \end{matrix} ) \} ⊕ ( \begin{matrix}& & & & 2 \\ & & & 1 \\ 1 & & 2 \\ & 1 \end{matrix} )

and

f_{2}Γ_{B} = ( \begin{matrix}2 \\ 1 \\ 2 \\ 1 \end{matrix} ) ⊕ \{ 0 \} ⊕ ( \begin{matrix}2 \\ 1 \\ 2 \end{matrix} ) ,

where e_{i} (resp. f_{i}) stands for the primitive idempotent element in the algebra A (resp. B) corresponding to the vertex in Q(A) (resp. Q(B)) indexed by i and n is the common number of non-isomorphic simple A- or B-modules.

Put P = ( \begin{matrix}a & c \\ b & d \end{matrix} ) ∈ Mat_{2}(ℤ), then we have

tP · C_{Λ} · P = ( \begin{matrix}8a^{2} + 6ab + 5b^{2} & 8ac + 3(ad + bc) + 5bd \\ 8ac + 3(ad + bc) + 5bd & 8c^{2} + 6cd + 5d^{2} \end{matrix} ) .

Hence, ^{t}P · C_{Λ} · P = C_{Γ} implies that

5c^{2} + 3(c + d)^{2} + 2d^{2} = 8c^{2} + 6cd + 5d^{2} = 4,

and this is impossible for integers c, d ∈ ℤ. Therefore, the algebras Λ and Γ are not derived equivalent by the previous lemma.

References

[1] Y. Usami, Derived equivalence and perfect isometry II, Proceedings of the 4th Symposium on Representation Theory of Algebras, 165-189, Saitama, 1994. (in Japanese)

[2] T. Wakamatsu, Stable equivalences induced from generalized tilting modules, Proceedinds of the 36th Symposium on Ring Theory and Representation Theory, 147-156, Yamanashi, 2004.

Faculty of Education, Saitama University Shimo-Okubo 255, Sakura, Saitama 338-8570 Japan