Homotopy category of N-complexes of projective modules

Journal of Pure and Applied Algebra 220 (2016) 2414–2433

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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Homotopy category of N -complexes of projective modules P. Bahiraei a,∗ , R. Hafezi b , A. Nematbakhsh b a

Department of Mathematics, University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 25 March 2015 Received in revised form 1 October 2015 Available online 14 December 2015 Communicated by S. Koenig MSC: 18E30; 16G99; 18G05; 18G35

a b s t r a c t In this paper, we show that the homotopy category of N -complexes of projective R-modules is triangle equivalent to the homotopy category of projective TN −1 (R)modules where TN −1 (R) is the ring of triangular matrices of order N −1 with entries in R. We also define the notions of N -singularity category and N -totally acyclic complexes. We show that the category of N -totally acyclic complexes of finitely generated projective R-modules embeds in the N -singularity category, which is a result analogous to the case of ordinary chain complexes. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Given an associative unitary ring R, by an N -complex X • , we mean a sequence of R-modules and R-linear maps · · · → X n−1 → X n → X n+1 → · · · such that composition of any N consecutive maps gives the zero map. The notion of N -complexes was introduced by Mayer [18] in his study of simplicial complexes and its homological theory was studied by Kapranov and Dubois-Violette in [16,5]. Besides their applications in theoretical physics [4,13], the homological properties of N -complexes have become a subject of study for many authors as in [8,10,9,20]. Iyama et al. studied the homotopy category KN (B) of N -complexes of an additive category B as well as the derived category DN (A) of an abelian category A. Recall that an abelian category A is an (Ab4)-category (resp. (Ab4)∗ -category) provided that it has any coproduct (resp. product) of objects, and that the coproduct (resp. product) of monomorphisms (resp. epimorphisms) is monic (resp. epic). In the paper, [15], they showed that the well known equivalences between homotopy category of chain complexes and their derived categories also generalize to the case of N -complexes. More precisely, if A is an abelian category satisfying the condition (Ab4), then we have triangle equivalence  KN (Prj-A) ∼ = DN (A),

* Corresponding author. E-mail addresses: [email protected] (P. Bahiraei), [email protected] (R. Hafezi), [email protected] (A. Nematbakhsh). http://dx.doi.org/10.1016/j.jpaa.2015.11.012 0022-4049/© 2015 Elsevier B.V. All rights reserved.

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where (, ) = (proj, nothing), (−, −), ((−, b), b) and Prj-A is the category of projective objects of A. As for chain complexes a similar statement is also true for the category Inj-A of injective objects of A provided that A satisfies the condition (Ab4)∗ . They also showed that there exists a triangle equivalence DN (A) ∼ = D(TN −1 (A)).

(1.1)

As a consequence of this equivalence they showed that there exist the following triangle equivalences between derived and homotopy categories. Corollary 1.1. For a ring R, we have the following triangle equivalences. DN (Mod-R) ∼ = D (Mod-TN −1 (R)), where  = −, b. KN (Prj-R) ∼ = K (Prj-TN −1 (R)), where  = −, b, (−, b) and also KN (prj-R) ∼ = K (prj-TN −1 (R)), where  = −, b, (−, b). In this paper, we show that the homotopy category KN (Prj-(R)) of N -complexes is embedded in the ordinary homotopy category K(Prj-TN −1 (R)). Having this embedding in hand we are able to recover (1.1) by using different techniques than those in [15]. We also show that KN (Prj-(R)) is equivalent to K(Prj-TN −1 (R)) whenever R is a left coherent ring. The explicit construction of such triangle equivalence allows us to prove an N -complex version of the following equivalence of triangulated categories given in [3,12,2]. Ktac (prj-R) → Dbsg (R) where Ktac (prj-R) is the homotopy category of totally acyclic complexes of finitely generated projective R-modules and Dbsg (R) is the singularity category. The paper is organized as follows. In section 2, we recall some generalities on N -complexes and provide any background information needed through this paper. Our main result appears in section 3 as Theorem 3.17. In that section, we show that the category KN (Prj-R) embeds as a triangulated subcategory in the category K(Prj-TN −1 (R)) see Proposition 3.9. As an application of this embedding we provide a different proof for the triangle equivalence in (1.1). At the end of this section we show that this embedding is also dense, hence an equivalence. In section 4 we define an N -totally acyclic complex as a complex X • in prj-R satisfying the property that for all P • ∈ KbN (prj-R), HomKN (prj-R) (P • , X • ) = HomKN (prj-R) (X • , P • ) = 0. Then we show that the homotopy category Ktac N (prj-R) of N -totally acyclic complexes in prj-R in this sense is triangle equivalent to the homotopy category of ordinary totally acyclic complexes in prj-TN −1 (R), i.e. Ktac (prj-TN −1 (R)). We also define a similar notion of singularity category for N -complexes Dsg N (R) and show that it contains Ktac (prj-R) as a triangulated subcategory. Furthermore, the embedding N sg Ktac N (prj-R) → DN (R)

is an equivalence of triangulated categories, when R is a Gorenstein ring.

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2. Preliminaries 2.1. The category of N -complexes Throughout, R is an associative ring with identity. Mod-R denotes the category of all right R-modules. We fix a positive integer N ≥ 2. An N -complex X • is a diagram

···

di−1 X•

Xi

diX •

X i+1

di+1 X•

···

with X i ∈ Mod-R and morphisms diX • ∈ HomR (X i , X i+1 ) satisfying dN = 0. That is, composition of any N -consecutive maps is 0. A morphism between N -complexes is a commutative diagram

···

di−1 X•

Xi

diX •

X i+1

fi

···

di−1 Y•

Yi

di+1 X•

···

f i+1 diY •

Y i+1

di+1 Y•

···

+ b We denote by CN (R) (resp. C− N (R), CN (R), CN (R)) the category of unbounded (resp. bounded above, bounded below, bounded) N -complexes over Mod-R. For any object M of Mod-R, j ∈ Z and 1 ≤ i ≤ N , let

Dij (M ) : · · ·

X j−i+1

0

j−i+1 dX •

···

dj−2 X•

X j−1

dj−1 X•

Xj

0

···

be an N -complex satisfying X n = M for all j − i + 1 ≤ n ≤ j and dnX • = 1M for all j − i + 1 ≤ n ≤ j − 1. For 0 ≤ r < N and i ∈ Z, we define diX • ,{r} := di+r−1 · · · diX • . X• In this notation diX • ,{1} = diX • and diX • ,{0} = 1Xi . Definition 2.1. Let f : X • −→ Y • be a morphism in CN (R). The mapping cone C(f ) of f is defined as follows ⎡

C(f )m = Y m ⊕

m+N −1 i=m+1

X i,

dm C(f )

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

dm Y•

f m+1

0

0 .. .

0 .. .

1 .. .

0

···

0

0

···

0

−dm+1 X • ,{N −1}

−dm+2 X • ,{N −2}

0 ..

.

..

.

···

··· .. . .. .

0 .. .

1

0

0

··· ···

0 1 m+N −1 −dX •

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Let SN (R) be the collection of short exact sequences in CN (R) with split short exact sequences in each degree. It is easy to see that the category (CN (R), SN (R)) is an exact category such that for every −i+N −1 M ∈ Mod-R and every i ∈ Z, DN (M ) is an SN -projective and SN -injective object of this category. Hence this category is a Frobenius category, see [15, Proposition 1.5].

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Definition 2.2. A morphism f : X • −→ Y • of N -complexes is called null-homotopic if there exists si ∈ HomR (X i , Y i−N +1 ) such that fi =

N −1

i−(N −1−j)

dY • ,{N −1−j} si+j diX • ,{j} .

j=0

We denote the homotopy category of unbounded N -complexes by KN (R). Definition 2.3. For X • = (X i , di ) ∈ CN (R), we define suspension functor Σ : KN (R) −→ KN (R) as follows 0

1

.. .

0 .. .

0 −dm+1 {N −1}

0 −dm+2 {N −2}

⎤ ··· 0 ⎥ .. .. .. ⎥ . . . . ⎥ ⎥ .. .. .. ⎥ . . . 0 ⎥ ⎥ .. .. ⎥ . . 0 ⎥ ⎥ ⎦ ··· ··· 0 1 · · · · · · · −dm+N −1

−dm−1

1

⎡

(ΣX • )m =

m+N −1 i=m+1

X i,

dm ΣX •

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡

(Σ−1 X • )m =

i=m−1 m−N +1

X i,

dm Σ−1 X •

⎢ ⎢ −dm−1 ⎢ {2} ⎢ ⎢ .. ⎢ . =⎢ ⎢ ⎢ ⎢ ⎢ m−1 ⎢ −d ⎣ {N −2} −dm−1 {N −1}

0

0 ..

⎤ ··· ··· 0 .⎥ .. .. . . .. ⎥ 0 1 ⎥ ⎥ ⎥ .. . . .. .. . . . . 0⎥ ⎥ ⎥ .. .. . . 0⎥ ⎥ ⎥ 0 ··· ··· 0 1⎥ ⎦ 0 ··· ··· · 0 0

It is known that KN (R) together with this suspension functor is a triangulated category, see [15, Theorem 1.7]. Let X • ,

···

di−1 X•

Xi

diX •

X i+1

di+1 X•

···

be an N -complex of R-modules. We define Zir (X • ) := KerdiX • ,{r} , Cir (X • ) := Cokerdi−r X • ,{r} ,

Bir (X • ) := Imdi−r X • ,{r} Hir (X • ) := Zir (X • )/BiN −r (X • ).

In each degree we have N − 1 cycles and clearly ZnN (X • ) = X n . Remark 2.4. For any X • ∈ CN (R) if Hi1 (X • ) = 0 for any i ∈ Z, then we have Hir (X • ) = 0 for any i ∈ Z and 0 < r < N . Definition 2.5. Let X • ∈ KN (R). We say X • is N -exact if Hir (X • ) = 0 for each i ∈ Z and all r = 1, 2, . . . , N − 1. We denote the full subcategory of KN (R) consisting of N -exact complexes by Kac N (R).

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 For a full subcategory B of Mod-R, we denote by K,b N (B) the full subcategory of KN (B) consisting of N -complexes X • satisfying Hir (X • ) = 0 for almost all but finitely many i and r, where  = nothing, −, +.

Definition 2.6. A morphism f : X • → Y • is called a quasi-isomorphism if the induced morphism Hir (f ) : Hir (X • ) → Hir (Y • ) is an isomorphism for any i and 1 ≤ r ≤ N − 1, or equivalently if the mapping cone C(f ) belongs to Kac N (R). The derived category DN (R) of N -complexes is defined as the quotient category ac KN (R)/KN (R). 2.2. Triangular matrix ring Let Mn (R) be the set of all n × n square matrices with coefficients in R for n ∈ N. Mn (R) is a ring with respect to the usual matrix addition and multiplication. The identity of Mn (R) is the matrix En = diag(1, ..., 1) ∈ Mn (R) with 1 on the main diagonal and zeros elsewhere. The subset ⎡

R ⎢ ⎢R Tn (R) = ⎢ ⎢ .. ⎣ . R

0 R .. . R

⎤ ··· 0 ⎥ ··· 0 ⎥ . ⎥ .. ⎥ . .. ⎦ ··· R

of Mn (R) consisting of all triangular matrices [aij ] in Mn (R) with zeros over the main diagonal is a subring of Mn (R). It is well known that if Q is the quiver An = 1

2

···

3

n

then RQ ∼ = Tn (R), where RQ is the path algebra of quiver Q. Let Q = (V, E) be a quiver. A representation of Q by a ring R is a correspondence which associates an object Mv to each vertex v and a morphism ϕa : Ms(a) → Mt(a) to each arrow a ∈ E. Let X and Y be two representations by left R-modules of the quiver Q. A morphism f : X → Y is a family of homomorphisms fv : Xv → Yv such that Ya ◦ fv = fw ◦ Xa for any arrow a : v → w. The representations of Q by R-modules and R-homomorphisms form a category denoted by Rep(Q, R). It is known that the category Rep(Q, R) is equivalent to the category of modules over path algebra RQ whenever Q is a finite quiver. Set Q = An . For 1 ≤ i ≤ n, let ei : Rep(Q, R) → Mod-R be the evaluation functor defined by ei (X ) = Xi , for any X ∈ Rep(Q, R). It is proved in [7] that ei has a right adjoint eiρ : Mod-R → Rep(Q, R), where eiρ (M ) is the following representation M

M

···

M

0

···

0

for an R-module M , where M ends in i-th position with identity morphisms beforehand. Moreover, for 1 ≤ j ≤ n, it is shown that ej also admits a left adjoint ejλ , with ejλ (M ) defined as 0

0

···

M

M

···

M

for an R-module M , where M starts in j-th position with identity morphisms afterward. It is proved

n in [6] that any projective (resp. injective) representation X in Rep(Q, R) is of the form i=1 eiλ (P i ) (resp.

n i i i i i=1 eρ (I )), where for any 1 ≤ i ≤ n, P (resp. I ), is the cokernel (resp. kernel) of the split monomorphism X i−1 → X i (resp. epimorphism X i → X i+1 ). Hence any projective object in Mod-Tn (R) is of the form

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P1 → P1 ⊕ P2 → P1 ⊕ P2 ⊕ P3 → ··· → P1 ⊕ P2 ⊕ ··· ⊕ Pn and any injective object in Mod-Tn (R) is of the form I 1 ⊕ I 2 ⊕ · · · ⊕ I n → I 1 ⊕ I 2 ⊕ · · · ⊕ I n−1 → · · · → I 1 ⊕ I 2 → I 1 . 3. Some triangle equivalences between homotopy categories In this section, we show that the homotopy category KN (Prj-(R)) of N -complexes is embedded in the ordinary homotopy category K(Prj-TN −1 (R)). As a result of this embedding we show that there exists a triangle equivalence between derived category of N -complexes and ordinary derived category of complexes of Mod-TN −1 (R). At the end of this section we show that KN (Prj-(R)) ∼ = K(Prj-TN −1 (R)). Let SN (R) be the collection of short exact sequence in CN (R) with split exact sequences in each degree. It is shown in [15] that (CN (R), SN (R)) is a Frobenius category. We need the following definition and lemma from [11]. Definition 3.1. Let (B, S) and (B  , S  ) be Frobenius categories. An additive functor F : B −→ B is called F (u) F (v) exact if 0 −→ F (X) −−−−→ F (Y ) −−−→ F (Z) −→ 0 is contained in S  whenever 0 −→ X −u→ Y −v→ Z −→ 0 is contained in S. If F transforms S-injectives into S  -injectives then F induces a functor F : B −→ B between stable categories. We denote by Σ (resp. Σ ) the translation functor on stable category B (resp. B ). Lemma 3.2. Let F be an exact functor between Frobenius categories B and B such that F transforms S-injectives into S  -injectives. If there exists an invertible natural transformation α : F Σ −→ Σ F then F is a triangle functor. In order to show that KN (Prj-(R)) embeds in K(Prj-TN −1 (R)), we explicitly construct the embedding functor. Construction 3.3. Define the functor F : CN (Prj-R) −→ C(Prj-TN −1 (R)) by the following rules. On objects: Let (P • , d• ) be an object in CN (Prj-R). Define i-th term of F(P • ) as follows: • For i = 2r, let m = N r and define F(P • )i as the following projective representation of AN −1 : 

Pm

1 0





P m ⊕ P m+1

E2 0





···

EN −2 0





P m ⊕ P m+1 ⊕ · · · ⊕ P m+N −3

EN −1 0



P m ⊕ · · · ⊕ P m+N −2

• For i = 2r + 1, let m = N r and define F(P • )i as the following projective representation of AN −1 : 

P m+N −1

1 0





P m+N −1 ⊕ P m+N

E2 0





···

EN −1 0



P m+N −1 ⊕ P m+N ⊕ · · · ⊕ P m+2N −3

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For the definition of differential of F(P • ), we consider the following two cases: (i) For i = 2r, define μi : F(P • )i → F(P • )i+1 by μi = (μij )1≤j≤N −1 where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ μij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

dm+j−1 {N −j}

dm {N −1}

dm+1 {N −2}

···

0

dm+1 {N −1}

· · · dm+j−1 {N −j+1}

0 .. .

0 .. .

..

0

0

···

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.. .

.

dm+j−1 {N −1}

(ii) For i = 2r + 1, define λi : F(P • )i → F(P • )i+1 by λi = (λij )1≤j≤N −1 where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ i λj = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

dm+N −1

−1

0

···

···

0

0

dm+N

−1

0

···

0

0 .. . .. .

0 .. . .. .

dm+N +1

−1

···

..

..

0 .. .

0

···

0

.

.

⎤

−1 ···

0

d

m+N +j−2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

On morphisms: Let f • : Q• −→ P • be a morphism in CN (Prj-R). We define F(f • ) as follows: (i) For i = 2r, let m = N r. Define ϕi : F(Q• )i −→ F(P • )i by ϕi = (ϕij )0≤j≤N −2 where ϕij = diag(f m , . . . , f m+j ). (ii) For i = 2r + 1, let m = N r. Define ϕi : F(Q• )i −→ F(P • )i by ϕi = (ϕij )0≤j≤N −2 where ϕij = diag(f m+N −1 , . . . , f m+N −j ). It is straightforward to show that this construction defines covariant functor from CN (Prj-R) to C(Prj-TN −1 (R)). Example 3.4. Let N = 3. Let P • P• = ···

d−2

P −1

d−1

P0

d0

P1

d1

P2

d2

d3

P3

···

be a 3-complex in C3 (Prj-R). The functor F maps P • to the following complex in C(Prj-T2 (R)) d−1

P −1 (1,0)

P −1 ⊕ P 0



d−1 0

−1 d0

d1 d0

P0

(1,0)

P0 ⊕ P1



d1 d0 0

d1 d2 d1

d2

P2

(1,0)

P2 ⊕ P3



d2 0

−1 d3

d4 d3

P3

(1,0)

P3 ⊕ P4



d4 d3 0

d4 d5 d4

P5

(1,0)

P5 ⊕ P6

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Now consider the morphism f • : (Q• , e• ) −→ (P • , d• ) in C3 (Prj-R), i.e., ···

e−2

e−1

Q−1

f −1

···

d−2

e1

Q1

f0

d−1

P −1

e0

Q0

Q2

f1 d0

P0

e2

f2 d1

P1

P2

e3

Q3

···

f3 d2

d3

P3

···

The image of f • under F is the following diagram in C(Prj-T2 (R)). Q−1

Q0

f −1

f0

Q−1 ⊕ Q0

⎡ ⎣

f −1 0

P −1

⎦

P −1 ⊕ P 0

f0 0

0 f1

Q3

f2

Q0 ⊕ Q1

⎤

0 f0

Q2

f3

Q2 ⊕ Q3



P0

P0 ⊕ P1

f2 0

0 f3

Q3 ⊕ Q4



P2

P2 ⊕ P3

f3 0

0 f4



P3

P3 ⊕ P4

Lemma 3.5. The functor F, defined above, induces a functor from the category KN (Prj-R) to the category K(Prj-TN −1 (R)) which we denote it again by F. Proof. We show that if f • : (Q• , e• ) −→ (P • , d• ) is a null homotopic map in CN (Prj-R) then F(f • ) is a null homotopic map in C(Prj-TN −1 (R)). Since f • ∼ 0• , by definition there exists sm ∈ HomR (Qm , P m−N +1 ) such that fm =

N −1

m−(N −1−k) m+k m s e{k} .

d{N −1−k}

k=0 i i+1 i i i+1 i We want to construct (ti )i∈Z , such that (F(f • ))i = λi−1 μQ• (resp. (F(f • ))i = μi−1 λQ• ) P• t + t P• t + t when i is even (resp. odd). We consider the following two cases: (i) Let i = 2r and m = N r. We define ti = (tij )1≤j≤N −1 where

⎡ N −2 tij

⎢ ⎢ ⎢ =⎢ ⎢ ⎣

k=0

m−(N −1−k) m+k m s e{k}

d{N −2−k} 0 0 .. .

N −2 k=1

N −2 k=0

m−(N −1−k) m+k m+1 s e{k−1}

···

m−(N −2−k) m+k+1 m+1 s e{k}

···

d{N −2−k} d{N −2−k} 0 .. .

0

0

..

N −2 k=j−1

N −2 k=l−2

m−(N −1−k) m+k m+j s e{k−j+1}

d{N −2−k}

m−(N −2−k) m+k+1 m+j s e{k−j+2}

d{N −2−k}

.. .

. ···

N −2

0

··· ···

0 0

..

.. .

k=0

m−(N −j−k) m+k+j−1 m+j s e{k}

d{N −2−k}

(ii) Let i = 2r + 1 and m = N r. We define ⎡

ti = (tij )1≤j≤N −1

where

ti+1 j

sm+N −1 ⎢ 0 ⎢ ⎢ 0 =⎢ ⎢ .. ⎢ ⎣ . 0

sm+N 0 .. . 0

. · · · sm+N +j−2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Lemma 3.6. The functor F : KN (Prj-(R)) −→ K(Prj-TN −1 (R)) is a fully faithful functor.

2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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Proof. Let P • and Q• be two objects in CN (Prj-R). We want to show that HomKN (Prj-R) (Q• , P • ) ∼ = HomK(Prj-TN −1 (R)) (F(Q• ), F(P • )). Let f • : (Q• , e• ) −→ (P • , d• ) be a morphism in CN (Prj-R) such that F(f • ) = 0 in K(Prj-TN −1 (R)). We want to show that f • = 0 in KN (Prj-R). We construct sm : Qm → P m−N +1 such that fm =

N −1

m−(N −1−k) m+k m s e{k}

d{N −1−k}

(3.1)

k=0

for all m ∈ Z. First of all, note that since F(f • ) ∼ 0• , there exists ⎡

tn = (tnj )1≤j≤N −1

n α11 ⎢ 0 ⎢ ⎢ 0 tnj = ⎢ ⎢ . ⎢ . ⎣ . 0

where

n α12 n α22 0 .. . 0

n · · · α1j n · · · α2j

.. . n αjj

..

. ···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

such that  (F(f

•

))nj

=

n−1 (μn−2 + tnj (λn−1 P • )j tj Q• )j

if n is even,

n−1 (λn−2 + tnj (μn−1 P • )j tj Q• )j

if n is odd.

(3.2)

For an r ∈ Z, consider the following diagram QN r−1

eN r−1

f N r−1

P N r−1

N r−1

d

QN r

eN r

f Nr

P Nr

QN r+1

eN r+1

···

d

P N r+1

N r+1

d

(3.3)

f N r+N −1

f N r+1 Nr

QN r+N −1

···

P N r+N −1

Let n = 2r. The first N − 1 vertical maps of diagram (3.3) sit on the diagonal of the matrix of (F(f • ))nN −1 . 2r Therefore for t2r−1 N −1 , tN −1 and −1 ≤ i < N − 3 by (3.2) we have r−N +i+1 2r−1 2r f N r+i = dN α(i+2)(i+2) + α(i+2)(i+2) eN r+i . {N −1}

(3.4)

2r The remaining relations of (3.2) provides us with the following equations for αxy , 1 ≤ x ≤ y ≤ N − 2.

2r αxy =

⎧ y N r−N +x−1+k 2r−1 2r α(k+1)(y+1) + αx(y+1) eN r+y−1 ⎨ k=x−1 d{N −1−k}

if x = N − 2,

⎩ dN r−3 α2r−1 N r−2 2r−1 2r N r+1 {N −1} (N −2)(N −1) + d{N −2} α(N −1)(N −1) + α(N −2)(N −1) e

if x = y = N − 2.

(3.5)

Similarly for n = 2r + 1 the last vertical maps in (3.3) appear on the diagonal in the matrix of (F(f • ))nN −1 . Therefore by taking t2r and t2r+1 in (3.2) we have 2r+1 r+l 2r f N r+l = dN r+l−1 α(l+1)(l+1) + α(l+1)(l+1) eN {N −1} ,

where 0 ≤ l ≤ N − 2 and also

(3.6)

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2r αpq =

⎧ q 2r+1 r(N +1)+q−1 2r + k=p−1 α1k e{N −1−q+k} ⎨ dN r−p+1 α(p−1)q ⎩

2423

if p = N − 1,

N r+N −2 rN +r+N −2 2r+1 2r+1 2r + α(N dN r+N −4 α(N −2)(N −1) + α(N −2)(N −2) e{N −2} −2)(N −1) e{N −1}

if p = q = N − 1, (3.7)

for any 2 ≤ p ≤ q ≤ N − 1. Notice that each f N r+i , i = 0, . . . , N −2 can be defined by two relations, one from (3.4) and one from (3.6). To construct our homotopy maps we need to fix one of the two relations for each f N r+i. Now we choose the following relations

f N r+s

⎧ N r−N +s+1 2r−1 2r α(s+2)(s+2) + α(s+2)(s+2) eN r+s d{N −1} ⎪ ⎪ ⎪ ⎪ ⎨ 2r+1 N r+N −2 2r = dN r+N −3 α(N −1)(N −1) + α(N −1)(N −1) e{N −1} ⎪ ⎪ ⎪ ⎪ 2(r+1) N r+N −1 ⎩ Nr 2r+1 d{N −1} α11 + α11 e

if 0 ≤ s < N − 2, if s = N − 2,

(3.8)

if s = N − 1.

In the relations above, the case s = N −2 is chosen from the relation (3.4) and the rest are chosen from (3.6). Now define the maps sN r+s , 0 ≤ s ≤ N − 1 as

sN r+s

⎧ 2r+1 α11 ⎪ ⎪ ⎪ ⎨ 2r = α1(N −1) ⎪ ⎪ ⎪ N −2−s N r+s−N 2r−1 ⎩ α2r−1 r d α(s+1)(s+k+1) eN k=1 {k−1} (s+2)(s+2) +

if s = N − 1, if s = N − 2, if 0 ≤ s ≤ N − 3.

As r varies in Z, the numbers N r give us a collection of morphisms as above. This gives us a map (sm )m∈Z of degree −(N − 1). By use of (3.5), (3.7) and (3.8) it is easy to show that f N r , f N r+1 , . . . , f N r+N −1 and the homotopy maps (sN r+s )0≤s≤2N −1 satisfy relation (3.1). Now let ϕ• ∈ HomK(Prj-TN −1 (R)) (F(Q• ), F(P • )). We want to find f • ∈ HomKN (Prj-R) (Q• , P • ) such that F(f • ) = ϕ• . We have ϕ• = (ϕi )i∈Z where ϕi = (ϕij )1≤j≤N −1 and ⎡

i β11 ⎢ 0 ⎢ ⎢ i 0 ϕj = ⎢ ⎢ . ⎢ . ⎣ . 0

i β12 i β22 0 .. . 0

i · · · β1j i · · · β2j

..

. ···

.. . i βjj

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2r−1 2r−1 2r−1 We again consider the image of diagram (3.3) under F. By assumption we have ϕ2r j (λQ• )j = (λP • )j ϕj 2r 2r and ϕ2r+1 (μ2r Q• )j = (μP • )j ϕj for any 1 ≤ j ≤ N − 1. This gives us the following set of relations: j

2r−1 2r−1 2r 2r N r+q−2 dN r+p−2 βpq − β(p+1)q = −βp(q−1) + βpq e ,

1 ≤ p ≤ q ≤ N − 1,

(3.9)

and q k=p

r+k−1 2r dN {N +p−k−1} βkq =

q k=p

2r−1 N r+q−1 βpk e{N +k−q−1} ,

1 ≤ p ≤ q ≤ N − 1.

(3.10)

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Our goal is to construct a set of maps (f N r+s )−1≤s
f N r+s

⎧ N −1 2r−1 N r−1 e{k−1} ⎪ k=1 β1k ⎪ ⎪ ⎪ ⎨ N −1 2r−1 N r 2r Nr e{k−2} + β1(N = k=2 β2k −1) e{N −2} ⎪ ⎪ ⎪ ⎪ s+1 N r+k−2 2r ⎩ N −1 2r−1 N r+s N r+s k=2+s β(2+s)k e{k−(s+2)} + k=1 d{s+1−k} βk(N −1) e{N −s−2}

if s = −1, if s = 0, if 1 ≤ s < N − 1.

By (3.9), (3.10) we have −1 ≤ s ≤ N − 1.

dN r+s f N r+s = f N r+s+1 eN r+s ,

Therefore as r varies in Z the set of maps (f N r+s )−1≤s
s2r = (s2r j )1≤j≤N −1

where s2r j

s2r 11 ⎢ 0 ⎢ ⎢ 0 =⎢ ⎢ . ⎢ . ⎣ . 0

s2r 12 s2r 22 0 .. . 0

· · · s2r 1j · · · s2r 2j .. . . · · · s2r jj ..

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

such that for any 1 ≤ p ≤ q ≤ j, N −(q+1)

s2r pq

=



N r+q−1 2r−1 βp(q+k) e{k−1} +

k=1

p−1

N r+q−1 r+k−1 2r dN {p−1−k} βk(N −1) e{N −q−1}

k=1

and also define 2

s2r+1 = 0.

• Lemma 3.7. Let (P • , d• ) ∈ Kac N (Prj-R). The image of P under F is an exact complex in K(Prj-TN −1 (R)).

Proof. Suppose i = 2r and m = N r. The following diagram shows the image of P • under F in degree i, i + 1, i + 2 and i + 3. Pm μi1

P m+N −1 λi+1 1

P m+N μi+2 1

P m+2N −1

P m ⊕ P m+1

···

μiN −1

μi2

P m+N −1 ⊕ P m+N

···

···

P m+N ⊕ · · · ⊕ P m+2N −2 μi+2 N −1

μi+2 2

P m+2N −1 ⊕ P m+2N

P m+N −1 ⊕ · · · ⊕ P m+2N −3 λi+1 N −1

λi+1 2

P m+N ⊕ P m+N +1

P m ⊕ · · · ⊕ P m+N −2

···

P m+2N −1 ⊕ · · · ⊕ P m+3N −3

P. Bahiraei et al. / Journal of Pure and Applied Algebra 220 (2016) 2414–2433

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We want to show that Imμij = Kerλi+1 for any 1 ≤ j ≤ N −1. Clearly Imμij ⊆ Kerλi+1 j j . Let (x1 , x2 , . . . , xj ) ∈ i+1 Kerλj . It is easy to show that there exists yt ∈ P m+t−1 for all 1 ≤ t ≤ j such that

xp =

j−p+1

i+p−1+(k−1)

d{N −k}

(yp+k−1 ),

k=1

for all 1 ≤ p ≤ t. Hence (x1 , x2 , . . . , xj ) = μij (y1 , y2 , . . . , yj ). = Kerμi+2 for any 1 ≤ j ≤ Likewise suppose that i = 2r + 1 and m = N r. We show that Imλi+1 j j i+1 i+2 i+2 N − 1. Clearly Imλj ⊆ Kerμj . Let (x1 , x2 , . . . , xj ) ∈ Kerμj . It is easy to show that there exists yt ∈ P m+N −2+t for all 1 ≤ t ≤ j such that xj = dm+2N −3 (yj ), and xq = −yq+1 + dm+N +q−2 (yq ) for any 1 ≤ q ≤ t and q = j. Hence (x1 , x2 , . . . , xj ) = λi+1 j (y1 , y2 , . . . , yj ).

2

Lemma 3.8. The functor F : KN (Prj-(R)) −→ K(Prj-TN −1 (R)) is a triangle functor. Proof. Clearly F : CN (Prj-(R)) −→ C(Prj-TN −1 (R)) is an exact functor. Since F preserves direct sum j it is enough to show that F transforms DN (P ) to a projective object of C(Prj-TN −1 (R)). By Lemma 3.7 j j F(DN (P )) is a bounded exact complex in C(Prj-TN −1 (R)), since DN (P ) is an N -exact complex. Hence j F(DN (P )) is a projective object in C(Prj-TN −1 (R)). Now we show that F(ΣP • ) ∼ = F(P • )[1]. Let P • be a complex in KN (Prj-(R)). For i ≡ 0(mod 2) let m = m+N −2+j l i αj,k : (ΣP • )m+k−1 → ⊕l=m+N −1 P be a morphism given as 1

⎛

dm+k {N −k−1}

⎜ ⎜ dm+k ⎜ {N −k} ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ ⎜ m+k = k ⎜ d{N −2} ⎜ k +1⎜ 0 ⎜ ⎜ .. .. ⎜ ⎜ . . ⎝ j 0 2

i αj,k

dm+k+1 {N −k−2} dm+k+1 {N −k−1} .. . dm+k+1 {N −3} 0 .. . 0

iN 2 .

−2 · · · dm+k+N {−k+1}

i αj,2

⎞

⎟ −2 ⎟ · · · dm+k+N ⎟ {−k+2} ⎟ ⎟ .. ⎟ . ⎟ ⎟ m+k+N −2 ⎟ · · · d{0} ⎟ ⎟ ⎟ ··· 0 ⎟ ⎟ .. ⎟ ⎟ . ⎠ ··· 0

These morphisms define a map αji : F (ΣP • )ij → (F(P • )[1])ij as i αji = [ αj,1

For 1 ≤ j ≤ N − 1 and 1 ≤ k ≤ j let

i · · · αj,j ].

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For i ≡ 1(mod 2) let m = be a morphism given as

(i+1)N . 2

i For 1 ≤ j ≤ N − 1 and 1 ≤ k ≤ j let αj,k : (ΣP • )m+k−1 → ⊕m+j−1 Pl l=m

⎛

1 .. .

i αj,k

0 ⎜ .. ⎜. ⎜ k−1⎜ ⎜0 ⎜ = k ⎜1 ⎜ k+1⎜ ⎜0 .. ⎜ .. ⎝. . j 0

⎞ 0 ··· 0 .. ⎟ . 0⎟ ⎟ 0 ··· 0⎟ ⎟ ⎟ 0 ··· 0⎟ ⎟ 0 ··· 0⎟ ⎟ .. ⎟ . 0⎠ 0 ··· 0

Likewise these morphisms give a map αji : F (ΣP • )ij → (F(P • )[1])ij defined by i αji = [ αj,1

i αj,2

i · · · αj,j ].

In the other direction for i ≡ 0(mod 2), define βji : (F(P • )[1])ij → F (ΣP • )ij as ⎡

i ⎤ βj,1 i ⎥ ⎢ βj,2 ⎢ ⎥ i βj = ⎢ . ⎥ ⎣ .. ⎦ i βj,j

where ⎛ i βj,k =

1 ··· k − 1

0 ··· ⎜ .. ⎜. ⎜. ⎜. ⎝. 0 ···

0 .. . .. . 0

k 0 .. . 0 1

k + 1 ··· 0 .. . .. . 0

j

⎞ ··· 0 .. ⎟ .⎟ .. ⎟ ⎟ .⎠ ··· 0

For i ≡ 1(mod 2) define βji : (F(P • )[1])ij → F (ΣP • )ij as i ⎤ βj,1 i ⎥ ⎢ βj,2 ⎥ ⎢ i βj = ⎢ . ⎥ ⎣ .. ⎦ i βj,j

⎡

where

i βj,k

=

 0j−k+1×k−1 0k−1×k−1

Ij−k+1 0k−1×j−k+1

in which, Ij−k+1 is the identity matrix of order j − k + 1, and the other three entries are zero matrices of given size. It is not so hard to see that the composition αi ◦ β i : (F(P • )[1])i → (F(P • )[1])i is the identity morphism. One can show that β ◦ α − 1 is null-homotopic where the homotopy maps are defined as follows: For i ≡ 0(mod 2), 1 ≤ j ≤ N − 1 and 1 ≤ k ≤ j let

P. Bahiraei et al. / Journal of Pure and Applied Algebra 220 (2016) 2414–2433

ψki =

2427



0k×N −k−1

0k×k

−IN −k−1

0N −k−1×k

and define ⎡ ⎢ ⎢ ⎢ sij = ⎢ ⎢ ⎢ ⎣

ψ1i

ψ2i

0

ψ1i

.. .

..

0

···

··· ..

.

. 0

ψji

⎤

⎥ ⎥ i ψj−1 ⎥ ⎥ .. ⎥ ⎥ . ⎦ ψ1i

For i ≡ 1(mod 2) and 1 ≤ j ≤ N − 1 define sij = 0. It is quite tedious to show that the morphisms α : F (ΣP • ) → F (P • )[1] and β : F (P • )[1] → F (ΣP • ) are natural in P • . 2 Putting it all together, we have Proposition 3.9. The functor F : KN (Prj-R) −→ K(Prj-TN −1 (R)) is a fully faithful triangle functor. Definition 3.10. We call an N -complex P • , K-projective, if P • ∈ CN (Prj-R) and HomKN (R) (P • , Y • ) = 0 Proj for all Y • ∈ Kac N (R). We denote by KN (Prj-R) the category of all K-projective N -complexes. Dually • we define the triangulated full subcategory KInj N (Inj-R) consisting of complexes I of injectives such that • • • ac HomKN (R) (X , I ) = 0 for all X ∈ KN (R).

Theorem 3.11. We have the following triangle equivalences: ∼ KProj N (Prj-R) = DN (R),

∼ KInj N (Inj-R) = DN (R).

Proof. See [15, Theorem 2.22.]. 2 Definition 3.12. For an additive category A with arbitrary coproducts, an object C is called compact in A if    the canonical morphism i HomA (C, Xi ) → HomA (C, i Xi ) is an isomorphism for any coproduct i Xi in A. We denote by Ac the subcategory of A consisting of all compact objects. Definition 3.13. Let T be a triangulated category. A non-empty subcategory S of T is said to be thick if it is a triangulated subcategory of T that is closed under retracts. If, in addition, S is closed under all coproducts allowed in T , then it is localizing; if it is closed under all products in T it is colocalizing. The following remark gives us a better understanding of the objects in a thick subcategory, see [17]. Remark 3.14. Let S be a class of objects of a triangulated category T . Then  • Thick(S) = n∈N S n , where – S 1 is the full subcategory of T containing S and closed under finite direct sums, direct summands and shifts. – For n > 1, S n is the full subcategory of T consisting of all objects S such that there is a distinguished triangle Y → X → Z → ΣY in T with Y ∈ S i , and Z ∈ S j such that i, j < n and S is a direct summand of shifting of X.

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The following theorem has been proved by Iyama et al. in [15]. As a result of Proposition 3.9, we present another proof for this theorem. Theorem 3.15. For a ring R, we have the following triangle equivalence: DN (R) ∼ = D(TN −1 (R)). Proj ∼ Proof. By Theorem 3.11, KProj (Prj-TN −1 (R)) ∼ = D(TN −1 (R)), and we have N (Prj-R) = DN (R) and K the following diagram:

KN (Prj-R)

F

KProj N (Prj-R)

K(Prj-TN −1 (R))

KProj (Prj-TN −1 (R))

In addition D(TN −1 (R))c ∼ = Kb (prj-TN −1 (R)). For 1 ≤ i ≤ N − 1 let Ri be the following projective representation of AN −1 : 0 → 0 → ··· → R → R → ··· → R where R start in i-th position with identity morphisms afterward. We can show that Kb (prj-TN −1 (R)) = Thick({R•1 , . . . , R•N −1 }) whenever R•i is a complex · · · → 0 → Ri → 0 → 0 → · · · concentrated in degree 0. Now we show that each R•i belong to ImF. Let R• be an N −2 • N -complex · · · → 0 → R → 0 → 0 → · · · concentrated in degree 0. It is easy to check that F(DN −i (R )) = • • • • Ri . Therefore Ri ∈ ImF for 1 ≤ i ≤ N −1. Hence Thick({R1 , . . . , RN −1 }) ⊆ ImF ⊆ D(TN −1 (R)). But ImF is closed under coproduct and contains compact objects, therefore the restriction of functor F to DN (R) is dense, hence DN (R) ∼ = D(TN −1 (R)). 2 According to the above theorem, we have a triangle equivalence ∼ − D− N (R) = D (TN −1 (R)). ∼

Thus F |K− (Prj-R) : K− −=−→ K− (Prj-TN −1 (R)), see [15, Corollaries 4.11, 2.17]. Moreover N (Prj-R) − N F |K− (Prj-R) induces some triangle equivalences between subcategories of K− N (Prj-R) and subcategories of N K− (Prj-TN −1 (R)), see [15, Corollary 4.15]. We summarize all of these equivalences in the following diagram. Note that the existence of the first row follows from Proposition 3.9. KN (Prj-R)

K− N (Prj-R)

−,b KN (Prj-R)

KbN (Prj-R)

KbN (prj-R)

F

∼ =

∼ =

∼ =

∼ =

K(Prj-TN −1 (R))

K− (Prj-TN −1 (R))

K−,b (Prj-TN −1 (R))

Kb (Prj-TN −1 (R))

Kb (prj-TN −1 (R))

P. Bahiraei et al. / Journal of Pure and Applied Algebra 220 (2016) 2414–2433

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At the end of this section we show that the functor F is dense, hence there exists a triangle equivalence between KN (Prj-R) and K(Prj-TN −1 (R)). Before we give the proof we need to introduce another functor. Let Q be a quiver of type An . Any projective representation P of Q is of the form P = ⊕ni=1 eiλ (P i ), where P i is the cokernel of split monomorphism Pi−1 → Pi . For any projective representation P = ⊕ni=1 eiλ (P i )  = ⊕n ei (P i ). Clearly P  is an object in the category Prj-op (An ), where Prj-op (An ) is the of Q, set P i=1 ρ category of all representations by projective modules with split epimorphism maps.  Now we define a functor  : Prj-An → Prj-op An such that any P ∈ Prj-An is mapped under  to P, j i i j as defined above, and for any morphism ϕ = (ϕt )1≤t≤n in Hom(eλ (P ), eλ (P )) define ϕ  by  ϕ =

0 (ϕt )t

i
It is easy to check that  is in fact an equivalence of categories. We also see that for a finite quiver Q,  is an equivalence of categories, see [1]. The functor  can be naturally extended to a functor K(Prj-An ) → • be the complex with X i as its K(Prj-op An ) which we denote again by . For any X • ∈ K(Prj-An ), let X  i-th term and di as its i-th differential. The functor  also is an equivalence of homotopy categories. We also need the following description of compact objects in KN (Prj-R). Neeman [19] showed that an object X • of K(Prj-R) is compact if and only if it is isomorphic, in K(Prj-R), to a complex Y satisfying (i) Y is a complex of finitely generated projective modules. (ii) Y i = 0 if i << 0. (iii) Hi (Y∗ ) = 0 if i << 0, where Y∗ = Hom(Y, R). He also showed that when the compact objects generate the category K(Prj-R). Proposition 3.16. If R is a left coherent ring, then the category K(Prj-R) is compactly generated. This idea enables us to prove our main theorem: Theorem 3.17. For a left coherent ring R, we have triangle equivalence KN (Prj-R) ∼ = K(Prj-TN −1 (R)). Proof. In view of Proposition 3.9, the functor F is full and faithful. Now we show that F is dense. Let T be a triangulated subcategory of KN (Prj-R) such that for all T ∈ T we have the following conditions (1) T ∈ K+ N (prj-R); (2) There exists an integer n ∈ Z such that for every i < n and 1 ≤ r ≤ N − 1, Hir T∗ = 0 where T∗ denotes the induced complex Hom(T, R). −,b Clearly the duality Hom(−, R) : prj-R −→ prj-Rop induces an equivalence T op ∼ = KN (prj-Rop ) of triangulated categories. On the other hand, by restricting the functor F to T , we have

KN (Prj-R)

T

F

F|T

K(Prj-TN −1 (R))

K(Prj-TN −1 (R))c

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We want to show that F|T is an equivalence. It is easy to check that we have the following commutative diagram F|T

T ∼ =

K(Prj-TN −1 (R))c 

Hom(−,R)

op K−,b N (prj-R )

Fop

◦Hom(−,R)

∼ =

K−,b (prj-TN −1 (Rop ))

∼ =

 where Fop is the functor F in construction 3.3 whenever we define it on CN (Prj-Rop ) and is the quasiinverse of the functor  . Therefore F|T = ◦ Hom(−, R) ◦ Fop ◦ Hom(−, R), hence it is an equivalence. It shows that K(Prj-TN −1 (R))c ⊆ ImF. On the other hand ImF is closed under coproduct and contains compact objects, therefore ImF = K(Prj-TN −1 (R)). 2 Corollary 3.18. If R is a left coherent ring, then the category KN (Prj-R) is compactly generated. In a dual manner, in view of construction 3.3 and Lemmas 3.5, 3.6 and 3.8 we can embed the category KN (Inj-R) into the category K(Inj-TN −1 (R)). Since the compact objects of K(Inj-TN −1 (R)) are different from K(Prj-TN −1 (R)), the proof of Theorem 3.17 dose not work. However, when R is an artin algebra the embedding is dense. Proposition 3.19. Let Λ be an artin algebra. We have a triangle equivalence KN (Inj-Λ) ∼ = K(Inj-TN −1 (Λ)). Proof. Let D denote the duality between right and left Λ-modules. The adjoint pair of functors − ⊗Λ D(Λ) and HomΛ (D(Λ), −) induces an equivalence between Prj-Λ and Inj-Λ, which restricts to an equivalence between prj-Λ and inj-Λ. Therefore the adjoint pair induces an equivalence between KN (Prj-Λ) and KN (Inj-Λ). So if we denote the embedding KN (Inj-R) → K(Inj-TN −1 (R)) by G, then we have the following diagram: KN (Prj-Λ)

∼ =

KN (Inj-Λ)

∼ =

K(Prj-TN −1 (Λ))

G ∼ =

K(Inj-TN −1 (Λ))

Hence G is an equivalence of categories. 2 As a final remark we will discuss about N -dualizing complex. Remark 3.20. Let R be a commutative noetherian ring with a dualizing complex D and Q be a finite quiver. In [1] they showed that Grothendieck duality is extendable to path algebra RQ. Therefore when Q = AN −1 , we have Db (mod-TN −1 (R))op ∼ = Db (mod-TN −1 (R)). On the other hand by [15] we have DbN (mod-R) ∼ = Db (mod-TN −1 (R)).

P. Bahiraei et al. / Journal of Pure and Applied Algebra 220 (2016) 2414–2433

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Hence DbN (mod-R)op ∼ = DbN (mod-R). So in case R has a dualizing complex, there exists a Grothendieck duality for derived category of N -complexes and the question is “What could be the definition of an N -dualizing complex?”. 4. N -totally acyclic complexes Let A be an additive category. We say that a complex X • in A is acyclic if the complex HomA (A, X • ) of abelian groups is acyclic for all A ∈ A. If in addition HomA (X • , A) is acyclic for all A ∈ A, then X • is called totally acyclic. Let Ctac (A) denote the full subcategory of C(A) consisting of totally acyclic complexes. Note that these definitions are up to isomorphism in K(A). The full triangulated subcategory of K(A) consisting of totally acyclic complexes, will be denoted by Ktac (A). For instance if A = Prj-R is the class of projective objects in Mod-R then the object X • of Ktac (Prj-R) is an exact complex such that HomR (X • , P ) is acyclic for all P ∈ Prj-R. The objects of Ktac (Prj-R) will be called totally acyclic complexes of projectives. Remark 4.1. It is easy to see that X • ∈ Ktac (Prj-R) if and only if HomK(Prj-R) (P • , X • ) = 0 and HomK(Prj-R) (X • , P • ) = 0 for all P • ∈ Kb (Prj-R). Let R be a Noetherian ring. For the rest of this section we are only considering the category prj-R, i.e. the category of finitely generated projective left R-modules. Given an integer n and a complex ···

dn−3

X n−2

dn−2

X n−1

dn−1

dn

Xn

X n+1

dn+1

X n+2

dn+2

···

in Mod-R, we denote its brutal truncation at degree n by dn−3

τ≤n (X • ) : · · ·

X n−2

dn−2

X n−1

dn−1

Xn

0

0

···

If X • ∈ Ktac (prj-R), then τ≤n (X • ) ∈ K−,b (prj-R). The brutal truncation at degree zero induces a map from the category Ktac (prj-R) to the category K−,b (prj-R). However, this map is not a functor. Now consider the singularity category Dbsg (R) = K−,b (prj-R)/Kb (prj-R). The brutal truncation induces a triangle functor τproj : Ktac (prj-R) −→ Dbsg (R). This functor is always full and faithful. If R is either an Artin ring or commutative Noetherian local ring, then the functor τproj is dense if and only if R is Gorenstein, see [3,12] and [2]. Motivated by the discussion above, in this section we introduce N -totally acyclic complexes and N -singularity category. We show that the restriction of functor F to this category is an equivalence. As a result of this equivalence, we show that there exists an equivalence between N -singularity category of Mod-R and usual singularity category of Mod-TN −1 (R). It is easy to see that an N -complex X • ∈ KN (prj-R) is N -acyclic if and only if HomKN (prj-R) (P • , X • ) = 0 for all P • ∈ Kb (prj-R).

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Definition 4.2. An N -complex X • ∈ KN (prj-R) is called N -totally acyclic if and only if HomKN (prj-R) (P • , X • ) = 0 and HomKN (prj-R) (X • , P • ) = 0 for all P • ∈ KbN (prj-R). We denote by Ktac N (prj-R) the category of all N -totally acyclic complexes in prj-R. • Clearly, if X • ∈ Ktac N (prj-R) then X is an N -acyclic complex.

Proposition 4.3. For a left coherent ring R, we have the following triangle equivalences. ∼ (i) Kac N (prj-R) = Kac (prj-TN −1 (R)). tac (ii) KN (prj-R) ∼ = Ktac (prj-TN −1 (R)). Proof. (i) By Theorem 3.17, the functor F induced an equivalence KN (Prj-R)

Kac N (prj-R)

F ∼ =

K(Prj-TN −1 (R))

Kac (prj-TN −1 (R))

• • • b Let P • be an object of Kac N (prj-R). Hence HomKN (prj-R) (Q , P ) = 0 for all Q ∈ KN (prj-R). Now let • b • b • • P ∈ K (prj-TN −1 (R)). There exists an object Q ∈ KN (prj-R) such that F(Q ) = P . We have

HomK(prj-TN −1 (R)) (P • , F(P • )) = HomK(prj-TN −1 (R)) (F(Q• ), F(P • )) ∼ = HomKN (prj-R) (Q• , P • ) = 0. It shows that F(P • ) ∈ Kac (prj-TN −1 (R)). Hence the functor F sends any object of the subcategory Kac N (prj-R) of KN (Prj-R) to an object of the subcategory Kac (prj-TN −1 (R)) of K(Prj-TN −1 (R)). • • Let P • ∈ Kac (prj-TN −1 (R)). In a similar way there exists P • ∈ Kac N (prj-R) such that F(P ) = P , hence is dense. F|Kac N (prj-R) (ii) It is similar to (i) by use of Remark 4.1. 2 We define the N -singularity category Dsg N (R) of R as a Verdier quotient b K−,b N (prj-R)/KN (prj-R).

Remark 4.4. The brutal truncation at degree zero induces a triangle functor sg Ktac N (prj-R) −→ DN (R),

and the following diagram shows that this functor is always full and faithful. Moreover it is a triangle equivalence of categories when R is a Gorenstein ring. −,b b Dsg N (R) = KN (prj-R)/KN (prj-R)

Ktac N (prj-R) ∼ =

∼ =

F

Ktac (prj-TN −1 (R))

τproj

 F

Dbsg (TN −1 (R)) = K−,b (prj-TN −1 (R))/Kb (prj-R)

 is induced from equivalences in the diagram after Theorem 3.15. The functor F

P. Bahiraei et al. / Journal of Pure and Applied Algebra 220 (2016) 2414–2433

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Remark 4.4 provides us with another interpretation of quotient category K∞,b (prj-R)/Kb (prj-R) where K∞,b (prj-R) is the homotopy category of unbounded complexes with bounded homologies. Iyama et al. showed that there is a triangle equivalence between the above quotient category and Dbsg(T2 (R)), whenever R is a Gorenstein ring, see [14]. Hence by Remark 4.4 we have the following equivalence of categories sg K∞,b (prj-R)/Kb (prj-R) ∼ = D3 (R).

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