aa r X i v : . [ m a t h . R T ] O c t THE PROJECTIVE LEAVITT COMPLEX
HUANHUAN LI
Abstract.
Let Q be a finite quiver without sources, and A be the correspond-ing radical square zero algebra. We construct an explicit compact generatorfor the homotopy category of acyclic complexes of projective A -modules. Wecall such a generator the projective Leavitt complex of Q . This terminologyis justified by the following result: the opposite differential graded endomor-phism algebra of the projective Leavitt complex of Q is quasi-isomorphic tothe Leavitt path algebra of Q op . Here, Q op is the opposite quiver of Q and theLeavitt path algebra of Q op is naturally Z -graded and viewed as a differentialgraded algebra with trivial differential. Introduction
Let A be a finite dimensional algebra over a field k . We denote by K ac ( A -Proj)the homotopy category of acyclic complexes of projective A -modules. This cate-gory is a compactly generated triangulated category whose subcategory of compactobjects is triangle equivalent to the opposite category of the singularity category[4, 15] of the opposite algebra A op .In this paper, we construct an explicit compact generator for the homotopycategory K ac ( A -Proj) in the case that A is an algebra with radical square zero. Thecompact generator is called the projective Leavitt complex . Recall from [7, Theorem6.2] that the homotopy category K ac ( A -Proj) was described in terms of Leavittpath algebra in the sense of [2, 3]. We prove that the opposite differential gradedendomorphism algebra of the projective Leavitt complex of a finite quiver withoutsources is quasi-isomorphic to the Leavitt path algebra of the opposite quiver. Here,the Leavitt path algebra is naturally Z -graded and viewed as a differential gradedalgebra with trivial differential.Let Q be a finite quiver without sources, and A = kQ/J be the correspondingalgebra with radical square zero. We introduce the projective Leavitt complex P • of Q in Definition 2.4. Then we prove that P • is acyclic; see Proposition 2.7.Denote by L k ( Q op ) the Leavitt path algebra of Q op over k , which is naturally Z -graded. Here, Q op is the opposite quiver of Q . We consider L k ( Q op ) as a differentialgraded algebra with trivial differential.The following is the main result of the paper. Theorem . Let Q be a finite quiver without sources, and A = kQ/J be the corre-sponding finite dimensional algebra with radical square zero. (1) The projective Leavitt complex P • of Q is a compact generator for the ho-motopy category K ac ( A - Proj) . (2) The opposite differential graded endomorphism algebra of the projectiveLeavitt complex P • of Q is quasi-isomorphic to the Leavitt path algebra L k ( Q op ) . (cid:3) Date : September 19, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Leavitt path algebra, projective Leavitt complex, compact generator,dg quasi-balanced module.
The Theorem is a combination of Theorem 3.7 and Theorem 5.2. We mentionthat for the construction of the projective Leavitt complex P • , we use the basis ofthe Leavitt path algebra L k ( Q op ) given by [1, Theorem 1].For the proof of (1), we construct subcomplexes of P • . For (2), we actuallyprove that the projective Leavitt complex has the structure of a differential graded A - L k ( Q op )-bimodule, which is right quasi-balanced. Here, we consider A as adifferential graded algebra concentrated in degree zero, and L k ( Q op ) is naturally Z -graded and viewed as a differential graded algebra with trivial differential.The paper is structured as follows. In section 2, we introduce the projectiveLeavitt complex P • of Q and prove that it is acyclic. In section 3, we recall somenotation and prove that the projective Leavitt complex P • is a compact generatorof the homotopy category of acyclic complexes of projective A -modules. In section4, we recall some facts of the Leavitt path algebra and endow the projective Leavittcomplex P • with a differential graded L k ( Q op )-module structure, which makes itbecome an A - L k ( Q op )-bimodule. In section 5, we prove that the opposite differ-ential graded endomorphism algebra of P • is quasi-isomorphic to the Leavitt pathalgebra L k ( Q op ).2. The projective Leavitt complex of a finite quiver without sources
In this section, we introduce the projective Leavitt complex of a finite quiverwithout sources, and prove that it is an acyclic complex of projective modules overthe corresponding finite dimensional algebra with radical square zero.2.1.
The projective Leavitt complex.
A quiver Q = ( Q , Q ; s, t ) consists of aset Q of vertices, a set Q of arrows and two maps s, t : Q −→ Q , which associateto each arrow α its starting vertex s ( α ) and its terminating vertex t ( α ), respectively.A quiver Q is finite if both the sets Q and Q are finite.A path in the quiver Q is a sequence p = α m · · · α α of arrows with t ( α k ) = s ( α k +1 ) for 1 ≤ k ≤ m −
1. The length of p , denoted by l ( p ), is m . The startingvertex of p , denoted by s ( p ), is s ( α ). The terminating vertex of p , denoted by t ( p ),is t ( α m ). We identify an arrow with a path of length one. For each vertex i ∈ Q ,we associate to it a trivial path e i of length zero. Set s ( e i ) = i = t ( e i ). Denote by Q m the set of all paths in Q of length m for each m ≥ Q is a sink if there is no arrow starting at it and a sourceif there is no arrow terminating at it. Recall that for a vertex i which is not a sink,we can choose an arrow β with s ( β ) = i , which is called the special arrow startingat vertex i ; see [1]. For a vertex i which is not a source, fix an arrow γ with t ( γ ) = i .We call the fixed arrow the associated arrow terminating at i . For an associatedarrow α , we set T ( α ) = { β ∈ Q | t ( β ) = t ( α ) , β = α } . (2.1) Definition 2.1.
For two paths p = α m · · · α α and q = β n · · · β β with m, n ≥ p, q ) an associated pair in Q if s ( p ) = s ( q ), and either α = β ,or α = β is not associated. In addition, we call ( p, e s ( p ) ) and ( e s ( p ) , p ) associatedpairs in Q for each path p in Q . (cid:3) For each vertex i ∈ Q and l ∈ Z , set Λ li = { ( p, q ) | ( p, q ) is an associated pair with l ( q ) − l ( p ) = l and t ( p ) = i } . (2.2) Lemma 2.2.
Let Q be a finite quiver without sources. The above set Λ li is notempty for each vertex i and each integer l .Proof. Recall that the opposite quiver Q op of the quiver Q has arrows with op-posite directions. For each vertex i ∈ Q , fix the special arrow of Q op starting HE PROJECTIVE LEAVITT COMPLEX 3 at i as the opposite arrow of the associated arrow of Q terminating at i . Ob-serve that for each vertex i and each integer l , Λ li is one-to-one corresponded to { ( q op , p op ) | ( q op , p op ) is an admissible pair in Q op with l ( p op ) − l ( q op ) = − l and s ( p op ) = i } . Here, refer to [12, Definition 2.1] for the definition of admissible pair.By [12, Lemma 2.2], the later set is not empty. The proof is completed. (cid:3)
Let k be a field and Q be a finite quiver. For each m ≥
0, denote by kQ m the k -vector space with basis Q m . The path algebra kQ of the quiver Q is defined as kQ = L m ≥ kQ m , whose multiplication is given by such that for two paths p and q in Q , if s ( p ) = t ( q ), then the product pq is their concatenation; otherwise, we setthe product pq to be zero. Here, we write the concatenation of paths from right toleft.We observe that for any path p and vertex i ∈ Q , pe i = δ i,s ( p ) p and e i p = δ i,t ( p ) p ,where δ denotes the Kronecker symbol. We have that P i ∈ Q e i is the unit of thepath algebra kQ . Denote by J the two-sided ideal of kQ generated by arrows.We consider the quotient algebra A = kQ/J ; it is a finite dimensional algebrawith radical square zero. Indeed, A = kQ ⊕ kQ as a k -vector space. The Jacobsonradical of A is rad A = kQ satisfying (rad A ) = 0. For each vertex i ∈ Q andeach arrow α ∈ Q , we abuse e i and α with their canonical images in the algebra A . Denote by P i = Ae i the indecomposable projective left A -module for i ∈ Q .We have the following observation. Lemma 2.3.
Let i, j be two vertices in Q , and f : P i −→ P j be a k -linear map.Then f is a left A -module morphism if and only if f ( e i ) = δ i,j λe j + P { β ∈ Q | s ( β )= j, t ( β )= i } µ ( β ) βf ( α ) = δ i,j λα with λ and µ ( β ) scalars for all α ∈ Q with s ( α ) = i . (cid:3) For a set X and an A -module M , the coproduct M ( X ) will be understood as L x ∈ X M ζ x such that each component M ζ x is M . For an element m ∈ M , we use mζ x to denote the corresponding element in the component M ζ x .For a path p = α m · · · α α in Q of length m ≥
2, we denote by b p = α m − · · · α and e p = α m · · · α the two truncations of p . For an arrow α , denote by b α = e s ( α ) and e α = e t ( α ) . Definition 2.4.
Let Q be a finite quiver without sources. The projective Leavittcomplex P • = ( P l , δ l ) l ∈ Z of Q is defined as follows:(1) the l -th component P l = L i ∈ Q P i ( Λ li ) .(2) the differential δ l : P l −→ P l +1 is given by δ l ( αζ ( p,q ) ) = 0 and δ l ( e i ζ ( p,q ) ) = ( βζ ( b p,q ) , if p = β b p ; P { β ∈ Q | t ( β )= i } βζ ( e s ( β ) ,qβ ) , if l ( p ) = 0,for any i ∈ Q , ( p, q ) ∈ Λ li and α ∈ Q with s ( α ) = i . (cid:3) Each component P l is a projective A -module. The differential δ l are A -modulemorphisms; compare Lemma 2.3. By the definition of the differential δ l , it is directto see that δ l +1 ◦ δ l = 0 for each l ∈ Z . In conclusion, P • is a complex of projective A -modules. HUANHUAN LI
The acyclicity of the projective Leavitt complex.
We will show that theprojective Leavitt complex is acyclic.In what follows,
V, V ′ are two k -vector spaces and f : V −→ V ′ is a k -linear map.Suppose that B and B ′ are k -bases of V and V ′ , respectively. We say that thetriple ( f, B, B ′ ) satisfies Condition (X) if f ( B ) ⊆ B ′ and the restriction of f on B is injective. In this case, we have Ker f = 0.We suppose further that there are disjoint unions B = B ∪ B ∪ B and B ′ = B ′ ∪ B ′ . We say that the triple ( f, B, B ′ ) satisfies Condition (W) if the followingstatements hold:(W1) f ( b ) = 0 for each b ∈ B ;(W2) f ( B ) ⊆ B ′ and ( f , B , B ′ ) satisfies Condition (X) where f is the restric-tion of f to the subspace spanned by B .(W3) For b ∈ B , f ( b ) = b + P c ∈ B ( b ) f ( c ) for some b ∈ B ′ and some finitesubset B ( b ) ⊆ B . Moreover, if b, b ′ ∈ B and b = b ′ , then b = b ′ .We have the following observation. The proof of it is similar as that of [12,Lemma 2.7]. We omit it here. Lemma 2.5.
Assume that ( f, B, B ′ ) satisfies Condition (W), then B is a k -basisof Ker f and f ( B ) ∪ { b | b ∈ B } is a k -basis of Im f . (cid:3) From now on, Q is a finite quiver without sources. We consider the differential δ l : P l −→ P l +1 in Definition 2.4. We have the following k -basis of P l : B l = { e i ζ ( p,q ) , αζ ( p,q ) | i ∈ Q , ( p, q ) ∈ Λ li and α ∈ Q with s ( α ) = i } . Denote by B l = { αζ ( p,q ) | i ∈ Q , ( p, q ) ∈ Λ li and α ∈ Q with s ( α ) = i } a subsetof B l . Set B l = { e i ζ ( e i ,q ) | i ∈ Q , ( e i , q ) ∈ Λ li } for l ≥
0. If l <
0, put B l = ∅ . Take B l = B l \ ( B l ∪ B l ). Then we havethe disjoint union B l = B l ∪ B l ∪ B l . Set B ′ l = { βζ ( e s ( q ) ,q ) | i ∈ Q , ( e s ( q ) , q ) ∈ Λ li such that q = e qβ and β is associated } for l ∈ Z . We mention that B ′ l = ∅ for l <
0. Take B ′ l = B l \ B ′ l for l ∈ Z . Then we have the disjoint union B l = B ′ l ∪ B ′ l for each l ∈ Z . Lemma 2.6.
For each l ∈ Z , the set B l is a k -basis of Ker δ l and the set B l +10 isa k -basis of Im δ l .Proof. For l <
0, we have B l = ∅ = B ′ l +10 . We observe that the triple ( δ l , B l , B l +1 )satisfies Condition (W). Indeed, δ l ( b ) = 0 for each b ∈ B l . The differential δ l induces an injective map δ l : B l −→ B ′ l +11 . Then (W1) and (W2) hold. To see(W3), for l ≥ i ∈ Q , e i ζ ( e i ,q ) ∈ B l , we have δ l ( e i ζ ( e i ,q ) ) = αζ ( e s ( α ) ,qα ) + X β ∈ T ( α ) δ l ( e s ( β ) ζ ( β,qβ ) ) , where α ∈ Q such that t ( α ) = i and α is associated. Here, recall T ( α ) from (2.1).Thus ( e i ζ ( e i ,q ) ) = αζ ( e s ( α ) ,qα ) and the finite subset B l ( e i ζ ( e i ,q ) ) = { e s ( β ) ζ ( β,qβ ) | β ∈ T ( α ) } .Recall that B l +10 = { αζ ( p,q ) | i ∈ Q , ( p, q ) ∈ Λ l +1 i and α ∈ Q with s ( α ) = i } . Now we prove that B l +10 = δ l ( B l ) ∪ { b | b ∈ B l } . We mention that theset { b | b ∈ B l } = { αζ ( e s ( α ) ,qα ) | q ∈ Q l and α is associated with t ( α ) = s ( q ) } .Clearly, δ l ( B l ) ∪ { b | b ∈ B l } ⊆ B l +10 . Conversely, for each i ∈ Q and ( p, q ) ∈ Λ l +1 i , we have αζ ( p,q ) = δ l ( e t ( α ) ζ ( αp,q ) ) ∈ δ l ( B l ) for α ∈ Q with s ( α ) = i but αζ ( p,q ) / ∈ { b | b ∈ B l } . Applying Lemma 2 . δ l , B l , B l +1 ), we aredone. (cid:3) HE PROJECTIVE LEAVITT COMPLEX 5
Proposition 2.7.
Let Q be a finite quiver without sources. Then the projectiveLeavitt complex P • of Q is an acyclic complex.Proof. The statement follows directly from Lemma 2 . (cid:3) Example 2.8.
Let Q be the following quiver with one vertex and one loop. · α e e The unique arrow α is associated. Set e = e and Λ l = Λ l for each l ∈ Z . It followsthat Λ l = { ( α − l , e ) } , if l < { ( e, e ) } , if l = 0; { ( e, α l ) } , if l > . The corresponding algebra A with radical square zero is isomorphic to k [ x ] / ( x ).Write A ( Λ l ) = Aζ l , where ζ l = ζ ( α − l ,e ) for l < ζ = ζ ( e,e ) and ζ l = ζ ( e,α l ) for l >
0. Then the projective Leavitt complex P • of Q is as follows · · · −→ Aζ l − δ l − −→ Aζ l δ l −→ Aζ l +1 −→ · · · , where the differential δ l is given by δ l ( eζ l ) = αζ l +1 and δ l ( αζ l ) = 0 for each l ∈ Z .Observe that A is a self-injective algebra. The projective Leavitt complex P • isisomorphic to the injective Leavitt complex I • as complexes; compare [12, Example2.11]. Example 2.9.
Let Q be the following quiver with one vertex and two loops. · α α e e We choose α to be the associated arrow terminating at the unique vertex. Set e = e and Λ l = Λ l for each l ∈ Z . A pair ( p, q ) of paths lies in Λ l if and only if l ( q ) − l ( p ) = l and p, q do not start with α simultaneously.We denote by A the corresponding radical square zero algebra. The projectiveLeavitt complex P • of Q is as follows: · · · δ − / / A ( Λ ) δ / / A ( Λ ) δ / / · · · We write the differential δ explicitely: δ ( α k ζ ( p,q ) ) = 0 and δ ( eζ ( p,q ) ) = ( α k ζ ( b p,q ) , if p = α k b p ; α ζ ( e,qα ) + α ζ ( e,qα ) , if p = e. for k = 1 , p, q ) ∈ Λ .3. The projective Leavitt complex as a compact generator
In this section, we prove that the projective Leavitt complex is a compact gen-erator of the homotopy category of acyclic complexes of projective A -modules.3.1. A cokernel complex and its decomposition.
Let Q be a finite quiverwithout sources and A be the corresponding algebra with radical square zero. Foreach i ∈ Q , l ∈ Z and n ≥
0, denote by Λ l,ni = { ( p, q ) | ( p, q ) ∈ Λ li with p ∈ Q n } . Refer to (2 .
2) for the definition of the set Λ li .Recall the projective Leavitt complex P • = ( P l , δ l ) l ∈ Z of Q . For each l ≥
0, wedenote by K l = L i ∈ Q P ( Λ l, i ) i ⊆ P l , where P i = Ae i . Observe that the differential δ l : P l −→ P l +1 satisfies δ l ( K l ) ⊆ K l +1 . Then we have a subcomplex K • of P • , HUANHUAN LI whose components K l = 0 for l <
0. Denote by φ • = ( φ l ) l ∈ Z : K • −→ P • be theinclusion chain map by setting φ l = 0 for l <
0. We set C • to be the cokernel of φ • .We now describe the cokernel C • = ( C l , e δ l ) of φ • . For each vertex i ∈ Q and l ∈ Z , set Λ l, + i = [ n> Λ l,ni . Observe that we have the disjoint union Λ li = Λ l, i ∪ Λ l, + i for l ≥ Λ li = Λ l, + i for l <
0. The component of C • is C l = L i ∈ Q P ( Λ l, + i ) i for each l ∈ Z . We have C l = P l for l < e δ l = δ l for l ≤ −
2. The differential e δ l : C l −→ C l +1 for l ≥ − e δ l ( αζ ( p,q ) ) = 0 and e δ l ( e i ζ ( p,q ) ) = ( , if l ( p ) = 1; δ l ( e i ζ ( p,q ) ) , otherwise , for any i ∈ Q , ( p, q ) ∈ Λ l, + i and α ∈ Q with s ( α ) = i . The restriction of e δ l to L i ∈ Q P ( Λ l, i ) i is zero for l ≥ −
1. We emphasize that the differentials e δ l for l ≥ − δ l in Definition 2.4.We observe the inclusions e δ l ( L i ∈ Q P ( Λ l,ni ) i ) ⊆ L i ∈ Q P ( Λ l +1 ,n − i ) i inside thecomplex C • for each l ∈ Z and n ≥
2. Then for each n ≥ C • n , · · · δ n − −→ M i ∈ Q P ( Λ n − , i ) i δ n − −→ M i ∈ Q P ( Λ n − , i ) i δ n − −→ M i ∈ Q P ( Λ n − , i ) i → C • satisfying C ln = 0 for l ≥ n . The differential δ l for l ≤ n − P • .We visually represent the projective Leavitt complex P • and the cokernel com-plex C • of φ • . For each l ∈ Z and n ≥
0, we denote L i ∈ Q P ( Λ l,ni ) i by P ( Λ l,n ) forsimplicity. .. . ... .. . ... .. . P ( Λ − , ) P ( Λ − , ) P ( Λ , ) P ( Λ , ) P ( Λ , ) ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ P ( Λ − , ) P ( Λ , ) P ( Λ , ) P ( Λ , ) P ( Λ , ) ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ ❅❅❅❅❅❘ P ( Λ , ) P ( Λ , ) P ( Λ , ) P ( Λ , ) · · · ✲ ✲ ✲ ✲ · · · − − · · · HE PROJECTIVE LEAVITT COMPLEX 7
Remark 3.1. (1) For each l ∈ Z , the l -th component of the projective Leavittcomplex P • is the coproduct of the objects in the l -th column of the abovediagram. The differentials of P • are coproducts of the maps in the diagram.(2) The horizontal line of the above diagram is the subcomplex K • , while theother part gives the cokernel C • of φ • : K • −→ P • . The diagonal lines (notincluding the intersection with the horizontal line) of the above diagramare the subcomplexes C • n of C • . For example, the first diagonal line on theleft of the above diagram (not including P ( Λ , ) ) is the subcomplex C • .We have the following observation immeadiately. Proposition 3.2.
The complex C • = L n ≥ C • n .Proof. Observe that for each n ≥
0, the l -th component of C • n is C ln = (L i ∈ Q P ( Λ l,n − li ) i , if l < n ;0 , otherwise . Then the l -th component of L n ≥ C • n is L n ≥ C ln = L i ∈ Q P ( Λ l, + i ) i = C l . Recallthe differential e δ l : C l −→ C l +1 of C • . The restriction of e δ l to L i ∈ Q P ( Λ l, i ) i is zeroand the restriction of e δ l to L i ∈ Q P ( Λ l,ni ) i for n > δ l . Thus e δ l : C l −→ C l +1 is thecoproduct of the differentials δ l : C ln −→ C l +1 n for n ≥
0. The proof is completed. (cid:3)
An explicit compact generator of the homotopy category.
We denoteby A -Mod the category of left A -modules. Denote by K ( A -Mod) the homotopycategory of A -Mod. We will always view a module as a stalk complex concentratedin degree zero.For X • = ( X n , d nX ) n ∈ Z a complex of A -modules, we denote by X • [1] the complexgiven by ( X • [1]) n = X n +1 and d nX [1] = − d n +1 X for n ∈ Z . For a chain map f • : X • −→ Y • , its mapping cone Con( f • ) is a complex such that Con( f • ) = X • [1] ⊕ Y • with the differential d n Con( f • ) = (cid:18) − d n +1 X f n +1 d nY (cid:19) . Each triangle in K ( A -Mod) isisomorphic to X • f • −−−−→ Y • −−−−→ Con( f • ) (cid:16) (cid:17) −−−−−→ X • [1]for some chain map f • .Denote by I i = D ( e i A A ) the injective left A -module for each vertex i ∈ Q , where( e i A ) A is the indecomposable projective right A -module and D = Hom k ( − , k )denotes the standard k -duality. Denote by { e ♯i } ∪ { α ♯ | α ∈ Q , t ( α ) = i } the basisof I i , which is dual to the basis { e i } ∪ { α | α ∈ Q , t ( α ) = i } of e i A .We denote by M • the following complex0 → M i ∈ Q I ( Λ , i ) i d −→ M i ∈ Q I ( Λ , i ) i −→ · · · −→ M i ∈ Q I ( Λ l, i ) i d l −→ M i ∈ Q I ( Λ l +1 , i ) i −→ · · · of A -modules satisfying M l = 0 for l <
0, where the differential d l for l ≥ d l ( e ♯i ζ ( e i ,q ) ) = 0 and d l ( α ♯ ζ ( e i ,q ) ) = e ♯s ( α ) ζ ( e s ( α ) ,qα ) for i ∈ Q , ( e i , q ) ∈ Λ l, i and α ∈ Q with t ( α ) = i . Consider the semisimple left A -module kQ = A/ rad A . Lemma 3.3.
The left A -module kQ = A/ rad A is quasi-isomorphic to the complex M • . In other words, M • is an injective resolution of the A -module kQ . HUANHUAN LI
Proof.
Define a left A -module map f : kQ −→ M such that f ( e i ) = e ♯i ζ ( e i ,e i ) for each i ∈ Q . Then we obtain a chain map f • = ( f l ) l ∈ Z : kQ −→ M • such that f l = 0 for l = 0. We observe the following k -basis of M l for l ≥ l = { e ♯i ζ ( e i ,q ) , α ♯ ζ ( e i ,q ) | i ∈ Q , ( e i , q ) ∈ Λ l, i and α ∈ Q with t ( α ) = i } . Set Γ l = { e ♯i ζ ( e i ,q ) | i ∈ Q , ( e i , q ) ∈ Λ l, i } , Γ l = Γ l \ Γ l , and Γ ′ l = Γ l . We have thedisjoint union Γ l = Γ l ∪ Γ l . The triple ( d l , Γ l , Γ l +1 ) satisfies Condition (W). ByLemma 2 . l is a k -basis of Ker d l and the set d l (Γ l ) is a k -basis of Im d l .For each l ≥ i ∈ Q and ( e i , q ) ∈ Λ l +1 , i , write q = e qα with α ∈ Q . Then wehave e ♯i ζ ( e i ,q ) = d l ( α ♯ ζ ( e t ( α ) , e q ) . Thus d l (Γ l ) = Γ l +10 . Hence Im d l = Ker d l +1 for each l ≥ d ∼ = kQ . The statement follows directly. (cid:3) For a triangulated category A , a thick subcategory of A is a triangulated sub-category of A that is closed under direct summands. Let S be a class of objectsin A . Denote by thick hSi the smallest thick subcategory of A containing S . If thetriangulated category A has arbitrary coproducts, we denote by Loc hSi the small-est triangulated subcategory of A which contains S and is closed under arbitrarycoproducts. By [5, Proposition 3.2], thick hSi ⊆ Loc hSi .For a triangulated category A with arbitrary coproducts, we say that an object M ∈ A is compact if the functor Hom A ( M, − ) commutes with arbitrary coproducts.Denote by A c the full subcategory consisting of compact objects; it is a thicksubcategory.A triangulated category A with arbitrary coproducts is compactly generated [9,13] if there exists a set S of compact objects such that any nonzero object X satisfies that Hom A ( S, X [ n ]) = 0 for some S ∈ S and n ∈ Z . This is equivalent tothe condition that A = Loc hSi , in which case A c =thick hSi ; see [13, Lemma 3.2].If the above set S consists of a single object S , we call S a compact generator ofthe triangulated category A .The following is [12, Lemma 3.9]. Lemma 3.4.
Suppose that A is a compactly generated triangulated category witha compact generator X . Let A ′ ⊆ A be a triangulated subcategory which is closedunder arbitrary coproducts. Suppose that there exists a triangle X −−−−→ Y −−−−→ Z −−−−→ X [1] such that Y ∈ A ′ and Z satisfies Hom A ( Z, A ′ ) = 0 for each object A ′ ∈ A ′ . Then Y is a compact generator of A ′ . (cid:3) Let A -Inj and A -Proj be the categories of injective and projective A -modules,respectively. Denote by K ( A -Inj) and K ( A -Proj) the homotopy categories of com-plexes of injective and projecitve A -modules, respectively. These homotopy cate-gories are triangulated subcategories of K ( A -Mod) which are closed under coprod-ucts. By [11, Proposition 2.3(1)] K ( A -Inj) is a compactly generated triangulatedcategory.Recall that the Nakayama functor ν = DA ⊗ A − : A -Proj −→ A -Inj is anequivalence, whose quasi-inverse ν − = Hom A ( D ( A A ) , − ). Thus we have a trian-gle equivalence K ( A -Inj) ∼ −→ K ( A -Proj). The category K ( A -Proj) is a compactlygenerated triangulated category; see [8, Theorem 2.4] and [14, Proposition 7.14]. Lemma 3.5.
The complex K • is a compact generator of K ( A - Proj) .Proof.
We first prove that K • ∼ = ( ν − ( M i ) , ν − ( d i )) i ∈ Z as complexes. Recall that ν − ( I i ) = Hom A ( D ( A A ) , I i ) ∼ = Ae i HE PROJECTIVE LEAVITT COMPLEX 9 for each i ∈ Q . Observe that all the sets Λ l, i are finite. Then for each l ≥ A -modules f l : M i ∈ Q P ( Λ l, i ) i ∼ −→ Hom A ( D ( A A ) , M i ∈ Q I ( Λ l, i ) i )such that f l ( e i ζ ( e i ,q ) )( e ♯j ) = δ ij e ♯j ζ ( e i ,q ) and f l ( e i ζ ( e i ,q ) )( β ♯ ) = δ i,t ( β ) β ♯ ζ ( e i ,q ) foreach i, j ∈ Q , ( e i , q ) ∈ Λ l, i and β ∈ Q . We have Hom A ( D ( A A ) , d l ) ◦ f l = f l +1 ◦ δ l by direct calculation for each l ≥ M • is an injective resolution of the left A -module kQ . It follows from [11, Proposition 2.3] that M • is a compact object in K ( A -Inj)and Loc hM • i = K ( A -Inj). Since K ( A -Inj) ∼ −→ K ( A -Proj) is a triangle equivalencewhich sends M • to K • , we have Loc hK • i = K ( A -Proj). (cid:3) Lemma 3.6.
Suppose that X • ∈ K ( A - Proj) is a bounded-above complex. Then wehave
Hom K ( A - Mod) (X • , Y • ) = 0 for any acyclic complex Y • of A -modules.Proof. Directly check that any chain map f • : X • −→ Y • is null-homotopic. (cid:3) Denote by K ac ( A -Proj) the full subcategory of K ( A -Mod) which is formed byacyclic complexes of projective A -modules. Applying [14, Propositions 7.14 and7.12] and the localization theorem in [10, 1.5], we have that the category is acompactly generated triangulated category with the triangle equivalence D sg ( A op ) op ∼ −→ K ac ( A -Proj) c . Here, for a category C , we denote by C op its opposite category; the category D sg ( A op ) is the singularity category of algebra A op in the sense of [4, 15]. Theorem 3.7.
Let Q be a finite quiver without sources. Then the projective Leavittcomplex P • of Q is a compact generator of the homotopy category K ac ( A - Proj) .Proof.
Recall from Proposition 2.7 that P • is an object of K ac ( A -Proj). The com-plex C • = Coker( φ • ), where φ • : K • −→ P • is the inclusion chain map. Then wehave the following exact sequence0 / / K • φ • / / P • / / C • / / , which splits in each component. This gives rise to a triangle K • φ • −−−−→ P • −−−−→ C • −−−−→ X [1] (3.1)in the category K ( A -Proj).By Proposition 3.2 and Lemma 3.6, the following equality holdsHom K ( A - Proj) ( C • , X • ) = Y n ≥ Hom K ( A - Proj) ( C • n , X • ) = 0for any X • ∈ K ac ( A -Proj). Recall from Lemma 3.5 that K • is a compact generatorof K ( A -Proj). By the triangle (3.1) and Lemma 3.4, the proof is completed. (cid:3) The projective Leavitt complex as a differential graded bimodule
In this section, we endow the projective Leavitt complex with a differentialgraded bimodule structure over the corresponding Leavitt path algebra.
A module structure of the Leavitt path algebra.
Let k be a field and Q be a finite quiver. We will endow the projective Leavitt complex of Q with aLeavitt path algebra module structure. Recall from [2, 3] the notion of the Leavittpath algebra. Definition 4.1.
The
Leavitt path algebra L k ( Q ) of Q is the k -algebra generatedby the set { e i | i ∈ Q } ∪ { α | α ∈ Q } ∪ { α ∗ | α ∈ Q } subject to the followingrelations:(0) e i e j = δ i,j e i for every i, j ∈ Q ;(1) e t ( α ) α = αe s ( α ) = α for all α ∈ Q ;(2) e s ( α ) α ∗ = α ∗ e t ( α ) = α ∗ for all α ∈ Q ;(3) αβ ∗ = δ α,β e t ( α ) for all α, β ∈ Q ;(4) P { α ∈ Q | s ( α )= i } α ∗ α = e i for i ∈ Q which is not a sink. (cid:3) Here, δ denotes the Kronecker symbol. The above relations (3) and (4) are called Cuntz-Krieger relations . The elements α ∗ for α ∈ Q are called ghost arrows .The Leavitt path algebra L k ( Q ) can be viewed as a quotient algebra of the pathalgebra as follows. Denote Q the double quiver obtained from Q by adding an arrow α ∗ in the opposite direction for each arrow α in Q . The Leavitt path algebra L k ( Q )is isomorphic to the quotient algebra of the path algebra kQ of the double quiver Q modulo the ideal generated by { αβ ∗ − δ α,β e t ( α ) , P { γ ∈ Q | s ( γ )= i } γ ∗ γ − e i | α, β ∈ Q , i ∈ Q which is not a sink } . If p = α m · · · α α is a path in Q of length m ≥
1, we define p ∗ = α ∗ α ∗ · · · α ∗ m .For convention, we set e ∗ i = e i for i ∈ Q . The Leavitt path algebra L k ( Q ) isspanned by the following set { p ∗ q | p, q are paths in Q with t ( p ) = t ( q ) } ; see [2,Lemma 1.5], [16, Corollary 3.2] or [6, Corollary 2.2]. By the relation (4), this set isnot k -linearly independent in general.For each vertex which is not a sink, we fix a special arrow starting at it. Thefollowing result is [1, Theorem 1]. Lemma 4.2.
The following elements form a k -basis of the Leavitt path algebra L k ( Q ) : (1) e i , i ∈ Q ; (2) p, p ∗ , where p is a nontrivial path in Q ; (3) p ∗ q with t ( p ) = t ( q ) , where p = α m · · · α and q = β n · · · β are nontrivialpaths of Q such that α m = β n , or α m = β n which is not special. (cid:3) From now on, Q is a finite quiver without sources. For notation, Q op is theopposite quiver of Q . For a path p in Q , denote by p op the corresponding path in Q op . The starting and terminating vertices of p op are t ( p ) and s ( p ), respectively.For convention, e op j = e j for each vertex j ∈ Q . The opposite quiver Q op has nosinks.For the opposite quiver Q op of Q , choose α op to be the special arrow of Q op starting at vertex i , where α is the associated arrow in Q terminating at i . ByLemma 4.2 there exists a k -basis of the Leavitt path algebra L k ( Q op ), denotedby Γ. Define a map χ : S l ∈ Z ,i ∈ Q Λ li −→ Γ such that χ ( p, q ) = ( p op ) ∗ q op . Here,( p op ) ∗ q op is the multiplication of ( p op ) ∗ and q op in L k ( Q op ). The map χ is abijection. We identify Γ with the set of associated pairs in Q . A nonzero element x in L k ( Q op ) can be written in the unique form x = m X i =1 λ i ( p op i ) ∗ q op i with λ i ∈ k nonzero scalars and ( p i , q i ) pairwise distinct associated pairs in Q . HE PROJECTIVE LEAVITT COMPLEX 11
In what follows, B = L k ( Q op ). We write ab for the multiplication of a and b in B for a, b ∈ B . Recall that the projective Leavitt complex P • = ( P l , δ l ) l ∈ Z and P l = L i ∈ Q P i ( Λ li ) .We define a right B -module action on P • . For each vertex j ∈ Q and eacharrow α ∈ Q , define right actions “ · ” on P l for any l ∈ Z as follows. For anyelement xζ ( p,q ) ∈ P i ζ ( p,q ) with i ∈ Q and ( p, q ) ∈ Λ li , we set xζ ( p,q ) · e j = δ j,t ( q ) xζ ( p,q ) ; (4.1) xζ ( p,q ) · α op = xζ ( e p,e t ( α ) ) − P β ∈ T ( α ) xζ ( e pβ,β ) , if l ( q ) = 0 , p = e pα, and α is associated; δ s ( α ) ,t ( q ) xζ ( p,αq ) , otherwise . (4.2) xζ ( p,q ) · ( α op ) ∗ = ( δ α,α xζ ( p, b q ) , if q = α b q ; δ s ( p ) ,t ( α ) xζ ( pα,e s ( α ) ) , if l ( q ) = 0 . (4.3)Here for the notation, a path p = α m · · · α α in Q of length m ≥ b p = α m − · · · α and e p = α m · · · α . For an arrow α , b α = e s ( α ) and e α = e t ( α ) . The set T ( α ) = { β ∈ Q | t ( β ) = t ( α ) , β = α } for an associated arrow α . We observe the following fact: ( xζ ( p,q ) · α op = 0 , If s ( α ) = t ( q ); xζ ( p,q ) · ( α op ) ∗ = 0 , If t ( α ) = t ( q ) . (4.4) Lemma 4.3.
The above actions make the projective Leavitt complex P • of Q aright B -module.Proof. We prove that the above right actions satisfy the defining relations of theLeavitt path algebra L k ( Q op ) of the opposite quiver Q op . We fix xζ ( p,q ) ∈ P i ζ ( p,q ) ⊆P l .For (0), we observe that xζ ( p,q ) · ( e j ◦ e j ′ ) = δ j,j ′ xζ ( p,q ) · e j . For (1), for each α ∈ Q we have xζ ( p,q ) · ( α op e t ( α ) ) = ( xζ ( p,q ) · α op ) · e t ( α ) = xζ ( p,q ) · α op We have xζ ( p,q ) · ( e s ( α ) α op ) = ( xζ ( p,q ) · e s ( α ) ) · α op = δ s ( α ) ,t ( q ) xζ ( p,q ) · α op = xζ ( p,q ) · α op , where the last equality uses (4.4). Similar arguments prove the relation (2).For (3), we have that for α, β ∈ Q xζ ( p,q ) · ( α op ( β op ) ∗ ) = ( xζ ( p,q ) · α op ) · ( β op ) ∗ = δ t ( α ) ,t ( β ) xζ ( e pβ,e s ( β ) ) − P γ ∈ T ( α ) δ γ,β xζ ( e pγ,e s ( γ ) ) , if l ( q ) = 0 , p = e pα and α is associated; δ s ( α ) ,t ( q ) δ α,β xζ ( p,q ) , otherwise;= δ s ( α ) ,t ( q ) δ α,β xζ ( p,q ) = xζ ( p,q ) · ( δ α,β e s ( α ) )Here, we use the fact that in the case that l ( q ) = 0, p = e pα and α is associated,if α = β , then s ( α ) = t ( q ) and γ = β for each γ ∈ T ( α ); and if α = β with t ( α ) = t ( β ), then there exists an arrow γ ∈ T ( α ) such that γ = β . For (4), for each j ∈ Q we have that: if α ∈ Q with t ( α ) = j is associated,then xζ ( p,q ) · (( α op ) ∗ α op ) = ( xζ ( p,q ) · ( α op ) ∗ ) · α op = δ α,α ( xζ ( p,q ) , if q = α b q ; δ j ′ ,s ( p ) ( xζ ( p,e s ( p ) ) − P β ∈ T ( α ) xζ ( pβ,β ) ) , if l ( q ) = 0 . If α ∈ Q with t ( α ) = j is not associated, then xζ ( p,q ) · (( α op ) ∗ α op ) = ( δ α,α xζ ( p,q ) , if q = α b q ; δ j,s ( p ) xζ ( pα,α ) , if l ( q ) = 0 . Thus, we have the following equality xζ ( p,q ) · ( X { α ∈ Q | t ( α )= j } ( α op ) ∗ α op )= (cid:26) δ j,t ( q ) xζ ( p,q ) , if q = α b q ; δ j,s ( p ) xζ ( p,e s ( p ) ) , if l ( q ) = 0 . = δ j,t ( q ) xζ ( p,q ) = xζ ( p,q ) · e j . (cid:3) The following observation gives an intuitive description of the B -module actionon P • . Lemma 4.4.
Let ( p, q ) be an associated pair in Q . (1) We have P i ∈ Q e i ζ ( e i ,e i ) · ( p op ) ∗ q op = e t ( p ) ζ ( p,q ) ;(2) For each arrow β ∈ Q , we have the following equality holds: βζ ( e s ( β ) ,e s ( β ) ) · ( p op ) ∗ q op = δ s ( β ) ,t ( p ) βζ ( p,q ) . Proof.
Since ( p, q ) is an associated pair in Q , we are in the second subcases in (4.3)and (4.2) for the right action of ( p op ) ∗ q op . Then the statements follow from directcalculation. (cid:3) A differential graded bimodule.
We recall some notion on differentialgraded modules; see [9]. Let A = L n ∈ Z A n be a Z -graded algebra. For a (left)graded A -module X = L n ∈ Z X n , elements x in X n are said to be homogeneous ofdegree n , denoted by | x | = n .A differential graded algebra (dg algebra for short) is a Z -graded algebra A witha differential d : A −→ A of degree one satisfying d ( ab ) = d ( a ) b + ( − | a | ad ( b ) forhomogenous elements a, b ∈ A .A (left) differential graded A -module (dg A -module for short) X is a graded A -module X = L n ∈ Z X n with a differential d X : X −→ X of degree one satisfying d X ( a · x ) = d ( a ) · x + ( − | a | a · d X ( x ) for homogenous elements a ∈ A and x ∈ X . Amorphism of dg A -modules is a morphism of A -modules which preserves degreesand commutes with differentials. A right differential graded A -module (right dg A -module for short) Y is a right graded A -module Y = L n ∈ Z Y n with a differential d Y : Y −→ Y of degree one satisfying d Y ( y · a ) = d Y ( y ) · a + ( − | y | y · d ( a ) forhomogenous elements a ∈ A and y ∈ Y . Here, central dots denotes the A -moduleaction.Let B be another dg algebra. Recall that a dg A - B -bimodule X is a left dg A -module as well as a right dg B -module satisfying ( a · x ) · b = a · ( x · b ) for a ∈ A , x ∈ X and b ∈ B . HE PROJECTIVE LEAVITT COMPLEX 13
Recall that Q is a finite quiver without sources. In what follows, we write B = L k ( Q op ), which is naturally Z -graded by the length of paths. We view B as adg algebra with trivial differential.Consider A = kQ/J as a dg algebra concentrated in degree zero. Recall theprojective Leavitt complex P • = L l ∈ Z P l . It is a left dg A -module. By Lemma4.3, P • is a right B -module. By (4.1), (4.2) and (4.3), we have that P • is a rightgraded B -module.There is a unique right B -module morphism φ : B −→ P • with φ (1) = X i ∈ Q e i ζ ( e i ,e i ) . Here, 1 is the unit of B . For each arrow β ∈ Q , there is a unique right B -modulemorphism φ β : B −→ P • with φ β (1) = βζ ( e s ( β ) ,e s ( β ) ) . By Lemma 4.4 we have φ (( p op ) ∗ q op ) = e t ( p ) ζ ( p,q ) and φ β (( p op ) ∗ q op ) = δ s ( β ) ,t ( p ) βζ ( p,q ) (4.5)for ( p op ) ∗ q op ∈ Γ. Here, we emphasize that Γ is the k -basis of B = L k ( Q op ). Then φ is injective and the restriction of φ β to e s ( β ) B is injective. Observe that both φ and φ β are graded B -module morphisms. Lemma 4.5.
For each i ∈ Q , l ∈ Z and ( p, q ) ∈ Λ li , we have ( δ l ◦ φ )(( p op ) ∗ q op ) = X { α ∈ Q | t ( α )= i } φ α ( α op ( p op ) ∗ q op ) . From this, we have that ( δ l ◦ φ )( b ) = P α ∈ Q φ α ( α op b ) for b ∈ B l .Proof. For each arrow α ∈ Q and ( p op ) ∗ q op ∈ Γ, we observe that α op ( p op ) ∗ q op = ( δ α,α ( b p op ) ∗ q op , if p = α b p ;( qα ) op , if l ( p ) = 0 , (4.6)which are combinations of basis elements of L k ( Q op ). Then we have that( δ l ◦ φ )(( p op ) ∗ q op ) = δ l ( e i ζ ( p,q ) )= ( α ζ ( b p,q ) , if p = α b p ; P { α ∈ Q | t ( α )= i } αζ ( e s ( α ) ,qα ) , if l ( p ) = 0 . = X { α ∈ Q | t ( α )= i } φ α ( α op ( p op ) ∗ q op ) . Here, the last equality uses (4.6). (cid:3)
It is evident that the projective Leavitt complex P • is a graded A - B -bimodule.The following result shows that P • is a dg A - B -bimodule. Recall from Definition2.4 for the differentials δ l of P • . Proposition 4.6.
For each l ∈ Z , let xζ ( p,q ) ∈ P i ζ ( p,q ) with i ∈ Q and ( p, q ) ∈ Λ li .Then for each vertex j ∈ Q and each arrow β ∈ Q we have (1) δ l ( xζ ( p,q ) · e i ) = δ l ( xζ ( p,q ) ) · e i ;(2) δ l +1 ( xζ ( p,q ) · β op ) = δ l ( xζ ( p,q ) ) · β op ;(3) δ l − ( xζ ( p,q ) · ( β op ) ∗ ) = δ l ( xζ ( p,q ) ) · ( β op ) ∗ . In other words, the right B -action makes P • a right dg B -module and thus a dg A - B -bimodule.Proof. Recall that δ l ( αζ ( p,q ) ) = 0 for α ∈ Q with s ( α ) = i . It follows that (1), (2)and (3) hold for x = α . It suffices to prove that (1), (2) and (3) hold for x = e i .We recall that ( p, q ) ∈ Λ li , and thus t ( p ) = i . For (1), we have that δ l ( e i ζ ( p,q ) · e j ) = δ l ( φ (( p op ) ∗ q op ) e j )= ( δ l ◦ φ )(( p op ) ∗ q op e j )= X { α ∈ Q | t ( α )= i } φ α ( α op ( p op ) ∗ q op e j )= X { α ∈ Q | t ( α )= i } φ α ( α op ( p op ) ∗ q op ) · e j = δ l ( e i ζ ( p,q ) ) · e j . Here, the second and the fourth equalities hold because φ and φ α are right B -modulemorphisms; the third and the last equalities use Lemma 4.5; Similar argumentsprove (2) and (3). (cid:3) The differential graded endomorphism algebra of the projectiveLeavitt complex
In this section, we prove that the opposite differential graded endomorphismalgebra of the projective Leavitt complex of a finite quiver without sources is quasi-isomorphic to the Leavitt path algebra of the opposite quiver. Here, the Leavittpath algebra is naturally Z -graded and viewed as a differential graded algebra withtrivial differential.5.1. The quasi-balanced dg bimodule.
We first recall some notion and factsabout quasi-balanced dg bimodules. Let A be a dg algebra and X, Y be (left) dg A -modules. We have a Z -graded vector space Hom A ( X, Y ) = L n ∈ Z Hom A ( X, Y ) n such that each component Hom A ( X, Y ) n consists of k -linear maps f : X −→ Y satisfying f ( X i ) ⊆ Y i + n for all i ∈ Z and f ( a · x ) = ( − n | a | a · f ( x ) for all homoge-nous elements a ∈ A . The differential on Hom A ( X, Y ) sends f ∈ Hom A ( X, Y ) n to d Y ◦ f − ( − n f ◦ d X ∈ Hom A ( X, Y ) n +1 . Furthermore, End A ( X ) := Hom A ( X, X )becomes a dg algebra with this differential and the usual composition as multipli-cation. The dg algebra End A ( X ) is usually called the dg endomorphism algebra ofthe dg module X .We denote by A opp the opposite dg algebra of a dg algebra A , that is, A opp = A as graded spaces with the same differential, and the multiplication “ ◦ ” on A opp isgiven such that a ◦ b = ( − | a || b | ba .Let B be another dg algebra. Recall that a right dg B -module is a left dg B opp -module. For a dg A - B -bimodule X , the canonical map A −→ End B opp ( X ) isa homomorphism of dg algebras, sending a to l a with l a ( x ) = a · x for a ∈ A and x ∈ X . Similarly, the canonical map B −→ End A ( X ) opp is a homomorphism of dgalgebras, sending b to r b with r b ( x ) = ( − | b || x | x · b for homogenous elements b ∈ B and x ∈ X .A dg A - B -bimodule X is called right quasi-balanced provided that the canonicalhomomorphism B −→ End A ( X ) opp of dg algebras is a quasi-isomorphism; see [7,2.2].We denote by K ( A ) the homotopy category of left dg A -modules and by D ( A )the derived category of left dg A -modules; they are both triangulated categorieswith arbitrary coproducts. For a dg A - B -bimodule X and a left dg A -module Y ,Hom A ( X, Y ) has a natural structure of left dg B -module.The following lemma is [7, Proposition 2.2]; compare [9, 4.3] and [11, AppendixA]. Lemma 5.1.
Let X be a dg A - B -bimodule which is right quasi-balanced. Recall that Loc h X i ⊆ K ( A ) is the smallest triangulated subcategory of K ( A ) which contains HE PROJECTIVE LEAVITT COMPLEX 15 X and is closed under arbitrary coproducts. Assume that X is a compact object in Loc h X i . Then we have a triangle equivalence Hom A ( X, − ) : Loc h X i ∼ −→ D ( B ) . (cid:3) In what follows, Q is a finite quiver without sources and A = kQ/J is thecorresponding algebra with radical square zero. Consider A as a dg algebra con-centrated in degree zero. Recall that the Leavitt path algebra B = L k ( Q op ) isnaturally Z -graded, and that it is viewed as a dg algebra with trivial differential.Recall from Proposition 4.6 that the projective Leavitt complex P • is a dg A - B -bimodule. There is a connection between the projective Leavitt complex and theLeavitt path algebra, which is established by the following statement. Theorem 5.2.
Let Q be a finite quiver without sources. Then the dg A - B -bimodule P • is right quasi-balanded.In particular, the opposite dg endomorphism algebra of the projective Leavittcomplex of Q is quasi-isomorphic to the Leavitt path algebra L k ( Q op ) of Q op . Here, Q op is the opposite quiver of Q ; L k ( Q op ) is naturally Z -graded and viewed as a dgalgebra with trivial differential. We will prove Theorem 5.2 in subsection 5.2. The following equivalence wasgiven by [7, Theorem 6.2].
Corollary 5.3.
Let Q be a finite quiver without sources. Then there is a triangleequivalence Hom A ( P • , − ) : K ac ( A - Proj) ∼ −→ D ( B ) such that Hom A ( P • , P • ) ∼ = B in D ( B ) .Proof. Recall from Theorem 3.7 that K ac ( A -Proj) = Loc hP • i . Then the triangleequivalence follows from Theorem 5.2 and Lemma 5.1. The canonical map B −→ End A ( P • ) opp is a quasi-isomorphism, which identifies Hom A ( P • , P • ) with B in D ( B ). (cid:3) The proof of Theroem 5.2.
In this subsection, we follow the notation insubsection 5.1.
Lemma 5.4. ([16, Theorem 4.8])
Let A be a Z -graded algebra and η : L k ( Q ) −→ A be a graded algebra homomorphism such that η ( e i ) = 0 for all i ∈ Q . Then η isinjective. (cid:3) Let Z n and C n denote the n -th cocycle and coboundary of the dg algebraEnd A ( P • ) opp . We have the following observation: Lemma 5.5.
Any element f : P • −→ P • in C n satisfies f ( P l ) ⊆ Ker δ n + l for eachinteger l and n .Proof. For any f ∈ C n there exists h = ( h l ) l ∈ Z ∈ End A ( P • ) opp such that f l = δ n + l − ◦ h l − ( − n − h l +1 ◦ δ l for each l ∈ Z . By Proposition 2.7 we have Im δ n + l − ⊆ Ker δ n + l and Im δ l ⊆ Ker δ l +1 . Then it suffices to prove h (Ker δ l +1 ) ⊆ Ker δ n + l . Re-call from Lemma 2.6 that { αζ ( p,q ) | i ∈ Q , ( p, q ) ∈ Λ l +1 i and α ∈ Q with s ( α ) = i } is a k -basis of Ker δ l +1 . By Lemma 2.3, we are done. (cid:3) Recall that the projective Leavitt complex P • is a dg A - B -bimodule; see Propo-sition 4.6. Let ρ : B −→ End A ( P • ) opp be the canonical map which is induced by theright B -action. We have that ρ ( L k ( Q op ) n ) ⊆ Z n for each n ∈ Z , since B is a dgalgebra with trivial differential. Taking cohomologies, the following graded algebrahomomorphism is obtained H ( ρ ) : B −→ H (End A ( P • ) opp ) . (5.1) Lemma 5.6.
The graded algebra homomorphism H ( ρ ) is an embedding.Proof. By Lemma 5.4, it suffices to prove that H ( ρ )( e i ) = 0 for all i ∈ Q . For eachvertex i ∈ Q , H ( ρ )( e i )( e i ζ ( e i ,e i ) ) = e i ζ ( e i ,e i ) . By Lemma 2.6 we have e i ζ ( e i ,e i ) / ∈ Ker δ . By Lemma 5.5, H ( ρ )( e i ) / ∈ C . This implies H ( ρ )( e i ) = 0. (cid:3) It remains to prove that the graded algebra homomorphism H ( ρ ) is surjective.For each element y ∈ Z n , we will find an element x ∈ B n such that y − ρ ( x ) ∈ C n .In what follows, we fix y ∈ Z n for n ∈ Z . Then y ∈ Z n implies δ • ◦ y − ( − n y ◦ δ • = 0. Recall that P l = L i ∈ Q P ( Λ li ) i for each l ∈ Z . The set { e i ζ ( p,q ) , αζ ( p,q ) | i ∈ Q , ( p, q ) ∈ Λ li and α ∈ Q with s ( α ) = i } is a k -basis of P l . For each i ∈ Q , l ∈ Z , and ( p, q ) ∈ Λ li , we have ( ( δ n + l ◦ y )( e i ζ ( p,q ) ) = ( − n ( y ◦ δ l )( e i ζ ( p,q ) )( δ n + l ◦ y )( αζ ( p,q ) ) = 0 , (5.2)where α ∈ Q with s ( α ) = i .Observe that y is an A -module morphism. By Lemma 2.3, we may assume that ( y ( e i ζ ( p,q ) ) = φ ( y ( p,q ) ) + P { γ ∈ Q | t ( γ )= i } φ γ ( µ γ ( p,q ) ) y ( αζ ( p,q ) ) = φ α ( y ( p,q ) ) , (5.3)where y ( p,q ) ∈ e i L k ( Q op ) n + l and µ γ ( p,q ) ∈ e s ( γ ) L k ( Q op ) n + l .By (5.3) and Lemma 4.5, we have that( δ n + l ◦ y )( e i ζ ( p,q ) ) = δ n + l ( φ ( y ( p,q ) )) = X { γ ∈ Q | t ( γ )= i } φ γ ( γ op y ( p,q ) )and that ( y ◦ δ l )( e i ζ ( p,q ) ) = y ( X { γ ∈ Q | t ( γ )= i } φ γ ( γ op ( p op ) ∗ q op )) . By (5 . X { γ ∈ Q | t ( γ )= i } φ γ ( γ op y ( p,q ) ) = ( − n y ( X { γ ∈ Q | t ( γ )= i } φ γ ( γ op ( p op ) ∗ q op )) . (5.4)We recall that for each arrow β ∈ Q , the restriction of φ β to e s ( β ) B is injective.If l ( p ) = 0, then by (5.4) and (5.3) for any γ ∈ Q with t ( γ ) = i we have y ( e s ( γ ) ,qγ ) = ( − n γ op y ( p,q ) . (5.5)If l ( p ) >
0, write p = a b p with a ∈ Q and t ( a ) = i . By (5.4) and (5.3) we have a op y ( p,q ) = ( − n y ( b p,q ) and γ op y ( p,q ) = 0 for γ ∈ Q with t ( γ ) = i and γ = a . Thefollowing equality holds: y ( p,q ) = X { γ ∈ Q | t ( γ )= i } ( γ op ) ∗ γ op y ( p,q ) = ( − n ( a op ) ∗ y ( b p,q ) . (5.6) Lemma 5.7.
Keep the notation as above. Take x = P j ∈ Q y ( e j ,e j ) ∈ L k ( Q op ) n .Then y ( p,q ) = ( − nl ( p op ) ∗ q op x in L k ( Q op ) for each ( p, q ) ∈ Λ li .Proof. Clearly, we have y ( e i ,e i ) = e i x in L k ( Q op ). For l ( p ) = 0 and l ( q ) > q = e qγ with γ ∈ Q . We use (5.5) to obtain y ( e s ( q ) ,q ) = ( − nl q op x in L k ( Q op ) by induction on l ( q ). For l ( p ) >
0, write p = β m · · · β with all β k arrows in Q . By (5 .
6) we have y ( p,q ) = ( − n ( β op m ) ∗ y ( b p,q ) . We obtain y ( p,q ) =( − nm ( β op m ) ∗ · · · ( β op1 ) ∗ y ( e s ( q ) ,q ) by iterating (5.6). Then we are done by y ( e s ( q ) ,q ) =( − n ( m + l ) q op x in L k ( Q op ). (cid:3) HE PROJECTIVE LEAVITT COMPLEX 17
We will construct a map h : P • −→ P • of degree n −
1, which will be used toprove y − ρ ( x ) ∈ C n . To define h , we first assign to each pair ( p, q ) ∈ Λ li an element θ ( p,q ) in L k ( Q op ).For each i ∈ Q , define θ ( e i ,e i ) = P { γ ∈ Q | s ( γ )= i } µ γ ( γ,e s ( γ ) ) ∈ e i L k ( Q op ) n − .Here, refer to (5.3) for the element µ γ ( γ,e s ( γ ) ) . We define θ ( e s ( q ) ,q ) inductively by θ ( e s ( q ) ,q ) = ( − n − ( γ op θ ( e s ( e q ) , e q ) − µ γ ( e s ( e q ) , e q ) ) ∈ e s ( q ) L k ( Q op ) n + l − , (5.7)where q = e qγ with l ( q ) = l and γ ∈ Q . Let ( p, q ) ∈ Λ li with l ( p ) >
0. We define θ ( p,q ) by induction on the length of p as follows: θ ( p,q ) = ( − n − ( β op ) ∗ θ ( b p,q ) + X { γ ∈ Q | t ( γ )= i } ( γ op ) ∗ µ γ ( p,q ) ∈ e i L k ( Q op ) n + l − , (5.8)where p = β b p with β ∈ Q is of length l ( q ) − l .We define a k -linear map h : P • −→ P • such that h ( e i ζ ( p,q ) ) = φ ( θ ( p,q ) ) and h ( αζ ( p,q ) ) = φ α ( θ ( p,q ) )for each i ∈ Q , l ∈ Z , ( p, q ) ∈ Λ li , and α ∈ Q with s ( α ) = i . Lemma 5.8.
Let x be the element in Lemma 5.7 and h be the above map. Foreach i ∈ Q , l ∈ Z , ( p, q ) ∈ Λ li , we have ( ( y − ρ ( x ))( e i ζ ( p,q ) ) = ( δ n + l − ◦ h − ( − n − h ◦ δ l )( e i ζ ( p,q ) )( y − ρ ( x ))( αζ ( p,q ) ) = ( δ n + l − ◦ h − ( − n − h ◦ δ l )( αζ ( p,q ) ) = 0 , where α ∈ Q with s ( α ) = i .Proof. Recall from (4.5) the right B -module morphisms φ and φ β for β ∈ Q . By(5.3) and Lemma 5.7, we have ρ ( x )( e i ζ ( p,q ) ) = ( − nl φ ( p op ∗ q op ) · x = φ ( y ( p,q ) ) ρ ( x )( αζ ( p,q ) ) = ( − nl φ α ( p op ∗ q op ) · x = φ α ( y ( p,q ) )( y − ρ ( x ))( e i ζ ( p,q ) ) = P { γ ∈ Q | t ( γ )= i } φ γ ( µ γ ( p,q ) );Recall that δ l ◦ φ β = 0 for each arrow β ∈ Q . It remains to prove ( δ n + l − ◦ h − ( − n − h ◦ δ l )( e i ζ ( p,q ) ) = P { γ ∈ Q | t ( γ )= i } φ γ ( µ γ ( p,q ) ).By the definition of δ l , we have( h ◦ δ l )( e i ζ ( p,q ) ) = ( φ β ( θ ( b p,q ) ) , if p = β b p ; P { γ ∈ Q | t ( γ )= i } φ γ ( θ ( e s ( γ ) ,qγ ) ) , if l ( p ) = 0 . By Lemma 4.5, we have ( δ n + l − ◦ h )( e i ζ ( p,q ) ) = P { γ ∈ Q | t ( γ )= i } φ γ ( γ op θ ( p,q ) ). Thenthe following equalities hold:( δ n + l − ◦ h )( e i ζ ( p,q ) ) − ( − n − ( h ◦ δ l )( e i ζ ( p,q ) )= (P { γ ∈ Q | t ( γ )= i } φ γ ( γ op θ ( p,q ) ) − ( − n − φ β ( θ ( b p,q ) ) , if p = β b p ; P { γ ∈ Q | t ( γ )= i } φ γ ( γ op θ ( p,q ) − ( − n − θ ( e s ( γ ) ,qγ ) ) , if l ( p ) = 0= X { γ ∈ Q | t ( γ )= i } φ γ ( µ γ ( p,q ) ) . Here, the last equality uses (5 .
7) and (5 . (cid:3) Proof of Theorem 5.2.
It suffices to prove that H ( ρ ) in (5.1) is an isomorphism.By Lemma 5.6, it remains to prove that H n ( ρ ) is surjective for any n ∈ Z . For anyelement y = y + C n with y ∈ Z n , take x = P j ∈ Q y ( e j ,e j ) ∈ B n = L k ( Q op ) n . By Lemma 5.8, we have y − ρ ( x ) ∈ C n . Then it follows that y = ρ ( x ) in H n (End A ( P • ) opp ). (cid:3) Acknowledgements
The author thanks Professor Xiao-Wu Chen for inspiring discussions. The authorthanks Australian Research Council grant DP160101481.
References [1] A. Alahmadi, H. Alsulami, S.K. Jain and E. Zelmanov, Leavitt path algebras of finiteGelfand-Kirillov dimension, J. Algebra Appl. (6), 6pp. (2012)[2] G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra (2)(2005), 319-334.[3] P. Ara, M.A. Moreno and E. Pardo, Nonstable K-theory for graph algebras, Algebr. Rep-resent. Theory (2) (2007), 157-178.[4] R.O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorensteinrings, unpublished manuscrip, 1987, available at: http://hdl. handle. net/1807/16682.[5] M. B¨okstedt and A. Neeman, Homotopy limits in triangulated categories, CompositioMath. (1993), 209-234.[6] X.W. Chen, Irreducible representations of Leavitt path algebras, Forum Math. (1)(2015), 549-574.[7] X.W. Chen and Dong Yang, Homotopy categories, Leavitt path algebras and Gorensteinprojective modules, Inter. Math. Res. Notices, (2015), 2597-2633.[8] P. Jørgensen, The homotopy category of complexes of projective modules, Adv. Math. (1) (2005), 223-232.[9] B. Keller, Deriving DG categories, Ann. Sci. ´Ecole Norm. Sup. (4) (1) (1994), 63-102.[10] B. Keller, On the construction of triangle equivalences, In: Derived Equivalences for GroupRings, Lecture Notes Math. , Springer-Verlag, Berlin, 1998.[11] H. Krause, The stable derived category of a noetherian scheme, Compositio Math. (2005), 1128-1162.[12] H.H. Li, The injective Leavitt complex, arXiv:1512.04178v1.[13] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown rep-resentability, J. Amer. Math. Soc. (1996), 205-236.[14] A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent.math. (2) (2008), 255-308.[15] D. Orlov, Triangulated categories of sigularities and D-branes in Landau-Ginzburg models,Trudy Steklov Math. Institute (2004), 240-262.[16] M. Tomforde, Uniqueness theorems and ideal structure for Leavitt path algebras, J. Algebra318