Derived equivalences for $\Phi$-Auslander-Yoneda algebras

In this paper, we introduce $\Phi$-Auslander-Yoneda algebras in a triangulated category with $\Phi$ a parameter set in $\mathbb N$, and provide a method to construct new derived equivalences between these $\Phi$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $\nu$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $\Phi$-Auslander-Yoneda algebras of $X$ and $Y$ are derived-equivalent for every admissible set $\Phi$. In particular, the Auslander algebras of $A$ and $B$ are both derived-equivalent and stably equivalent. (2) For a self-injective Artin algeba $A$ and an $A$-module $X$, the $\Phi$-Auslander-Yoneda algebras of $A\oplus X$ and $A\oplus \Omega_A(X)$ are derived-equivalent for every admissible set $\Phi$, where $\Omega$ is the Heller loop operator. Motivated by these derived equivalences between $\Phi$-Auslander-Yoneda algebras, we consider constructions of derived equivalences for quotient algebras, and show, among others, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factorizing out socles.


Introduction
Derived categories and derived equivalences were introduced by Grothendieck and Verdier in [13]. As is known, they have widely been used in many branches of mathematics and physics. One of the fundamental problems in the study of derived categories and derived equivalences is: how to construct derived equivalences ? On the one hand, Rickard's beautiful Morita theory for derived categories can be used to find all rings that are derived-equivalent to a given ring A by determining all tilting complexes over A (see [10] and [11]). On the other hand, a natural course of investigation on derived equivalences should be constructing new derived equivalences from given ones. In this direction, Rickard used tensor products and trivial extensions to produce new derived-equivalences in [10,12], Barot and Lenzing employed one-point extensions to transfer certain a derived equivalence to a new one in [2]. Up to now, however, it seems that not much is known for constructing new derived equivalences based on given ones.
In this paper, we continue the consideration in this direction and provide, roughly speaking, two methods to construct new derived equivalences from given ones. One is to form Φ-Auslander-Yoneda algebras (see Section 3.1 for definition) of generators, or cogenerators over derived-equivalent algebras, and the other is to form quotient algebras of derived-equivalent algebras. We point out that our family of Φ-Auslander-Yoneda algebras include Auslander algebras, generalized Yoneda algebras and some of their quotients. Thus our method produces also derived equivalences between infinitedimensional algebras.
To state our results, we first introduce a few terminologies. Suppose that F is a derived equivalence between two Artin algebras A and B, with the quasi-inverse functor G. Further, suppose that is a radical tilting complex over A associated to F, and suppose that T • : · · · −→ 0 −→T 0 −→T 1 −→ · · · −→T n −→ 0 −→ · · · is a radical tilting complex over B associated to G. The functor F is called almost ν-stable if add( −n i=−1 T i ) = add( −n i=−1 ν A T i ), and add( n where ν A is the Nakayama functor for A. We have shown in [6] that an almost ν-stable functor F induces an equivalence functorF between the stable module categories A-mod and B-mod. For further information on almost ν-stable derived equivalences, we refer the reader to [6]. For a module M over an algebra A, the generalized Yoneda algebra of M is defined by Ext * A (M) := i≥0 Ext i A (M, M). In case M = A/rad(A), the algebra Ext * A (M) is called the Yoneda algebra of A in literature. We shall extend this notion to a more general situation, and introduce the Φ-Auslander-Yoneda algebras with Φ a parameter set in N (for details see Subsection 3.1 below). We notice that a Φ-Auslander-Yoneda algebra may not be an Artin algebra in general.
Our main result on Φ-Auslander-Yoneda algebras of modules reads as follows: A dual version of Theorem 1.1 can be seen in Corollary 3.17 below. Since Auslander algebra and generalized Yoneda algebra are two special cases of Φ-Auslander-Yoneda algebras, Theorem 1.1 provides a large variety of derived equivalences between Auslander algebras, and between generalized Yoneda algebras, or their quotient algebras. Note that Theorem 1.1 (2) extends a result in [6, Proposition 6.1], where algebras were assumed to be finite-dimensional over a field, in order to employ two-sided tilting complexes in proofs, and where only endomorphism algebras were considered instead of general Auslander-Yoneda algebras. The existence of two-sided tilting complexes is guaranteed for Artin R-algebras that are projective as R-modules [11]. For general Artin algebras, however, we do not know the existence of two-sided tilting complexes. Hence, in this paper, we have to provide a completely different proof to the general result, Theorem 1.1.

Theorem 1.1. Let A and B be two Artin algebras, and letF : A-mod −→ B-mod be the stable equivalence induced by an almost ν-stable derived equivalence F between A and B. Suppose that X is an
As a direct consequence of Theorem 1.1, we have the following corollary concerning the Auslander algebras of self-injective algebras.

Preliminaries
In this section, we shall recall basic definitions and facts on derived categories and derived equivalences, which are elementary elements in our proofs. Throughout this paper, R is a fixed commutative Artin ring. Given an R-algebra A, by an Amodule we mean a unitary left A-module; the category of all finitely generated A-modules is denoted by A-mod, the full subcategory of A-mod consisting of projective (respectively, injective) modules is denoted by A-proj (respectively, A-inj). The stable module category A-mod of A is, by definition, the quotient category of A-mod modulo the ideal generated by homomorphisms factorizing through projective modules. An equivalence between the stable module categories of two algebras is called a stable equivalence An R-algebra A is called an Artin R-algebra if A is finitely generated as an R-module. For an Artin R-algebra A, we denote by D the usual duality on A-mod, and by ν A the Nakayama functor For an A-module M, we denote by Ω A (M) the first syzygy of M, and call Ω A the Heller loop operator of A. In this paper, we mainly concentrate us on Artin algebras and finitely generated modules.
Let C be an additive category.
For two morphisms f : X → Y and g : Y → Z in C , we write f g for their composition. But for two functors F : C → D and G : D → E of categories, we write GF for their composition instead of FG. For an object X in C , we denote by add(X ) the full subcategory of C consisting of all direct summands of finite direct sums of copies of X . An object X in C is called an additive generator for C if add(X ) = C .
By a complex X • over C we mean a sequence of morphisms d i X between objects X i in C : · · · → . For a complex X • , the brutal truncation σ <i X • of X • is a subcomplex of X • such that (σ <i X • ) k is X k for all k < i and zero otherwise. Similarly, we define σ i X • . For a fixed n ∈ Z, we denote by X • [n] the complex obtained from X • by shifting n degrees, that is, The category of all complexes over C with chain maps is denoted by C (C ). The homotopy category of complexes over C is denoted by K (C ). When C is an abelian category, the derived category of complexes over C is denoted by D(C ). The full subcategory of K (C ) and D(C ) consisting of bounded complexes over C is denoted by K b (C ) and D b (C ), respectively. As usual, for an algebra A, It is well-known that, for an R-algebra A, the categories K (A) and D(A) are triangulated categories. For basic results on triangulated categories, we refer the reader to the excellent books [3] and [9].
Let A be an Artin algebra. Recall that a homomorphism f : X → Y of A-modules is called a radical map if, for any module Z and homomorphisms h : Z → X and g : Y → Z, the composition h f g is not an isomorphism. A complex over A-mod is called a radical complex if all of its differential maps are radical maps. Every complex over A-mod is isomorphic to a radical complex in the homotopy category K (A). If two radical complexes X • and Y • are isomorphic in K (A), then X • and Y • are isomorphic in C (A).
Two R-Artin algebras A and B are said to be derived-equivalent if their derived categories D b (A) and D b (B) are equivalent as triangulated categories. By a result of Rickard (see Lemma 2.2 below), two algebras A and B are derived-equivalent if and only if B is isomorphic to the endomorphism algebra It is known that, given a derived equivalence F between A and B, there is a unique (up to iso- The following lemma, proved in [6, Lemma 2.2], will be used frequently in our proofs below. (1) X i is projective for all i > m and Y j = 0 for all j < m, (2) Y j is injective for all j < m and X i = 0 for all i > m, then the localization functor θ : Thus, for the complexes X • and Y • given in Lemma 2.1, the computation of morphisms from For later reference, we cite the following fundamental result on derived equivalences by Rickard (see [10,Theorem 6.4]) as a lemma. (e) Γ is isomorphic to End(T ), where T is a tilting complex in K b (Λ-proj).
Here Λ-Proj stands for the full subcategory of Λ-Mod consisting of all projective Λ-modules.
Two rings Λ and Γ are called derived-equivalent if one of the above conditions (a)-(e) is satisfied. For Artin algebras, the two definitions of a derived equivalence coincide with each other.

Derived equivalences for Φ-Auslander-Yoneda algebras
As is known, Auslander algebra is a key to characterizing representation-finite algebras, and Yoneda algebra plays a role in the study of the graded module theory of Koszul algebras. In this section, we shall first unify the two notions and introduce the so-called Φ-Auslander-Yoneda algebra of an object in a triangulated category, where Φ is a parameter subset of N, and then construct new derived equivalences between these Φ-Auslander-Yoneda algebras from a given almost ν-stable derived equivalence. In particular, Theorem 1.1 will be proved, and a series of its consequences will be deduced in this section.

Admissible sets and Auslander-Yoneda algebras
First, we introduce some special subsets of the set N := {0, 1, 2, · · · , } of the natural numbers, and then define a class of algebras called Auslander-Yoneda algebras.
A subset Φ of N containing 0 is called an admissible subset of N if the following condition is satisfied: For instance, the sets {0, 3, 4}, {0, 1, 2, 3, 4} are admissible subsets of N. The following is a family of admissible subsets of N.
Let n be a positive integer, and let m be a positive integer or positive infinity. We define  (2) If Φ 1 and Φ 2 are admissible subsets of N, then so is Φ 1 ∩ Φ 2 . Moreover, the intersection of a family of admissible subsets of N is admissible.
(3) For a subset Φ ⊆ N with 0 ∈ Φ, the set Φ m := {x m | x ∈ Φ} is an admissible subset of N for every integer m ≥ 3.
Proof. The statements (1) and (2) follow easily from the definition of an admissible subset. Now we consider (3). We pick an integer m ≥ 3. Let i m , j m , k m and l m be in Φ m such that i m + j m + k m = l m . If i m + j m ∈ Φ m , then i m + j m = t m for some t ∈ Φ. By Fermat's Last Theorem, one of the integers i and j is zero.
Hence the set Φ m is an admissible subset of N.
Let Φ be a subset of N. Given an N-graded R-algebra Λ = ∞ i≥0 Λ i , where R is a commutative ring and each Λ i is an R-module with Λ i Λ j ⊆ Λ i+ j for all i, j ∈ N, we define an R-module Λ(Φ) := i∈Φ Λ i , and a multiplication in A(Φ): for a i ∈ Λ i and b j ∈ Λ j with i, j ∈ Φ, we define a i · b j = a i b j if i + j ∈ Φ, and zero otherwise. Then one can easily check that Λ(Φ) is an associative algebra if Φ is an admissible subset of N.
This procedure can be applied to a triangulated category, in this special situation, the details which are needed in our proofs read as follows: Let T be a triangulated R-category over a commutative Artin ring R, and let Φ be a subset in N containing 0. We denote by Let X ,Y and Z be objects in T . For each i ∈ Φ, let ι i denote the canonical embedding of Hom for each i ∈ Φ. In particular, for f ∈ Hom T (X ,Y [i]) and g ∈ Hom T (X ,Y [ j]) with i, j ∈ Φ, we have The next proposition explains further why we need to introduce admissible subsets. Proof. If Φ is an admissible subset of N, then it is straightforward to check that the multiplication on E Φ T (V ) defined above is associative for all objects V ∈ T . Now we assume that Φ is not an admissible subset, that is, there are integers i, j, k ∈ Φ satisfying: i + j + k ∈ Φ, i + j ∈ Φ, and j + k ∈ Φ. Let X be a non-zero object in T , and let V := i+ j+k s=0 X [s]. We consider the multiplication on E Φ T (V ). By definition, the object where π is the canonical projection and λ is the canonical inclusion. Similarly, we define g : where the maps are canonical maps. Hence the map ι i ( f )ι j (g) ι k (h) is non-zero. Since j + k ∈ Φ, we have ι j (g)ι k (h) = 0, and consequently ι i ( f ) ι j (g)ι k (h) = 0. This shows that the multiplication of E Φ T (V ) is not associative, and the proof is completed.
Note that E N T (X ) is an N-graded associative R-algebra with Hom T (X , . From now on, we consider exclusively admissible subsets Φ of N. Thus, for objects X and Y in T , one has an R-algebra E Φ T (X , X ) (which may not be artinian), and a left E Φ T (X , X )-module E Φ T (X ,Y ). For simplicity, we write E Φ T (X ) for E Φ T (X , X ). In case Φ = Φ(1, 0), we see that E Φ T (X ) is the endomorphism algebra of the object X in T . In case Particularly, let us take T = D b (A) with A an Artin R-algebra. If A is representation-finite and if X is an additive generator for A-mod, then E T (X ) is the usual Yoneda algebra of A. Thus the algebra E Φ T (X ) is a generalization of both Auslander algebra and Yoneda algebra. For this reason, the algebra E Φ T (X ) of X in a triangulated category T is called the Φ-Auslander-Yoneda algebra of X in T in this paper.
Note also that the algebra E is neither a subalgebra nor a quotient algebra of the generalized Yoneda algebra of X .
Let us remark that one may define the notion of an admissible subset of Z (or of a monoid M with an identity e), and introduce Φ-Auslander-Yoneda algebra of an object in an arbitrary R-category C with an additive self-equivalence functor (or a family of additive functors {F g } g∈M from C to itself, . For our goals in this paper, we just formulate the admissible subsets for N.

Almost ν-stable derived equivalences
We briefly recall some basic facts on almost ν-stable derived equivalences from [6], which are needed in proofs.
Let A and B be Artin algebras, and let F : be a derived equivalence between A and B. Suppose that Q • andQ • are the tilting complexes associated to F and to a quasi-inverse G of F, respectively. Now, we assume that Q i = 0 for all i > 0, that is, the complex Q • is of the form In this case, the complexQ • may be chosen of the following form (see [6, Lemma 2.1], for example) i . The functor F is called an almost ν-stable derived equivalence provided add( A Q) = add(ν A Q) and add( BQ ) = add(ν BQ ). A crucial property is that an almost ν-stable derived equivalence induces an equivalence between the stable module categories A-mod and B-mod. Thus A and B share many common properties, for example, A is representation-finite if and only if B is representation-finite.
In the following lemma, we collect some basic facts on almost ν-stable derived equivalences, which will be used in our proofs.
be an almost ν-stable derived equivalence between Artin algebras A and B. Suppose that Q • andQ • are the tilting complexes associated to F and to its quasi-inverse G, respectively. Then: Proof. The statement (1)  For an Artin algebra A, let A E be the direct sum of all non-isomorphic indecomposable projective A-modules P with the property:

Derived equivalences for Auslander-Yoneda algebras
Our main result in this section is the following theorem on derived equivalences between Φ-Auslander-Yoneda algebras.
be an almost ν-stable derived equivalence between two Artin algebras A and B, and letF be the stable equivalence defined in Lemma 3.3 (3). For an A-module X , we set M := A ⊕ X and N := B ⊕F(X ). Suppose that Φ is an admissible subset in N. Then we have the following: are also stably equivalent. In particular, there is an almost ν-stable derived equivalence and a stable equivalence between End A (M) and End B (N).
Thus, under the assumptions of Theorem 3.4, if Φ is finite, then the algebras E Φ A (M) and E Φ B (N) share many common invariants: for example, finiteness of finitistic and global dimensions, representation dimension, Hochschild cohomology, representation-finite type and so on.
The rest of this section is essentially devoted to the proof of Theorem 3.4. First of all, we need some preparations. Let us start with the following lemma that describes some basic properties of the algebra E Φ A (V ), where V is an A-module and is considered as a complex concentrated on degree zero.

Lemma 3.5. Let A be an Artin algebra, and let V be an
Then

projective and finitely generated, and there is an isomorphism
is also projective. To show that µ is an isomorphism, we can assume that V 1 is indecomposable by additivity. Let π 1 : V −→ V 1 be the canonical projection, and let λ 1 : . This shows that γ µ = id. Similarly, one can check that µ γ = id. Hence µ is an isomorphism. The rest of (1) can be verified easily.
(2) Using definition, one can check that the map is the composition of the embedding ι 0 : (1), the statement (3) follows.
(4) This follows directly from the following isomorphisms . Thus we have finished the proof.
From now on, we assume that F : is an almost ν-stable derived equivalence with a quasi-inverse functor G, that Q • andQ • are tilting complexes associated to F and G, respectively, and thatF : A-mod −→ B-mod is the stable equivalence defined by Lemma 3.3 (3). For an A-module X , we may assume that F(X ) =Q • X as in Lemma 3.3 (1), and define A M = A ⊕ X and B N = B ⊕F(X ).
Lemma 3.6. Keeping the notations above, we have the following: we get an exact sequence Using the above exact sequence, we only need to show that the induced map

It follows that Hom
Since the vertical maps of the above diagram are all isomorphisms, the map ) as a triangulated category. By Lemma 3.3,Q 0 X =F(X ) and all the terms ofQ • X other thanQ 0 X are in add( B B). HenceF(X ) is in the triangulated subcategory generated by add(Q • ⊕Q • X ), and consequently add(Q • ⊕Q • X ) generates K b (add(B ⊕F(X ))) as a triangulated category. Thus, the statement (2) follows.
Proof. Let i = 0, and let f • be a morphism in Note that the term E Φ B (N,T i ) is zero if i < 0. Since all the terms ofT • other thanT 0 are projectiveinjective, and since i = 0, we see from Lemma 3.5 (3) that f k = E Φ B (N, g k ) for some g k :T k −→T k+i for all integers k. It follows from the above commutative diagram that, for each integer k, we have proj is a faithful functor by Lemma 3.5 (2), we have dg k+1 − g k d = 0 for all integers k, and consequently g • := (g k ) is in By Lemma 3.6 (1), the map g • is null-homotopic, and consequently In the following, we shall prove that the endomorphism algebra of the complex For this purpose, we first prove the following lemma.

Lemma 3.8. Keeping the notations above, for each A-module V , we have:
(1) For each positive integer k, there is an isomorphism Here we denote the image of g under θ k by θ k (g).
(2) For each pair of positive integers k and l, the θ k and θ l in (1) satisfy Proof. By Lemma 3.3, we may assume that F(V ) is the complexQ • V defined in Lemma 3.3 (1), and thereforeF(V ) =Q 0 V . As before, the complex σ >0Q The map b f exists because the composition f , that is, the map b f is uniquely determined by the above commutative diagram. Thus, we can define a morphism θ k : by sending f to b f . We claim that this θ k is an isomorphism. In fact, it is surjective: Since k > 0, and since both g and h can be chosen to be chain maps, we see immediately that f = gh = 0. This shows that the map θ k is injective, and therefore θ k is an isomorphism.
Remark: Let f be in Hom D b (A) (V,V ), and let g be in is a morphism such that π V t f = F( f )π V , then, by a proof similar to Lemma 3.8 (3), we have For instance, by Lemma 2.1, we can assume that the map Since the map π V is the canonical map fromQ • V toQ 0 V , we see that the map p 0 :F(V ) −→F(V ) satisfies the condition π V p 0 = F( f )π V . Therefore, by the above discussion, we have .
where µ is the isomorphism defined in Lemma 3.5 (1) and ι k is the embedding from Indeed, by the proof of Lemma 3.5 (2), we have E Φ B (N, d) = µ (ι 0 (d)). Thus, Thus, the map µ ι k (θ k ( f k )) gives rise to an endomorphism of E Φ• B (N,T • ): We denote this endomorphism byθ k ( f k ). Now, we define a map We claim that η is an algebra homomorphism. This will be shown with help of the next lemma.
Proof. (1). By Lemma 3.8 (2), we have (2) and (3). By definition, the map By the remark just before Lemma 3.9, we have Applying µ to these equalities, one can easily see that These are precisely the (2) and (3). Now, we continue the proof of Lemma 3.9: With Lemma 3.10 in hand, it is straightforward to check that η is an algebra homomorphism. In the following we first show that η is injective.
Pick an ( f i ) in E Φ A (M), let p • := η(( f i )). Then we have and

This yields that
Since µ is an isomorphism, and since E Φ is a direct sum, we get F( f 0 ) 0 = du 1 and θ k ( f k ) = 0 for all k ∈ Φ + . Since θ k is an isomorphism by Lemma 3.8, we have f k = 0 for all k ∈ Φ + . Now for each i > 0, we have (N, d).
Finally, we show that η is surjective. .
This gives rise to a morphism α • in End K b (B) (T • ) by defining α 0 := s 0 and α i := t i for all i > 0. By Lemma 2.1 and the fact that F is an equivalence, we conclude that α Hence η is surjective. This finishes the proof of Lemma 3.9.
Lemma 3.11. Let F : D b (Λ) −→ D b (Γ) be a derived equivalence between Artin R-algebras Λ and Γ, and let P • be a tilting complex associated to F. Suppose that the following two conditions are satisfied.
(1) All the terms of P • in negative degrees are zero, and all the terms of P • in positive degrees are in add( Λ W ) for some projective Λ-module Λ W with add(ν Λ W ) = add( Λ W ).
(2) For the module Λ W in (1), the complex F( Λ W ) is isomorphic to a complex in K b (add( Γ V )) for some projective Γ-module Γ V with add(ν Γ V ) = add( Γ V ).

Then the quasi-inverse of F is an almost ν-stable derived equivalence.
Proof. Let G be a quasi-inverse of F. By the definition of almost ν-stable equivalences, we need to consider the tilting complex associated to G. This is equivalent to considering F(Λ).
Since P • is a tilting complex over Λ, it is well-known that Λ Λ is in add( i∈Z P i ) which is contained in add(P 0 ⊕ W ) by the assumption (1). Hence F( Λ Λ) is in add(F(P 0 ⊕ W )). Let P + be the complex σ >0 P • . There is a distinguished triangle in D b (Λ). Applying F, we get a distinguished triangle [1] in D b (Γ). By definition, there is an isomorphism F(P • ) ≃ Γ in D b (Γ). By the assumption (1), we have P + ∈ K b (add( Λ W )), and consequently F( Since P i = 0 for all i < 0, we see from [6, Lemma 2.1] thatP • has zero homology in all positive degrees. Hence we can assume thatP i = 0 for all i > 0. Thus, the complexP • ≃ F(Λ) is a tilting complex associated to G and satisfies thatP i = 0 for all i > 0 andP i ∈ add( Γ V ) for all i < 0. The complex P • is a tilting complex associated to F and satisfies that P i = 0 for all i < 0 and P i ∈ add( Λ W ) for all i > 0. Since add(ν Λ W ) = add( Λ W ), and since add(ν Γ V ) = add( Γ V ), it follows from [6, Proposition 3.8 (3)] that the functor G is an almost ν-stable derived equivalence.
Note thatQ • M is just the complexT • considered in Proposition 3.9. Now we consider the isomorphism η in the proof of Proposition 3.9. Let e be the idempotent in End A (M) corresponding to the direct summand A E.
) induces by the isomorphism η in the proof of Proposition 3.9 sends . By [6, Lemma 3.9], the functor F induces an equivalence between the triangulated categories K b (add( A E)) and K b (add( BĒ )). (N,Ē)) and consequentlyF induces a full, faithful triangle functorF : (N,Ē))). Since add( A E) clearly generates K b (add( A E)) as a triangulated category, we see immediately that add(Q • E ) generates K b (add( BĒ )) as a triangulated category. This implies that add( (N,Ē))) as a triangulated category. This shows that is dense, and therefore an equivalence. LetĜ be a quasi-inverse of the derived equivalenceF. Then the functorĜ also induces an equivalence between the triangulated categories K b (add(E Φ B (N,Ē))) and K b (add(E Φ A (M, E))). This implies that the complexĜ(E Φ B (N,Ē)) is isomorphic to a complex in E))). Now we use Lemma 3.11 to complete the proof. In fact, the complex E Φ• B (N,T • ) is a tilting complex associated to the derived equivalenceĜ : (N,Ē)) for all i > 0, and it follows from Lemma 3.5 (4) that (N,Ē)). E)). Hence, by Lemma 3.11, the functor F is an almost ν-stable derived equivalence.

Similarly, we have add(ν E
The statements on stable equivalence in Theorem 3.4 follow from [6,Theorem 1.1]. This finishes the proof. Note that the proof of Theorem 3.4 (2) shows also that if both E Φ A (M) and E Φ B (N) are Artin Ralgebras, then the conclusion of Theorem 3.4 (2) is valid.
Let us remark that, in case of finite-dimensional algebras over a field, the special case for Φ = Φ(1, 0) in Theorem 3.4 about stable equivalence was proved in [6, Proposition 6.1] by using two-sided tilting complexes, and the conclusion there guarantees a stable equivalence of Morita type. But the proof there in [6] does not work here any more, since we do not have two-sided tilting complexes in general for Artin algebras.
As a consequence of Theorem 3.4, we have the following corollary.

a derived equivalence between self-injective Artin algebras A and B, and let φ be the stable equivalence induced by F. Then, for each A-module X and each admissible subset
Proof. There is an integer i such that F[i] is an almost ν-stable derived equivalence. Let φ 1 be the stable equivalence induced by where Ω i is the i-th syzygy operator of A. By the definition of an almost ν-stable derived equivalence, either ) are derived-equivalent by Theorem 3.4. Thus, by Theorem 3.4 again, the algebras E Φ A (A⊕Ω i (X )) and E Φ B (B⊕φ 1 Ω i (X )) are derived-equivalent. The stable equivalence follows from [6, Theorem 1.1]. Thus the proof is completed.
As a direct consequence of Corollary 3.12, we have the following corollary concerning Auslander algebras.

Corollary 3.13. Suppose that A and B are self-injective Artin algebras of finite representation type. If A and B are derived-equivalent, then the Auslander algebras of A and B are both derived and stably equivalent.
Let us remark that the notion of a stable equivalence of Morita type for finite-dimensional algebras can be formulated for Artin R-algebras. But, in this case, we do not know if a stable equivalence of Morita type between Artin algebras induces a stable equivalence since we do not know whether a projective A-A-bimodule is projective as a one-sided module when the ground ring is a commutative Artin ring. So, Theorem 3.4 (2), Corollary 3.12 (1) and Corollary 3.13 ensure a stable equivalence between the endomorphism algebras of generators over Artin algebras, while the main result in [6, Section 6] ensures a stable equivalence of Morita type between the endomorphism algebras of generators over finite-dimensional algebras.
Note that if A and B are not self-injective, then Corollary 3.13 may fail. For a counterexample, we just check the following two algebras A and B, where A is given by the path algebra of the quiver • → • → •, and B is given by • α −→ • β −→ • with the relation αβ = 0. Clearly, B is the endomorphism algebra of a tilting A-module. Note that the Auslander algebras of A and B have different numbers of non-isomorphic simple modules, and therefore are never derived-equivalent since derived equivalences preserve the number of non-isomorphic simple modules [7]. Notice that, though A and B are derivedequivalent, there is no almost ν-stable derived equivalence between A and B since A and B are not stably equivalent. This example shows also that Theorem 3.4 may fail if we drop the almost ν-stable condition.
The following question relevant to Corollary 3.13 might be of interest. We remark that Asashiba in [1] gave a complete classification of representation-finite self-injective algebras up to derived equivalence.

Question. Let
For a self-injective Artin R-algebra A, we know that the shift functor [−1]: is an almost ν-stable derived equivalence, and this functor induces a stable functorF : A-mod−→ A-mod, which is isomorphic to Ω A (−), the Heller loop operator. Thus we have the following corollary to Theorem 3.4, which extends [5,Corollary 3.7] in some sense. Corollary 3.14. Let A be a self-injective Artin algebra. Then, for any admissible subset Φ of N and for any A-module X , we have a derived equivalence between ). Thus they are stably equivalent. Let us mention the following consequence of Corollary 3.14.

Corollary 3.15. Let A be a self-injective Artin algebra, and let J be the Jacobson radical of A with the nilpotency index n. Then:
(1) For any 1 ≤ j ≤ n − 1 and for any admissible subset Φ of N, the Φ-Auslander-Yoneda algebras A/soc i ( A A)) is at most n.
(4) The global dimension of End A ( A A ⊕ soc( A A) ⊕ · · · ⊕ soc n−1 ( A A)) is at most n.
Proof. Since the syzygy of j i=1 A/J i is j i=1 J i up to a projective summand, we have (1) immediately from Corollary 3.14. The statement (2) follows from [6,Corollary 4.3] together with a result of Auslander, which says that, for any Artin algebra A, the global dimension of End Since A A is injective, we know that add( A A) = add(D(A A )). It follows from D(

The latter is Morita equivalent to End
op . This shows (4). The statement (3) follows from (4), Corollary 3.14 and [6,Corollary 4.3]. Finally, we state a dual version of Theorem 3.4, which will produce a derived equivalence between the endomorphism algebras of cogenerators. First, we point out the following facts. Proof. (1) Suppose that Q • andQ • are tilting complexes associated to F and G, respectively. We assume that Q • andQ • are radical complexes. There is an isomorphism are tilting complexes associated to DGD and DFD, respectively. Since (Hom B (Q, B B))) = add(Hom B (Q, B B)). Similarly, we have Hom A (Q, A A) = n i=1P i and add(ν A op (Hom A (Q, A A))) = add (Hom A (Q, A A)), and consequently DGD is an almost ν-stable derived equivalence. Clearly, the functors DGD and DFD are mutually quasiinverse functors. This proves (1).
(2) For each A-module X , we have DFD(D(X )) = DF(X ). By Lemma 3.3 (2), the complex DFD(D(X )) is isomorphic to a complex P • D(X) of the form

Derived equivalences for quotient algebras
In the previous section, we have seen that there are many derived equivalences between quotient algebras of Φ-Auslander-Yoneda algebras that are derived-equivalent (see Theorem 3.4 and Subsection 3.1). This phenomenon gives rise to a general question: How to construct a new derived equivalence for quotient algebras from the given one between two given algebras ? In this section, we shall consider this question and provide methods to transfer a derived equivalence between two given algebras to a derived equivalence between their quotient algebras. In particular, we shall prove Theorem 1.3

Derived equivalences for algebras modulo ideals
Let us start with the following general setting.
Suppose that A is an Artin R-algebra over a commutative Artin ring R, and suppose that I is an ideal in A. We denote by A the quotient algebra A/I of A by the ideal I. The category A-mod can be regarded as a full subcategory of A-mod. Also, there is a canonical functor from A-mod to A-mod which sends each X ∈ A-mod to X := X /IX . This functor induces a functor − : C (A) −→ C (A), which is defined as follows: for a complex X • = (X i ) i∈Z of A-modules, let IX • be the sub-complex of X • in which the i-th term is the submodule IX i of X i ; we define X • to be the quotient complex of X • modulo IX • . The action of − on a chain map can be defined canonically. Thus − is a well-defined functor. For each complex X • of A-modules, we have the following canonical exact sequence of complexes: For a complex Y • of A-modules, this sequence induces another exact sequence of R-modules: Since Y • is a complex of A-modules, the map i * must be zero, and consequently π * is an isomorphism. Now we show that π * actually induces an isomorphism between Hom K (A) (X • ,Y • ) and Hom K (A) (X • ,Y • ).

Lemma 4.1. Suppose that A is an Artin algebra and I is an ideal in A. Let A be the quotient algebra of A modulo I. If X • is a complex of A-modules and Y • is a complex of A-modules, then we have a natural isomorphism of R-modules
Proof. Note that we have already an isomorphism Clearly, π * sends null-homotopic maps to null-homotopic maps. This means that π * induces an epimorphism π * : .
It follows that f i = g i d i−1 Y + d i X g i+1 since π i is surjective for each i. Therefore, the map f • is nullhomotopic. Thus π * is injective.
For any complexes X • and X ′• over A-mod, we have a natural map which is the composition of π • * : In particular, if X • = X ′• , then we get an algebra homomorphism Now, let T • be a tilting complex over A, and let B = End K (A) (T • ). Further, suppose that I is an ideal in A. By the above discussion, there is an algebra homomorphism Let J I be the kernel of η, which is an ideal of B. Since (π * ) −1 is an isomorphism, we see that J I is the kernel of the map π • * : In fact, J I is also the set of all endomorphisms of T • which factorize through the embedding IT • −→ T • . We denote quotient algebra B/J I by B.
In the following, we study when the complex T • is a tilting complex over the quotient algebra A and induces a derived equivalence between A and B. The following result supplies an answer to this question.

Theorem 4.2. Let A be an Artin algebra, and let T • be a tilting complex over
. Therefore, the algebra homomorphism η : Consequently, the tilting complex T • induces a derived equivalence between A and B.
Conversely, we assume that T Proof. Since A and B are basic self-injective algebras, soc(P) and soc(P ′ ) are ideals in A and B, respectively. In the following, we shall verify that the conditions of Theorem 4.2 are satisfied by the ideal soc(P) in A and the tilting complex T • associated to F.
Since F(soc(P)) is isomorphic to soc(P ′ ), we can assume that P = s i=1 P i and P ′ = s i=1 P ′ i , where P and P ′ are indecomposable such that F(soc(P i )) is isomorphic to soc(P ′ i ) for all i = 1, · · · , s. Let D i be the endomorphism ring of soc(P i ), which is a division ring. Since F(soc(P i )) ≃ soc(P ′ i ), we see that D i is isomorphic to End B (soc(P ′ i )). Note that a radical map f : M 1 → M 2 between two projective modules M 1 and M 2 has image contained in rad(M 2 ). Since all the differential maps of T • are radical maps, the image of d k T is contained in rad(T k+1 ) for all integers k. It follows that , soc(P ′ i )) = 0 for all n = 0. Hence, for each integer n = 0, the module ν −1 A P i is not a direct summand of T n . Since T n ≃ ν A T n (Lemma 4.3(2)), we infer that P i is not a direct summand of T n for all n = 0. Recall that i . It follows that ν −1 A P i is a direct summand of T 0 with multiplicity 1. Since ν A T 0 ≃ T 0 , we see that P i is a direct summand of T 0 with multiplicity 1. Note that soc(P i )X = 0 for any A-module X if P i is not a direct summand of X . Hence soc(P i )T • is isomorphic to the stalk complex soc(P i )P i = soc(P i ). Therefore Let T • be the quotient complex T • /(soc(P)T • ). There is a canonical triangle in D b (A): Applying Hom D b (A) (T • , −) to this triangle, we have an exact sequence of B-modules: We claim that λ * is a monomorphism. Since soc(P)T • is isomorphic to s i=1 soc(P i )T • , the map λ can be written as (λ 1 , · · · , λ s ) tr , where λ i : soc(P i )T • → T • is the canonical map, and where tr stands for the transpose of a matrix. Now we consider the following commutative diagram of B-modules: Hom B (B, B).
Since λ i = 0, we see that F(λ i ) is nonzero. Moreover, F(soc(P i )T • ) ≃ F(soc(P i )) ≃ soc(P ′ i ). This implies that F(soc(P i )) is a simple B-module for all i. Hence F(λ i ) * must be injective. To show that λ * is injective, it suffices to show that F(λ) * is injective. This is equivalent to proving that (F(λ 1 ) * , · · · , F(λ s ) * ) tr is injective. For this, we use induction on s. If s = 1, the foregoing discussion shows that this is true. Now we assume s > 1. Then the kernel K of (F(λ 1 ) * , · · · , F(λ s ) * ) tr is the pull-back of (F(λ 1 ) * , · · · , F(λ s−1 ) * ) tr and F(λ s ) * both of which are monomorphisms by induction hypothesis. Thus K is isomorphic to a submodule of both Hom D b (A) (T • , s−1 i=1 soc(P)) and Hom D b (A) (T • , soc(P s )). However, the B-modules Hom D b (A) (T • , soc(P i )) ≃ soc(P ′ i ), i = 1, · · · , s, are pairwise non-isomorphic simple B-modules since B is basic. This implies that K = 0. Hence λ * is injective, and therefore We give a criterion to judge when a derived equivalence satisfies the condition in Theorem 4.4.

Proposition 4.5. Let T • = (T i , d i ) be a tilting complex associated to a derived equivalence F between self-injective basic Artin algebras A and B, and let P be an indecomposable projective A-module.
Suppose we have the following: (2) the multiplicity of P as a direct summand of ν A T 0 is one. Let T • P be the indecomposable direct summand of T • such that P is a direct summand of ν A (T 0 P ), and letP be the projective B-module ν B (Hom K b (A-proj) (T • , T • P )). Then F(soc( A P)) ≃ soc( BP ).
Proof. We know that the Nakayama functor sends P to the injective envelope of top( A P). From (1) it follows that Hom A (T i , soc( A P)) = 0 for all i = 0. Consequently, Hom D b (A) (T • , soc( A P)[i]) = 0 for all i = 0. This means that F(soc( A P)) is isomorphic in D b (B) to a B-module X that is indecomposable. Now we have the following isomorphisms: Hom B (B, X ) ≃ Hom D b (A) (T • , soc( A P)) ≃ Hom D b (A) (T • P , soc( A P)) ≃ Hom B (ν − BP , X ).
Hence soc( BP ) is the only simple B-module which occurs as a composition factor of X . If X were not simple, then we would get a nonzero homomorphism X → top(X ) → soc(X ) → X , which is not an isomorphism. This is a contradiction since End B (X ) ≃ End D b (B) (F(soc( A P)) ≃ End A (soc( A P)) is a division ring. Hence X is simple and isomorphic to soc( BP ). This finishes the proof.

Derived equivalences for algebras modulo annihilators
Now, we turn to another construction for derived-equivalent quotient algebras by using idempotent elements, which can be regarded as another consequence of Theorem 4.2. Proof. Note that such a left ideal I in A exists, and any left ideal L in A with eL = 0 is contained in I. Clearly, I is a left ideal in A. We have to show that I is a right ideal in A. Let x ∈ A and a ∈ I.
Since the right multiplying with x is a homomorphism ϕ from A A to A A, we see that the image ϕ(I) of I under ϕ is a left ideal in A. Since eI = 0, we have ϕ(I) ⊆ I, and ax ∈ I.
Let A be an Artin algebra and e an idempotent of A such that add(Ae) = add(D(eA)). By a result in [4], there is a tilting complex T • associated to e, which is defined in the following way: suppose ϕ is a minimal right add(Ae)-approximation of A. Then we form the following complex: with A in degree zero. Let T • e := (Ae) [1]. The tilting complex T • associated with e is defined to be the direct sum of T We point out that there is another type of construction by passing derived equivalences between two given algebras to that between their quotient algebras, namely, forming endomorphism algebras first, and then passing to stable endomorphism algebras. For details of this construction, we refer the reader to [5, Corollary 1.2, Corollary 1.3]. Now, we end this paper by two simple examples to illustrate our results. Example 1. Let k be a field, and let A be a k-algebra given by the quiver This algebra is isomorphic to the group algebra of the alternative group A 4 if k has characteristic 2. Let e 2 be the idempotent corresponding to the vertex 2, and let T • be the tilting complex T • associated with e 2 . Then the endomorphism algebra B of T • is given by the quiver o o with relations αδ = γβ = δαβγ−βγδα = 0. Note that B is isomorphic to the principal block of the group algebra of A 5 if k has characteristic 2. It is easy to see that the idempotentẽ 2 is the idempotent correspond to the vertex 2 in the quiver of B. Thus, by Proposition 4.7, the algebras A/∇(e 2 ) and B/∇(ẽ 2 ) are derived-equivalent. A calculation shows that A/∇(e 2 ) = A/ α 2 β 3 and B/∇(ẽ 2 ) = B/ βγδα . Note that the quotient algebras A/ α 2 β 3 and B/ βγδα are stably equivalent of Morita type by a result in [8]. Thus A/ α 2 β 3 and B/ βγδα are not only derived-equivalent, but also stably equivalent of Morita type.

Example 2. Let m ≥ 3 be an integer, and let A = k[t]/(t m )
, the quotient algebra of the polynomial algebra k[t] over a field k in one variable t modulo the ideal generated by t m . Let X be the simple A-module k. Then E N A (A ⊕ X ) and E N A (A ⊕ Ω A (X )) are infinite-dimensional k-algebras which can be described by quivers with relations: