Algebr Represent TheorDOI 10.1007/s10468-014-9467-x

A Note on Derived Equivalences for �-Green Algebras

Shengyong Pan ·Zhen Peng

Received: 7 June 2013 / Accepted: 4 February 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this note, we construct derived equivalences for �-Green algebras from a classof triangles in algebraic triangulated categories, where � is a finite admissible set of a setN of the natural numbers.

Keywords Derived equivalence · �-Green algebras

Mathematics Subject Classifications (2010) 18E30 · 16G10 · 16S10 · 18G15

1 Introduction

Derived categories have been one of the important tools in the study of many branches ofmathematics: Lie theory, algebraic geometry, mathematical physics, stable homotopy the-ory, representation theory, etc. In the representation theory of algebras, derived equivalencesof algebras or rings are one of the central themes and are extensively investigated. Derivedequivalences have been shown to preserve many algebraic and geometric invariants and pro-vide new connections. For instance, Hochschild homology and cohomology [14], the centerof algebra [12], finiteness of finitistic dimension [10] have been shown to be preservedunder derived equivalences.

S. Pan (�)Department of Mathematics, Beijing Jiaotong University, Beijing 100044,People’s Republic of Chinae-mail: shypan@bjtu.edu.cn

S. PanBeijing Center for Mathematics and Information Interdisciplinary Sciences,Beijing, People’s Republic of China

Z. PengSchool of Mathematics and Statistics, Anyang Normal University,Anyang 455000, People’s Republic of Chinae-mail: pzhen2002@163.com

S. Pan, Z. Peng

As is known, Rickard’s Morita theory for derived categories states that two rings arederived equivalent if and only if there is a tilting complex over one ring with endomorphismring isomorphic to the other ring [12, 13]. In general, it is a hard problem to construct tiltingcomplexes. In order to understand derived equivalences, it is an important problem to giveconcrete methods to calculate endomorphism rings of tilting complexes. In [6], Hu and Xiconstructed a new derived equivalence between �-Auslander–Yoneda algebras from a givenalmost ν-stable derived equivalence. In [9], Pan constructed derived equivalences between�-Cohen–Macaulay Auslander–Yoneda algebras from a given derived equivalence. In thispaper, we focus on constructing tilting complexes using direct products. The endomorphismrings of these complexes are isomorphic to matrix rings called �-Green algebras, allowingus to obtain derived equivalences that have not been observed by using previous techniques.

Suppose R is a commutative Artinian ring. Our main result is as follows:

Theorem 1.1 (Theorem 3.3) Let � be a finite admissible subset of N and let M be anobject in an algebraic triangulated R-category T . Given a triangle

Xf−→ M1

g−→ Yh−→ X[1]

in T such that f is a left (add (M),�⋃{i − 1|i ∈ �})-approximation of X and g is a right

(add (M),−�⋃{−i+1|i ∈ �})-approximation of Y . Then the �-Green algebras G �(X⊕

M) and G �(M ⊕ Y) are derived equivalent.

If � = {0, 1, 2, · · · , m}, then we have the following corollary.

Corollary 1.2 Let � = {0, 1, 2, · · · , m} and let M be an object in an algebraictriangulated R-category T . Given a triangle

Xf−→ M1

g−→ Yh−→ X[1]

in T such that f is a left (add (M),�⋃{−1})-approximation of X and g is a right

(add (M),−�⋃{1})-approximation of Y . Then the �-Green algebras G �(X ⊕ M) and

G �(M ⊕ Y) are derived equivalent.

This paper is organized as follows. In Section 2, we review some basic facts on derivedequivalences and algebraic triangulated categories and give some notions. In Section 3, wefirstly introduce the class of algebras called �-Green algebras with � a finite admissible setof N. Finally, we prove Theorem 1.1.

2 Preliminaries

In this section, we shall review basic definitions and facts which will be useful in the proofslater on.

2.1 Conventions

We begin by briefly recalling some definitions and notations on derived categories andderived equivalences.

A Note on Derived Equivalences for �-Green Algebras

Let A be an additive category. For two morphisms α : X → Y and β : Y → Z, theircomposition is denoted by αβ. A complex X• = (

Xi, diX

)over A is a sequence of objects

Xi and morphisms diX in A of the form:

· · · → XidiX→ Xi+1 di+1

X→ Xi+1 → · · · ,

such that diXdi+1X = 0 for all i ∈ Z. If X• = (Xi, diX) and Y • = (

Y i, diY

)are two complexes,

then a morphism f • : X• → Y • is a sequence of morphisms f i : Xi → Y i of Asuch that diXf

i+1 = f idiY for all i ∈ Z. The map f • is called a chain map between X•and Y •. The category of complexes over A with chain maps is denoted by C (A ). Letf • : X• → Y • be a morphism of complexes. We say that f • is null-homotopic if we getf i = diri+1 + ridi−1, where ri : Xi → Y i−1. The homotopy category of complexesover A is denoted by K (A ). If A is an abelian category, then a morphism of complexesf • : X• → Y • is a quasi-isomorphism if Hi(f •) : Hi(X•) ∼→ Hi(Y •) for i ∈ Z,where Hi(X•) denotes the i-th cohomology group of the complexX•. Denote by D(A ) thederived category of complexes over A . It is well known that, for an abelian category A , thecategories K (A ) and D(A ) are triangulated categories. For basic results on triangulatedcategories, we refer the reader to [5] and [8].

Let R be a commutative artinian ring with identity and let A be an Artin R-algebra [1].Denote by A-Mod and A-mod the category of left A-modules and finitely generated left A-modules, respectively. The full subcategory of A-Mod and A-mod consisting of projectivemodules is denoted by A-Proj and A-proj, respectively. Let K b(A) denote the homotopycategory of bounded complexes of A-modules and let Db(A) denote the bounded derivedcategory of A-mod, respectively.

The following theorem is a key ingredient of Morita theory on derived equivalences formodule categories of rings or algebras which was established by Rickard [12]. For moredetails on derived equivalences, we refer to [12].

Theorem 2.1 [12, Theorem 6.4] Let A and B be rings with identities. The followingconditions are equivalent.

(i) Db(A-Mod) and Db(B-Mod) are equivalent as triangulated categories.(ii) K −(A-Proj) and K −(B-Proj) are equivalent as triangulated categories.

(iii) K b(A-Proj) and K b(B-Proj) are equivalent as triangulated categories.(iv) K b(A-proj) and K b(B-proj) are equivalent as triangulated categories.(v) B is isomorphic to End Db(A-Mod)(T

•) for some complex T • in K b(A-proj) satisfying

(a) Hom Db(A-Mod)(T•, T •[n]) = 0 for all n �= 0.

(b) add (T •), the category of direct summands of finite direct sums of copies of T •,generates K b(A-proj) as a triangulated category.

Remarks

(1) The rings A and B are said to be derived equivalent if A and B satisfy the conditionsof the above theorem.

(2) The complex T • ∈ K b(A-proj) in Theorem 2.1 (v) which satisfies the conditions (a)and (b) is called a tilting complex for A.

S. Pan, Z. Peng

2.2 Algebraic Triangulated Categories

We start by reviewing some properties of algebraic triangulated categories and lemmas thatwill be essential in the proof of our main result.

Recall that a triangulated category is said to be algebraic if it is triangle equivalent to thestable category of an exact Frobenius category (B, S), for further details we refer to Happel’sbook [5]. Following Happel’s notation, we denote B an extension closed full subcategoryof an abelian category, we also denote S the set of short exact sequences all of whose termsbelong to B. If B has enough S-projective and S-injective objects and that these objectscoincide, then we call (B, S) an Frobenius category. The stable category of (B, S), thatis, the objects are the same as (B, S) and the morphisms obtained by factor through S-projectives, and will be denoted by B. Then the category B is a triangulated category and thesuspension functor is given by the cosyzygy functor �−1. Note that the definition requires

to fix sequences 0 → XiX−→ I (X)

πX−→ �−1(X) → 0 for each object X of B.Recall that a standard triangle in B is defined by completing a morphism u : X → Y in

B by a pushout diagram in B as below.

0 �� XiX ��

u

��

I (X)πX ��

a

��

�−1(X)h �� 0

0 �� Yv �� Cu

w �� �−1(X) �� 0

Then the distinguished triangle is defined to be that that is isomorphic to a standard tri-

angle Xu−→ Y

v−→ Cuw−→ X[1]. In this section, we concern the axiom (TR3) of triangulated

categories, which states that any commutative square can be completed to a morphism oftriangles.

X

(∗)

u ��

f

��

Yv ��

g

��

Cu

h

��

w �� X[1]f [1]

��X′ u′ �� Y ′ v′ �� Cu′

w′�� X′[1]

But such completion is usually not unique. In [2], Dugas constructed a good map relative tothe pair (f, g) such that the completion is unique. Now we recall the construction and somelemmas of good maps as follows.

We are now working with morphisms in B. Following the notation of the above diagram,a commutative square (∗) implies that ug − f u′ = iXα for some α : I (X) → Y ′. We canlift f to a map If : I (X) → I (X′) such that uIf = f u′. Now two maps gv′ : Y → Cu′and αv′ + If b

′ : I (X) → Cu′ induces a unique map h : Cu → Cu′ such that gv′ = vh andαv′ + If b

′ = bh. One can check that hw′ = w�f such that (f, g, h) gives a morphism of

A Note on Derived Equivalences for �-Green Algebras

triangles. Alternatively, chosen α and If , h is the unique map making the following diagramcommutative in B,

0 �� X

(−iXu

)

��

f

��

I (X)⊕ Y(a,v) ��

(If 0α g

)

��

Cu

h

�����

�� 0

0 �� X′(−iX′

u′)

�� I (X′)⊕ Y ′(a′,v′)

�� Cu′ �� 0.

We refer to the maps h constructed in this way as good maps relative to the pair (f, g). Withmaking choices of α and If , Dugas [2] described that the ambiguity in the construction ofa good map h : Cu → Cu′ in B comes from the maps of the form wβv′ with β : �X → Y ′.The following lemmas are now immediate.

Lemma 2.2 ( [2, Lemma 3.1] ) Suppose we have a commutative square (∗) in B, we extendthis commutative square to a morphism of triangles as in the diagram below.

X

(∗)

u ��

f

��

Yv ��

g

��

Cu

h

��

w �� X[1]f [1]

��X′ u′ �� Y ′ v′ �� Cu′

w′�� X′[1]

If B(w, Y ′) = 0, then there is a unique good map h in B making the diagram commute.

Lemma 2.3 ( [2, Lemma 3.2] ) Suppose that the commutative square (∗) in B is null-homotopic in the sense that there is a map ϕ : Y → X′ with f = uϕ and g = ϕu′. Then α

can be chosen such that the above construction yields a good map h with h = 0.

Lemma 2.4 ( [2, Lemma 3.3] ) If the commutative square (∗) in B can be completed bya good map h : Cu → C ′

u which is zero in B, then the square (∗) is null-homotopic (asdefined in Lemma 2.3).

Lemma 2.5 ( [2, Lemma 3.4] ) Let Xu−→ Y

v−→ Cuw−→ X[1] and X′ u′−→ Y ′ v′−→ Cu′

w′−→ X[1]be standard triangles, and let g : Y → Y ′ and h : Cu → Cu′ be maps in B for whichgv′ = vh in B . Then there exist maps f : X → X′ and g′ : Y → Y ′ such that h is goodrelative to the pair (f, g′).

3 Derived Equivalences Between �-Green Algebras

In this section, we shall introduce the �-Green algebras, where � is an admissible set ofN, then we construct derived equivalences between these �-Green algebras algebras fromgiven triangles in an algebraic triangulated category.

S. Pan, Z. Peng

3.1 �-Green Algebras

Let N be the set of natural numbers, and let Z be the set of all integers. Recall that a subsetof Z containing 0 is called an admissible subset of Z ([6]) if the following condition issatisfied:

If i, j, k ∈ � satisfy that i + j + k ∈ �, then i + j ∈ � if and only if j + k ∈ �.Let � be a subset of Z. For a triangulated R-category T with the shift functor [1] and a

triangulated functor F from T to T , we recall the definition of �-Yoneda algebras from [6]in the following.

Let Ei,F,�T be the bi-functor

Hom T (−, F i−) : T × T −→ Z-Mod

(X, Y ) −→ Ei,F,�T :=

{Hom T (X,F iY ), if i ∈ �.

0, if i /∈ �.

Xf−→ X′ → Hom T (f, F

iY ), Yg−→ Y ′ → Hom T (X,F ig),

and let

E�,i,FT (X, Y ) :=

⊕

j∈ZEj−i,F,�T (X,Y ).

Suppose that X,Y and Z are objects in T . Let fi ∈ Ei,F,�T (X, Y ) and gj ∈ Ej,F,�

T (Y,Z).The composition of fi, gj is defined as follows:

Ei,F,�T (X, Y )× Ej,F,�

T (Y,Z) −→ Ei+j,F,�T (X,Z)

(fi , gj ) → fi ◦ gj ={fi(F

igj ), if i ∈ �, j ∈ � and i + j ∈ �.

0, others.(1)

If X = Y , we write EF,i,�T (X) for EF,i,�

T (X,X). Set EF,�T (X) = EF,0,�

T (X) =⊕

i∈Z Ei,F,�T (X). In case F = [1], we denote EF,�

T (X,Y ) and EF,�T (X) by E�

T (X, Y ) andE�T (X), respectively. In [6], Hu and Xi proved that � is an admissible subset in Z if and

only if E�T (X) is an algebra. It is called the �-Auslander–Yoneda algebra of X [6].

Let � be a finite admissible subset of N and let m = max{i|i ∈ �}. We define the �-Green algebras G �(X) for an object X in T , and mention some basic properties of thesealgebras. Firstly, let us define an R-module G �(X) as follows:

G �(X) =

⎛

⎜⎜⎜⎜⎜⎝

EF (X)0,0 0 · · · 0 0EF (X)1,0 EF (X)1,1 · · · 0 0

......

......

...

EF (X)m−1,0 EF (X)m−1,1 · · · EF (X)m−1,m−1 0EF (X)m,0 EF (X)m,1 · · · EF (X)m,m−1 EF (X)m,m

⎞

⎟⎟⎟⎟⎟⎠

A Note on Derived Equivalences for �-Green Algebras

where EF (X)i,j = Ei−j,F,�T (X) is defined as above, 0 ≤ i, j ≤ m. Secondly, for any

x = (xij )0≤i,j≤m, y = (yij )0≤i,j≤m ∈ G �(X), the multiplication of x and y is defined asfollows: xy = (zst )0≤s,t≤m, where

zst ={

0 if 0 ≤ s < t ≤ m

�sk=t xskF

s−kykt if 0≤ t≤ s≤m=

{0 if 0 ≤ s < t ≤ m∑

k−t∈�s−k∈�

xskFs−kykt if 0 ≤ t ≤ s ≤ m.

The following lemma tells us thatG �(X) is an associative ring by the multiplication definedabove.

Lemma 3.1 [11] With the notations above, � is admissible if and only if G �(X) is anassociative ring.

Remark Let � = ⊕i≥0�i be a positive graded algebra. Let c ≥ 0 be a non-negative integer.

Recall that the Green algebra G (�) of the graded algebra � to be the following lowertriangular matrix algebra ([4])

G (�) =

⎛

⎜⎜⎜⎜⎜⎜⎝

�0 0 0 · · · 0 0�1 �0 0 · · · 0 0�2 �1 �0 · · · 0 0· · · · · · · · · · · · · · · · · ·

�c−1 �c−2 �c−3 · · · �0 0�c �c−1 �c−2 · · · �1 �0

⎞

⎟⎟⎟⎟⎟⎟⎠

.

We regard �-Auslander–Yoneda algebra EF,�T (X) as a positive graded algebra. If � is a

finite admissible subset of N and m = max{i|i ∈ �}, then G �(X) is the Green algebra of�-Auslander–Yoneda algebra EF,�

T (X).

Example [15, Example 3.18] Let n be a positive integer and A = k[x]/(xn+1), where k

is a field. Define a grading on A by degx := 1. Then A is a positive graded self-injectivealgebra of Gorenstein parameter n. Then the Green algebra G (A) of the graded algebra A

is isomorphic to the n× n lower triangular matrix algebra over k.The following fact of the �-Green algebra G �(X) is useful, which can be easily

check. Let ei =

⎛

⎜⎜⎜⎜⎜⎝

0 0 0 · · · 0 00 0 0 · · · 0 0......... 1End T (X)

......

0 0 0 · · · 0 00 0 0 · · · 0 0

⎞

⎟⎟⎟⎟⎟⎠

. Thus, we have that 1G�(X) = �mi=0ei is the

identity element of G �(X) and as a left G �(X)-module, G �(X)ei ∼= E�,i,F (X),whereE�,i,F (X) = ⊕m

j=0EF (X)i,j . Then G �(X) ∼= ⊕mi=0E�,i,F (X) as left G �(X)-modules.

The following lemma is essentially taken from [6, Lemma 3.5], the proof given therecarries over to the present situation.

S. Pan, Z. Peng

Lemma 3.2 [11] Let X be an object in T . Assume that X1, X2, X3 ∈ addX. Then we havethe following:

(1) The G �(X)-module E�,i,F (X,Xk) is finitely generated projective, for any 0 ≤ i ≤ m

and k = 1, 2, 3.(2) There is a natural isomorphism

μ : Ei−j,F (X1,X2) −→ Hom G�(X)(E�,i,F (X,X1),E�,j,F (X,X2)),

which sends x ∈ Ei−j,F (X1,X2) to the morphism (x)μ : E�,i,F (X,X1) −→E�,i,F (X,X2), which maps (fk) to (fkF

k−i (x)).(3) If x ∈ Ei−j,F (X1,X2) and y ∈ Ej−k,F (X2,X3), then (xF i−j (y))μ =

(x)μ(F i−j(y))μ.

3.2 Proof of the Main Result

Recall that the left and right cohomological approximations with respective to � in a trian-gulated category T have been defined in [7], where � is an admissible subset of Z. Supposethat T is a triangulated category with shift functor [1]. Let D be a full subcategory of T and a non-empty subset of Z. Suppose X is an object of T . A morphism f : X → D is calleda left (D, )-approximation of X if D ∈ D, and for any morphism f ′ : X → D′[i] withD′ ∈ D and i ∈ , there is a morphism f ′′ : D → D′[i] such that f ′ = ff ′′. Similarly, wehave the notion of a right (D, )-approximation of X.

Let � be a finite admissible subset of N and let m = max{i|i ∈ �}. With the notationsin hands, we can give the following theorem which is the main result of this paper.

Theorem 3.3 Let M be an object in an algebraic triangulated R-category T . Given atriangle

Xf−→ M1

g−→ Yh−→ X[1]

in T such that f is a left (add (M),�⋃{i − 1|i ∈ �})-approximation of X and g is a right

(add (M),−�⋃{−i + 1|i ∈ �})-approximation of Y . Then the algebras G �(X⊕M) and

G �(M ⊕ Y) are derived equivalent.

To prove the theorem, we need to construct a tilting complex overG �(X⊕M), accordingto Rickard’s theorem. Let M ∈ T . Assume that there is a triangle in T

Xf−→ M1

g−→ Yh−→ X[1] (∗)

with M1 ∈ addM , where f is a left (addM,�⋃{i − 1|i ∈ �})-approximation and g is a

right (addM,−�⋃{−i+1|i ∈ �})-approximation. Let U = X⊕M and V = M⊕Y . Set

f =(f, 0) : X → M1 ⊕M, g =(

0 g1 0

)

: M1 ⊕M → M ⊕ Y, h =(

0h

)

: M ⊕ Y → X[1];

f =(f 00 1

)

: X ⊕M → M1 ⊕M, g=(g0

)

: M1 ⊕M → Y, h=(h, 0) : Y → (X ⊕M)[1].Thus, we have two triangles

Xf−→ M1 ⊕M

g−→ Vh−→ X[1] (2)

Y [−1] −h[−1]−−−−→ Uf−→ M1 ⊕M

g−→ Y (3)

A Note on Derived Equivalences for �-Green Algebras

in T induced by the triangle (∗). For each i ∈ �, let T •i be the complex

T •i : 0 −→ E�,i (U,X)

E�,i(U,f )−−−−−−→ E�,i (U,M1 ⊕M) −→ 0,

where the term E�,i(U,X) is in degree 0. Then it is the direct sum of the following twocomplexes:

T •i,1 :0 −→E�,i (U,X)

E�,i(U,f )−−−−−−→ E�,i (U,M1) −→ 0,

T •i,2 :0 −→ 0 −→ E�,i(U,M) −→ 0.

Let T • be a complex with the following form:

T • : 0 −→ G �(U,X) −→ G �(U,M1 ⊕M) −→ 0.

It follows from Lemma 3.2 that each term of T • is a finitely generated projective G �(U)-module.

Remark T • is isomorphic to the complex ⊕mi=0T

•i in C b(G �(U)). In fact, by the definition,

we get

G �(U,X) =

⎛

⎜⎜⎜⎜⎝

Hom (U,X) 0 0 · · · 0E1(U,X) Hom (U,X) 0 · · · 0E2(U,X) E1(U,X) Hom (U,X) · · · 0

· · · · · · · · · · · · · · ·Em(U,X) Em−1(U,X) Em−2(U,X) · · · Hom (U,X)

⎞

⎟⎟⎟⎟⎠.

Hence, we have G �(U,X)ei = E�,i(U,X), where ei is a primitive idempotent element ofG �(U). Thus, we get T •ei ∼= T •

i and T • ∼= ⊕mi=0T

•i .

Lemma 3.4 T • is a tilting complex over G �(U).

Proof Since f is a left (add (M),�⋃{i − 1|i ∈ �})-approximation of X, we get epi-

morphisms T (f,M[i]) : T (M1,M[i]) → T (X,M[i]) → 0 for all i ∈ �. For all i ∈ �,applying the functor T (−,M[i]) to the given triangle (∗), then it follows that the long exactsequence splits up into the short exact sequences

0 → T (Y,M[i]) → T (M1,M[i]) → T (X,M[i]) → 0 ().

Similarly, g is a right (add (M),−�⋃{−i+1|i ∈ �})-approximation of Y , yields the short

exact sequences

0 → T (M,X[i]) → T (M,M1[i]) → T (M, Y [i]) → 0 (),

for all i ∈ �. Since T • is a two-term complex, it is easy to see that

Hom K b(G�(U)-proj)(T•, T •[k]) = 0 f or k ≤ −2 and f or k ≥ 2.

We have to check that

Hom K b(G�(U)-proj)(T•, T •[1]) = 0 and Hom K b(G�(U)-proj)(T

•, T •[−1]) = 0.

S. Pan, Z. Peng

According to the remark we have that

Hom K b(G�(U)-proj)(T•, T •[k]) ∼= Hom K b(G�(U)-proj)

(⊕m

i=0T•i ,⊕m

j=0T•j [k]

)

∼= ⊕mi=0 ⊕m

j=0 Hom K b(G�(U)-proj)

(T •i , T

•j [k]

).

Thus, it suffices to check HomK b(G�(U)-proj)

(T •i , T

•j [k]

)= 0 for k = 1,−1 and for all

i, j ∈ �.

Let f • ∈ Hom K b(G�(U)-proj)

(T •i , T

•j [1]

). We have the following commutative dia-

gram:

0 ��

0

��

0 ��

0��

E�,i (U,X)E�,i(U,f )��

f 0

��

E�,i (U,M1 ⊕M)

0

��

�� 0

0 �� E�,j (U,X)E�,j (U,f )�� E�,j (U,M1 ⊕M) �� 0 �� 0.

By Lemma 3.2(2), there exist two isomorphisms:

Ei−j (X,M1 ⊕M)∼−→ Hom ��(U)(E

�,i(U,X),E�,j (U,M1 ⊕M)),

Hom T (X,M1 ⊕M)∼−→ Hom ��(U)(E

�,i(U,X),E�,i (U,M1 ⊕M)).

Hence, there is uij : X → (M1⊕M)[i−j ] such that (uij )μ = f 0 and (f )μ = E�,i (U, f ).Since f is a left (add (M),�)-approximation, there exists a morphism u′ij : M1 ⊕ M →(M1 ⊕M)[i − j ] such that uij = f u′ij . Thus, we have

f 0 = (uij )μ =(f u′ij

)μ = (f )μ

(u′ij

)μ = E�,i(U, f )

(u′ij

)μ.

This means that f • = 0 in K b(G �(U)-proj). Therefore, Hom K b(G�(U)-proj)(T •, T •[1]) = 0.

Let f • ∈ Hom K b(G�(U)-proj)

(T •i , T

•j [−1]

). Consider the following commutative

diagram:

0 ��

0

��

E�,i (U,X) ��

0

��

E�,i(U,M1 ⊕M)

f 1

��

�� 0

0��

�� 0

0 �� 0 �� E�,j (U,X) �� E�,j (U,M1 ⊕M) �� 0.

By Lemma 3.2(1), we see that E�,i (U,X) is a projective G �(U)-module. It follows froman isomorphism

Ei−j (M1 ⊕M,X)∼−→ Hom G�(U)(E

�,i(U,M1 ⊕M),E�,j (U,X))

that there exists a morphism β ∈ Ei−j (M1 ⊕ M,X) such that (β)μ = f 1. Since f 1 isa chain map, we get f 1Ei−j (U, f ) = 0. By the exact sequence (), we see that β = 0.Therefore, we have f 1 = 0.

It is clearly that add (T •) generates K b(G �(U)-Proj) as a triangulated category.Therefore, T • is a tilting complex over G �(U).

A Note on Derived Equivalences for �-Green Algebras

To prove Theorem 3.3, it suffices to show that the endomorphism ring of T • isisomorphic to G �(V ).

Lemma 3.5 The two rings G �(V ) and End K b(G�(U)-proj)(T•) are isomorphic.

Proof To show the lemma, we will construct an isomorphism from End K b(G�(U)-proj)(T•)

to G �(V ). Since

End K b(G�(U)-proj)(T•) = Hom K b(G�(U)-proj)

(⊕m

i=0T•i ,⊕m

j=0Tj

)

∼= � :=⎛

⎝· · · · · · · · ·· · · Hom K b(G�(U)-proj)(T

•i , T

•j ) · · ·

· · · · · · · · ·

⎞

⎠

(m+1)×(m+1)

,

we will construct the isomorphism by calculating Hom K b(G�(U)-proj)

(T •i , T

•j

). As we

know, T •i is a direct sum of the following two complexes:

T •i,1 :0 −→E�,i (U,X)

E�,i(U,f )−−−−−−→ E�,i (U,M1) −→ 0,

T •i,2 :0 −→ 0 −→ E�,i (U,M) −→ 0.

Thus, we have T •i = T •

i,1 ⊕ T •i,2 and Tj = T •

j,1 ⊕ T •j,2. Therefore, we get

Hom K b(G�(U)-proj)

(T •i , T

•j

)

=⎛

⎝Hom K b(G�(U)-proj)

(T •i,1, T

•j,1

)Hom K b(G�(U)-proj)

(T •i,1, T

•j,2

)

Hom K b(G�(U)-proj)

(T •i,2, T

•j,1

)Hom K b(G�(U)-proj)

(T •i,2, T

•j,2

)

⎞

⎠

We claim that

ϕ1,1ij : Hom K b(G�(U)-proj)

(T •i,1, T

•j,1

) Ei−j (Y, Y ).

By Lemma 3.2(2), we see that a chain map from T •i,1 to T •

j,1 is induced by a pair of maps

xi−j ∈ Ei−j (X,X) and yi−j ∈ Ei−j (M1,M1). Furthermore, such a pair of maps commuteswith the differentials of T •

i,1 and T •j,1. Assuming this commutativity for a fix i−j , we obtain

a map zi−j completing a map of triangles in T ,

Xf ��

xi−j

��

M1g ��

yi−j

��

Y

zi−j

��

h �� X[1]xi−j [1]

��X[i − j ] f [i−j ]�� M1[i − j ] g[i−j ] �� Y [i − j ] h[i−j ]�� X[i − j + 1].

Since T (h,M[k]) = 0 for k ∈ �, it follows from Lemma 2.2 that there is a unique goodmap zi−j making the above diagram commute. Thus, ϕ1,1

ij is well-defined. Since g is a right(add (M),−�

⋃{−i + 1|i ∈ �})-approximation of Y , any map zi−j : Y → Y [i − j ] willbe lifted to two maps yi−j : M1 → M1[i − j ] and xi−j : X → X[i − j ] making the rightsquare of the above diagram commute. This shows that ϕ1,1 is surjective. Furthermore, thekernel of ϕ1,1 consists of those pairs (x, y), where the maps x and y are null-homotopic.Then we get our claim.

S. Pan, Z. Peng

Now consider any map α : T •i,1 → T •

j,2 in K b(G�(U)-proj). Since α is a map ofcomplexes, it must be induced by a unique map b ∈ T (M1,M[i − j ]) such that f b = 0.Then by the exact sequence (), we get

ϕ1,2ij : Hom K b(G�(U)-proj)

(T •i,1, T

•j,2

) Ker T (f,M[i − j ]) Ei−j (Y,M).

Similarly, observe that the map β : T •i,2 → T •

j,1 in K b(G�(U)-proj), this map is inducedby a unique map b′ ∈ T (M,M1[i − j ]) such that b′ factor through f [i − j ]. Then by theexact sequence (), we have

ϕ2,1ij : Hom K b(G�(U)-proj)

(T •i,2, T

•j,1

) Coker T (M, f [i − j ]) Ei−j (M, Y ).

By Lemma 2.2(2), we have

ϕ2,2ij : Hom K b(G�(U)-proj)

(T •i,2, T

•j,2

)= Hom K b(G�(U)-proj)(E

�,i(U,M),E�,j (U,M))

Ei−j (M,M).

Then we define the following correspondences induced by the above argument:

ϕij =(ϕ1,1ij ϕ1,2

ij

ϕ2,1ij ϕ2,2

ij

)

:⎛

⎝Hom

(T •i,1, T

•j,1

)Hom

(T •i,1, T

•j,2

)

Hom(T •i,2, T

•j,1

)Hom

(T •i,2, T

•i,2

)

⎞

⎠

−→(

Ei−j (Y, Y ) Ei−j (M, Y )

Ei−j (Y,M) Ei−j (M,M)

)

= Ei−j (V, V ),

ϕ :=⎛

⎝· · · · · · · · ·· · · ϕij · · ·· · · · · · · · ·

⎞

⎠ : End K b(G�(U)-proj)(T•) −→ G �(V ).

Therefore, ϕ is well-defined and bijective. Thus, it suffices to show that ϕ is an algebra

homomorphism. For any xij , x′ij ∈ Hom

(T •i , T

•j

), we have

ϕ

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎝

x00 0 0 · · · 0x10 x11 0 · · · 0x20 x21 x22 · · · 0· · · · · · · · · · · · · · ·xm0 xm1 xm2 · · · xmm

⎞

⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎝

x′00 0 0 · · · 0

x′10 x′11 0 · · · 0

x′20 x′21 x′22 · · · 0· · · · · · · · · · · · · · ·x′m0 x′

m1 x′m2 · · · x′mm

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

ϕ00(x00x

′00

)0 0 · · · 0

ϕ10(x10x

′00 + x11x

′10

)ϕ11

(x11x

′11

)0 · · · 0

ϕ20(x20x

′00 + x21x

′10 + x22x

′20

)ϕ21

(x21x

′11 + x21x

′11

)ϕ22

(x22x

′22

) · · · 0· · · · · · · · · · · · · · ·

ϕm0(�0≤k≤mxmkx

′k0

)ϕm1

(�1≤k≤mxmkx

′k1

)ϕm2

(�2≤k≤mxmkx

′k2

) · · · ϕmm

(xmmx

′mm

)

⎞

⎟⎟⎟⎠

=ϕ

⎛

⎜⎜⎜⎝

⎛

⎜⎜⎜⎝

x00 0 0 · · · 0x10 x11 0 · · · 0x20 x21 x22 · · · 0· · · · · · · · · · · · · · ·xm0 xm1 xm2 · · · xmm

⎞

⎟⎟⎟⎠

⎞

⎟⎟⎟⎠ϕ

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

x′00 0 0 · · · 0

x′10 x′11 0 · · · 0

x′20 x′21 x′22 · · · 0· · · · · · · · · · · · · · ·x′m0 x′

m1 x′m2 · · · x′mm

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠,

where, ϕii

(xiix

′ii

) = ϕii(xii)ϕii

(x ′ii

)and ϕij

(�0≤j≤ixikx ′kj

)=

�0≤j≤iϕik(xik)ϕkj

(x ′kj

), for i �= j . Then ϕ is an algebra homomorphism.

A Note on Derived Equivalences for �-Green Algebras

Proof of Theorem 3.3 By Lemmas 3.4 and 3.5, the theorem is straightforward fromTheorem 2.1.

Finally, we give an explicit example which satisfies all conditions in Theorem 3.3.

Example Let k be an algebraic closed field of characteristic 2 and A = kA4 be the groupalgebra of alternating group A4. Then there are three simple A-modules, which are denotedk,ω and � , respectively. It was shown in [3, §V2.4.1, p. 129] that A = kA4 is Moritaequivalent to the following algebra given by quiver

� �

�

w w

k

��

��

����������

��

������

���

β2

β3β1

α3 α2

α1

and relation αiβi+1−βiαi+2 = αiαi+1 = βiβi−1 = 0, where the subscripts are consideredmodulo 3.

Take the Auslander-Reiten sequence 0 → �3(ω) → �2(k) ⊕ �2(�) → �(ω) → 0considered in [7]. Let X = �3(ω), M = �2(k) ⊕ �2(�) and Y = �(ω). Then thissequence provides an Auslander-Reiten triangle in A-mod:

Xf−→ M

g−→ Yh−→ X[1]

We can choose � = {0, 1}. Since it is an Auslander-Reiten triangle in A-mod, f is a left (add (M),�

⋃{−1})-approximation of X and g is a right(add (M),−�

⋃{1})-approximation of Y . Then by Theorem 3.3, the algebras(

End (X⊕M) 0Hom (X ⊕M,�−1(X ⊕M)) End (X⊕M)

)

and (End (Y ⊕M) 0

Hom (Y ⊕M,�−1(Y ⊕M)) End (Y ⊕M)

)

are derived equivalent.

Acknowledgments S. Y. Pan is supported by National Natural Science Foundation of China (No.11201022), the Fundamental Research Funds for the Central Universities (2013JBM096, 2013RC027) andMERIT SCHOLARSHIP PROGRAM FOR FOREIGN STUDENTS–Postdoctoral research fellowship (V2)of Quebec, during which this work was carried out. This revision of the first draft was done when S. Y. Pan isa postdoctoral student of Bishop’s University, he would like to thank Professor Thomas Brustle for his warmhospitality. Both the authors thank Hongxing Chen for some useful suggestions.

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S. Pan, Z. Peng

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