TILTING COMPLEXES ASSOCIATED WITH A SEQUENCE OF IDEMPOTENTS

MITSUO HOSHINO AND YOSHIAKI KATO

This note is a summary of our paper ([HK3]).

Rickard [Ri2] showed that the Brauer tree algebras with the same numerical invariants are derived equivalent to each other. Let A be a Brauer tree algebra corresponding to a Brauer tree whose edges are labelled 1, 2, …, n. Note that there exists a partition of the edges {1, …, n} = E_{0} ∪ ⋯ ∪ E_{l}, where E_{s} consists of the edges i for which there exists a sequence of edges i_{0}, i_{1}, …, i_{s} = i such that i_{0} is adjacent to the exceptional vertex and for any 0 ≤ r < l, i_{r} ≠ i_{r+1} and i_{r}, i_{r+1} have a vertex in common. He constructed a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{j} = 0 for j > 0 and j < − l, P^{−j} ∈ add(⊕_{i∈E_{l−j}}e_{i}A), where e_{i} ∈ A is a local idempotent corresponding to the edge i, for 0 ≤ j ≤ l and End_{K(Mod-A)}(P^{•}) is a Brauer “star” algebra with the same numerical invariants as A. On the other hand, Okuyama [Ok] pointed out recently that for Brauer tree algebras A, B with the same numerical invariants there exists a sequence of Brauer tree algebras B_{0} = A, B_{1}, …, B_{l} = B such that B_{r+1} is the endomorphism algebra of a tilting complex for B_{r} of term length two defined by an idempotent. See König and Zimmermann [KZ] for another example of derived equivalences which are iterations of derived equivalences induced by tilting complexes of term length two. We will formulate these results.

Let A be a noetherian ring and e_{0}, e_{1}, …, e_{l} ∈ A a sequence of idempotents such that add(e_{0}A_{A}) = P_{A}, e_{i+1} ∈ e_{i}Ae_{i} for 0 ≤ i < l and Ext_{A}^{j}(A/Ae_{i}A, e_{i}A) = 0 for 0 ≤ j < i ≤ l. First, we will show that there exists a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{i} = 0 for i > 0 and i < − l, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l (Proposition 1.1), and that such a tilting complex P^{•} is essentially unique. Next, we will show that there exists a sequence of rings B_{0} = A, B_{1}, …, B_{l} = End_{K(Mod-A)}(P^{•}) such that for any 0 ≤ i < l, B_{i+1} is the endomorphism ring of a tilting complex for B_{i} of term length two defined by an idempotent (Theorem 1.2). Furthermore, in case A is a selfinjective artin algebra over a commutative artin ring R and add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l, where D = Hom_{R}(−, E(R/rad R)), we will show that End_{K(Mod-A)}(P^{•}) is a selfinjective artin R-algebra whose Nakayama permutation coincides with that of A (Proposition 2.3). Finally, we deal with the case where A is a finite dimensional algebra over a field k and add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l, where D = Hom_{k}(−, k). We will construct a two-sided tilting complex which corresponds to P^{•} (Section 3). Simultaneously, we will provide a sufficient condition for an algebra B containing A as a subalgebra to be derived equivalent to A (Theorem 4.1).

Throughout this note, rings are associative rings with identity and modules are unitary modules. Unless otherwise stated, modules are right modules. For a ring A, we denote by A^{op} the opposite ring of A and consider left A-modules as A^{op}-modules. In case A is a finite dimensional algebra over a field k, we denote by A^{e} the enveloping algebra A^{op} ⊗_{k} A. Sometimes, we use the notation X_{A} (resp., _{A}X) to signify that the module X considered is a right (resp., left) A-module. We denote by Mod-A the category of Amodules and by P_{A} the full additive subcategory of Mod-A consisting of finitely generated projective modules. For an object X in an additive category A, we denote by add(X) the full additive subcategory of A consisting of objects isomorphic to direct summands of finite direct sums of copies of X. For an additive category A, we denote by K(A) the homotopy category of cochain complexes over A and by K^{b}(A) the full subcategories of K(A) consisting of bounded complexes. In case A is an abelian category, we denote by D(A) the derived category of cochain complexes over A. Also, we denote by Z^{i}(X^{•}), Z′^{i}(X^{•}) and H^{i}(X^{•}) the i-th cycle, the i-th cocycle and the i-th cohomology of a complex X^{•}, respectively. Finally, we use the notation Hom^{•}(−, −) (resp., − ⊗^{•} −) to denote the single complex associated with the double hom (resp., tensor) complex. We refer to [RD], [Ve] and [BN] for basic results in the theory of derived categories. Also, we refer to [Ri1, Ri3] for definitions and basic properties of tilting complexes and two-sided tilting complexes, and to e.g. [Br], [Ri3], [Ri4], [Ro], [RZ] and so on for recent progress.

The detailed version has been published in another place.

1. General case

In this section, we will show that a certain sequence of idempotents e_{0}, e_{1}, …, e_{l} in a ring A defines a tilting complex P^{•} ∈ K^{b}(P_{A}) of term length l + 1 and that there exists a sequence of rings B_{0} = A, B_{1}, …, B_{l} = End_{K(Mod-A)}(P^{•}) such that for any 0 ≤ i < l, B_{i+1} is the endomorphism ring of a tilting complex for B_{i} of term length two defined by an idempotent.

Let e_{0}, e_{1}, … be idempotents in A such that add(e_{0}A) = P_{A} and e_{i+1} ∈ e_{i}Ae_{i} for all i ≥ 0.

Proposition 1.1. Assume A is right noetherian. Let l ≥ 0 and assume Ext_{A}^{j}(A/Ae_{i}A, e_{i}A) = 0 for 0 ≤ j < i ≤ l . Then there exists a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{i} = 0 for i > 0 and i < − l, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l.

Theorem 1.2. Let l ≥ 0 and assume Ext_{A}^{j}(A/Ae_{i}A, e_{i}A) = 0 for 0 ≤ j < i ≤ l. Let P^{•} ∈ K^{b}(P_{A}) be a tilting complex such that P^{i} = 0 for i > 0 and i < − l − 1, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l + 1 and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l + 1. Then the following hold.

(1) There exists a tilting complex P̄^{•} ∈ K^{b}(P_{A}) such that P̄^{i} = 0 for i > 0 and i < − l, P̄^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l, H^{−j}(P̄^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l and e_{l+1}A[l] is a direct summand of P̄^{•}. Furthermore, we have a distinguished triangle in K^{b}(P_{A}) of the form

P [l] → P̄^{•} → P^{•} →

with P = P^{−l−1} ⊕ e_{l+1}A.

(2) Let B = End_{K(Mod-A)}(P̄^{•}) and f ∈ B the composite of canonical homomorphisms P̄^{•} → e_{l+1}A[l] → P̄^{•}. Then Hom_{B}(B/BfB, fB) = 0 and there exists a tilting complex Q^{•} ∈ K^{b}(P_{B}) such that Q^{i} = 0 for i ≠ 0, −1, Q^{−1} ∈ add(fB), H^{0}(Q^{•}) ∈ Mod-(B/BfB) and End_{K(Mod-B)}(Q^{•}) ≅ End_{K(Mod-A)}(P^{•}).

2. The case of artin algebras

In this section, we will apply the results of the preceding section to the case where A is an artin algebra over a commutative artin ring R. We set D = Hom_{R}(−, E(R/rad R)).

According to Proposition 1.1, we have the following.

Proposition 2.1. Let e_{0}, e_{1}, e_{2}, … be idempotents in A such that add(e_{0}A) = P_{A} and e_{i+1} ∈ e_{i}Ae_{i} for i ≥ 0. Let l ≥ 0 and assume add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l. Then there exists a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{i} = 0 for i > 0 and i < − l, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l.

Proposition 2.2. Let e_{0}, e_{1}, e_{2}, … be idempotents in A such that add(e_{0}A) = P_{A} and e_{i+1} ∈ e_{i}Ae_{i} for i ≥ 0. Let l ≥ 0 and assume add(e_{i}A_{A}) = add(D(_{A}Ae_{i})) for 1 ≤ i ≤ l + 1. Let P^{•} ∈ K^{b}(P_{A}) be a tilting complex such that P^{i} = 0 for i > 0 and i < − l − 1, P^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l + 1 and H^{−j}(P^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l + 1. Then the following hold.

(1) There exists a tilting complex P̄^{•} ∈ K^{b}(P_{A}) such that P̄^{i} = 0 for i > 0 and i < − l, P̄^{−i} ∈ add(e_{i}A) for 0 ≤ i ≤ l, H^{−j}(P̄^{•}) ∈ Mod-(A/Ae_{i}A) for 0 ≤ j < i ≤ l and e_{l+1}A[l] is a direct summand of P̄^{•}. Furthermore, we have a distinguished triangle in K^{b}(P_{A}) of the form

P [l] → P̄^{•} → P^{•} →

with P = P^{−l−1} ⊕ e_{l+1}A.

(2) Let B = End_{K(Mod-A)}(P̄^{•}) and f ∈ B the composite of canonical homomorphisms P̄^{•} → e_{l+1}A[l] → P̄^{•}. Then add(fB_{B}) = add(D(_{B}Bf)) and there exists a tilting complex Q^{•} ∈ K^{b}(P_{B}) such that Q^{i} = 0 for i ≠ 0, −1, Q^{−1} ∈ add(fB), H^{0}(Q^{•}) ∈ Mod-(B/BfB) and End_{K(Mod-B)}(Q^{•}) ≅ End_{K(Mod-A)}(P^{•}).

Consider next the case of A being selfinjective. Let {e_{1}, …, e_{n}} be a basic set of orthogonal local idempotents in A and I_{0} = {1, …, n}. Set ν = D ∘ Hom_{A}(−, A). Then there exists a permutation σ of I_{0}, called the Nakayama permutation, such that ν(e_{i}A) ≅ e_{σ(i)}A for all i ∈ I_{0}.

Proposition 2.3. Let I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ ⋯ be a descending sequence of nonempty σstable subsets of I_{0} and e^{(i)} = ∑_{j∈I_{i}} e_{j} for i ≥ 0. Then for any l ≥ 0 there exists a tilting complex P^{•} ∈ K^{b}(P_{A}) such that P^{i} = 0 for i > 0 and i < − l, P^{−i} ∈ add(e^{(i)}A) for 0 ≤ i ≤ l and H^{−j}(P^{•}) ∈ Mod-(A/Ae^{(i)}A) for 0 ≤ j < i ≤ l. Furthermore, End_{K(Mod-A)}(P^{•}) is a selfinjective artin algebra whose Nakayama permutation coincides with σ.

3. Two-sided tilting complexes

Let A be a finite dimensional algebra over a field k and D = Hom_{k}(−, k). Our aim is to construct two-sided tilting complexes which correspond to tilting complexes constructed in Proposition 2.1. According to Proposition 2.2, we have only to deal with tilting complexes of term length two.

We will first construct a two-sided tilting complex T^{•} corresponding to the following tilting complex S^{•}. Recall that an idempotent e ∈ A is called local if eAe is a local ring. Let {e_{1}, …, e_{n}} be a basic set of orthogonal local idempotents in A and J the Jacobson radical of A. We fix a nonempty subset I_{0} of I = {1, …, n} and define S^{•} as the mapping cone of the multiplication map

ρ : ⨁_{i ∈ I_{0}}Ae_{i} ⊗_{k} e_{i}A → A.

We set e = ∑_{i∈I_{0}} e_{i}, B = End_{K(Mod-A)}(S^{•}) and d_{ij} = dim_{k} e_{i}Ae_{j} for i, j ∈ I_{0}. We assume the following conditions are satisfied:

(a_{1}) there exists a permutation σ of I_{0} such that e_{i}A_{A} ≅ D(_{A}Ae_{σ(i)}) for all i ∈ I_{0}; (a_{2}) e_{i}Je_{σ(i)} ≠ 0 for all i ∈ I_{0}; and (a_{3}) e_{i}Ae_{i}/e_{i}Je_{i} ≅ k for all i ∈ I_{0}.

Proposition 3.1. The following hold.

(1) S^{•} ∈ K^{b}(P_{A}) is a tilting complex with H^{0}(S^{•}) ∈ Mod-(A/AeA).

(2) The left multiplication of A on each homogeneous component of S^{•} gives rise to an injective k-algebra homomorphism φ : A → B.

(3) _{A}(B/A)_{A} ≅ ⊕_{i, j∈I_{0}}(_{A}Ae_{i} ⊗_{k} e_{j}A_{A})^{(α_{ij})}, where

α_{ij} = \begin{cases}d_{ji} − 2 & if i = j = σ(j), \\ d_{ji} − 1 & if j ≠ σ(j) and i ∈ \{j, σ(j)\}, \\ d_{ji} & otherwise. \end{cases}

(4) For any i ∈ I_{0}, e_{i}B_{B} ≅ ⊕_{j∈I_{0}}Hom_{K(Mod-A)}(S^{•}, e_{σ(j)}A[1])^{(μ_{ij})}, where

μ_{ij} = \begin{cases}d_{ji} − 1 if i = σ(j), \\ d_{ji} otherwise. \end{cases}

Proposition 3.2. For any i ∈ I_{0} there exists a local idempotent f_{i} ∈ e_{i}Be_{i} such that f_{i}B_{B} ≅ Hom_{K(Mod-A)}(S^{•}, e_{σ(i)}A[1]). Furthermore, the following hold.

(1) f_{i}B_{B} ≇ f_{j}B_{B} unless i = j.

(2) f_{i}B_{B} ≅ D(_{B}Bf_{σ(i)}) for all i ∈ I_{0}.

(3) f_{i}Bf_{j} ≅ e_{i}Ae_{j} for all i, j ∈ I_{0}.

(4) e_{i}B_{B} ≅ ⊕_{j∈I_{0}}f_{j}B_{B}^{(μ_{ij})} for all i ∈ I_{0}.

(5) f_{i}B_{A} ≅ ⊕_{j∈I_{0}}e_{j}A_{A}^{(μ_{ji})} for all i ∈ I_{0}.

4. Derived equivalent extension algebras

Let A be the same as in Section 3. We will show that an algebra B containing A as a subalgebra satisfying (3) of Proposition 3.1 and (1) ∼ (5) of Proposition 3.2 is derived equivalent to A.

More precisely, let B be a finite dimensional k-algebra containing A as a subalgebra and for each i ∈ I_{0} take a local idempotent f_{i} ∈ e_{i}Be_{i}. We assume the following conditions are satisfied:

(b_{1}) _{A}(B/A)_{A} ≅ ⊕_{i, j∈I_{0}}(_{A}Ae_{i} ⊗_{k} e_{j}A_{A})^{(α_{ij})};

(b_{2}) f_{i}B_{B} ≇ f_{j}B_{B} unless i = j and f_{i}B_{B} ≅ D(_{B}Bf_{σ(i)}) for all i ∈ I_{0}; (b_{3}) f_{i}Bf_{j} ≅ e_{i}Ae_{j} for all i, j ∈ I_{0}; (b_{4}) e_{i}B_{B} ≅ ⊕_{j∈I_{0}}f_{j}B_{B}^{(μ_{ij})} for all i ∈ I_{0}; and (b_{5}) f_{i}B_{A} ≅ ⊕_{j∈I_{0}}e_{j}A_{A}^{(ν_{ij})} for all i ∈ I_{0}.

Theorem 4.1. Denote by T^{•} the mapping cone of the multiplication map

δ : ⨁_{i∈I_{0}}_{B}Bf_{i} ⊗_{k} e_{i}A_{A} → _{B}B_{A}.

Then T^{•} is a two-sided tilting complex with T^{•} ≅ S^{•} in K(Mod-A) if

α_{ij} = \begin{cases}d_{ji} − 2 if i = j = σ(j), \\ d_{ji} − 1 if j ≠ σ(j) and i ∈ \{j, σ(j)\}, \\ d_{ji}  otherwise, \end{cases} μ_{ij} = ν_{ji} = \begin{cases}d_{ji} − 1 if i = σ(j), \\ d_{ji}  otherwise. \end{cases}

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M. Hoshino Institute of Mathematics University of Tsukuba Ibaraki, 305-8571, Japan E-mail address: hoshino@math.tsukuba.ac.jp

Y. Kato Institute of Mathematics University of Tsukuba Ibaraki, 305-8571, Japan E-mail address: ykato@math.tsukuba.ac.jp