On the First Hochschild Cohomology of Admissible Algebras?

Fang LI Dezhan TAN

1 Introduction

The Hochschild cohomology of algebras is invariant under Morita equivalence.Hence it is enough to consider basic connected algebras when the algebras are Artinian.Let Γ=(V,E)be a finite connected quiver,whereV(resp.E)is the set of vertices(resp.arrows)in Γ.Letkbe an arbitrary field andkΓ be the corresponding path algebra.Denote byRthe two-sided ideal ofkΓ generated byE.Recall that an idealIis called admissible if there existsm≥2 such thatRm?I?R2(see[2]).According to the Gabriel theorem,a finite dimensional basick-algebra over an algebraically closed fieldkis in the form ofkΓ/Ifor a finite quiver Γ and an admissible ideaI.

An Artinian algebra is called a monomial algebra(see[3])if it is isomorphic to a quotientkΓ/Iof a path algebrakΓ for a finite quiver Γ and an ideaIofkΓ generated by some paths in Γ.In particular,denote byknΓ the ideal ofkΓ generated by all paths of lengthn.Then the monomial algebrakΓ/knΓ is called then-truncated quiver algebra.

The study of Hochschild cohomology of quiver related algebras started with the paper of Happel in 1989(see[11]),who gave the dimensions of Hochschild cohomology of arbitrary orders of path algebras for acyclic quivers.Afterwards,there have been extensive studies on the Hochschild cohomology of quiver-related algebras such as truncated quiver algebras,monomial algebras,schurian algebras and 2-nilpotent algebras(see[1,4,6–7,12–15,17–19]).In[11],a minimal projective resolution of a finite-dimensional algebraAover its enveloping algebra is described in terms of the combinatorics when the fieldkis an algebraically closed field.In these papers listed above,the authors used this kind of projective resolution or its improving version to compute the Hochschild cohomology.

In[10],the authors applied an explicit and combinatorial method to studyHH1(kΓ).In this paper,we improve the method in[10]to the case of algebras with relations in order to study theHH1(kΓ/I),wherekΓ/Iis an admissible algebra.This way does not depend on projective resolution and the requirement ofkbeing an algebraically closed field.Using this method,we can obtain some structural results which would not arise by the classical method in the above listed papers.

IfI?R2holds for a two-sided idealI,we callkΓ/Ian admissible algebra(see Definition 2.1).So finite-dimensional basic algebras are always admissible algebras.We will give Proposition 2.1,which shows that admissible algebras,including basic algebras,possess the similar characterization of monomial algebras and truncated quiver algebras,although they are not graded.From this point of view,the admissible algebra is motivated to unify and generalize the basic algebra and the monomial algebra.

In the following,we always assume thatkΓ/Iis an admissible algebra.This paper includes three sections except for the introduction.In Section 2,we introduce the basic definitions which are used in this paper.In particular,we define the notion of an acyclic admissible algebra,which can be thought as a generalization of the notion of an acyclic quiver.A sufficient and necessary condition is obtained for a linear operator fromkΓ tokΓ/Ito be a differential operator.Next,we give a standard basis of Dif f(kΓ,kΓ/I).

In Section 3,we investigateH1(kΓ,kΓ/I).In(3.3),a dimension-formula ofH1(kΓ,kΓ/I)is given for a finite-dimensional admissible algebra.Moreover,in Theorem 3.1,we construct a basis ofH1(kΓ,kΓ/I)when Γ is planar andkΓ/Iis an acyclic admissible algebra.

In Section 4,we characterizeHH1(kΓ/I).In(4.2),we give the dimension-formula ofHH1(kΓ/I)for any finite-dimensional admissible algebraskΓ/I.Moreover,we apply this method to complete monomial algebras and truncated quiver algebras.In Theorems 4.1–4.2,we constructk-linear bases of their first cohomology groups under certain conditions.The Hochschild cohomology of monomial algebras and truncated quiver algebras has been studied in[7,12,16,18–19].Our results in Section 4 can be seen as the generalization of those corresponding conclusions in the listed references above.In the same section,two examples of admissible algebras are given which are not monomial algebras.Their first Hochschild cohomology is characterized using our theory.

2 The k-Linear Basis of Dif f(kΓ,kΓ/I)

We always assume Γ=(V,E)to be a finite connected quiver,whereV(resp.E)is the set of vertices(resp.arrows)in Γ.For a pathp,denote its starting vertex byt(p),called the tail ofp,and the ending point byh(p),called the head ofp.For two pathspandq,ift(p)=t(q)andh(p)=h(q),we saypandqare parallel,denoted bypq.Denote byP=PΓthe set of paths in a quiver Γ including its vertices;denote byPAthe set of its acyclic paths.Trivially,Γ is acyclic if and only ifPΓV=PA.Throughout this paper,we always assume quivers to be finite and connected.

Definition 2.1SupposeΓ =(V,E)is a quiver and I is a two-sided ideal of kΓ.We call the quotient algebra kΓ/I an admissible algebra if IR2,where R denotes the two-sided ideal of kΓgenerated by E.

Proposition 2.1Suppose kΓ/I is an admissible algebra,and then there exists a subsetof P such that V∪E?and Q={|x∈forms a basis of kΓ/I for=x+I.

ProofLetXbe ak-linear basis ofI.Denote byP≥2the set of all paths of length≥2.De fine

Tbecomes a partial set due to the order of inclusion between subsets ofkΓ.It is easy to see thatT?andTsatis fies the upper-bound condition of chains.So by the famous Zorn’s lemma,Thas a maximal element,denoted byZ.

We claim thatZis linearly equivalent toP≥2.Otherwise,there existsp∈P≥2such thatpcan not be linearly expressed byZ,and thenZ∪{p}is linearly independent inkΓ,which contradicts the maximal property ofZ.

SinceZis linearly equivalent toP≥2,it follows thatV∪E∪Zis linearly equivalent toP=V∪E∪P≥2.By the definition ofT,Z?X∪P≥2.Iis generated byX.HenceV∪E∪(ZX)forms a basis of the complement space ofIinkΓ.It means thatQ=V∪E∪(ZX)}forms a basis ofkΓ/I.It is clear thatV∪E∪(ZX)?Pand it is thewe want.

WhenI?R2is finite dimensional,we have an explicit way to determine theConcretely,suppose that{x1,x2,···,xm}is a basis ofI.Then there exists a finite subset{p1,p2,···,pn}ofPsuch thatxican be expressed by the linear combinations ofpj.Supposexi=fori=1,2,···,m,and then we obtain anm×nmatrixA=(aij).We can transform the matrixAinto a row-ladder matrixB=(bij)through only row transformations.Suppose thatbi,c(i)is the first nonzero number of thei-th row ofB.SinceBis a row-ladder matrix,we havecickforik.Then{x1,x2,···,xm}∪{pl|lc1,c2,···,cm}is linearly equivalent to{p1,p2,···,pn}.Hence(P{p1,p2,···,pn})∪{pl|lc1,c2,···,cm}is a basis of the complement space ofIinkΓ.Then the residue classes inkΓ/Iof all elements in this basis form a basis ofkΓ/I.

On the other hand,in some special cases,e.g.,whenkΓ/Iis a monomial algebra,even ifIis not finite dimensional,the choice ofis also given in the same way.IfkΓ/Iis a monomial algebra andIis generated by a set of paths of length≥2,the set of paths that does not belong toIis just therequired.

Definition 2.2Let A be a k-algebra and M an A-bimodule.A differential operator(or say,derivation)from A into M is a k-linear map D:A→M such that

In particular,when M=A,this coincides with the differential operator of algebras.

Lemma 2.1Suppose that D is a differential operator from kΓinto kΓ/I.Then D is determined by its action on the set V of vertices ofΓand the set E of arrows ofΓ.

Lemma 2.2LetΓbe a quiver.Denote by kV(resp.kE)the linear space spanned by the set V of the vertices ofΓ (resp.the set E of the arrows ofΓ).Assume that we have a pair of linear maps D0:kV→kΓ/I and D1:kE→kΓ/I satisfying that

Then,the pair of linear maps(D0,D1)can be uniquely extended to a differential operator D:kΓ →kΓ/I satisfying that

for any path p=p1p2···pl,pi∈E,1≤i≤l,l≥2.

ProofOne only need to prove thatDis indeed a differential operator.For this,we need to check(2.1)in the next four cases:

(a)x,y∈V;(b)x∈V,y∈PV;(c)x∈PV,y∈V;(d)x,y∈PV.

However,the checking process is routine,so we omit it here.

In the sequel,we always supposekΓ/Ito be an admissible algebra for the given idealIand the notations in Definition 2.1 are used.From Definition 2.1 and Proposition 2.1,there exists a basis ofkΓ/Iwhich consists of residue classes of some paths including that ofVandE.Denote the fixed basis ofkΓ/IbyQ.Suppose thatD:kΓ →kΓ/Iis a linear operator,then for anyp∈P,D(p)is a unique combination of the basisQofkΓ/I.Write this linear combination by

where all∈k.We will use this notation throughout this paper.As a convention,for the empty set?,we say

Lemma 2.3Suppose that q1,q2∈P,q1,q2/∈I andin kΓ/I,then t(q1)=t(q2),h(q1)=h(q2),i.e.,q1q2.

ProofIft(q1)t(q2),thena contradiction,sot(q1)=t(q2).Similarly,h(q1)=h(q2).

According to the lemma above,for∈Q,we can de fine:=t(q)(resp.:=h(q))for any pathq∈PsatisfyinginkΓ/I.For a paths∈Pandp∈Q,ift(s)=andh(s)=we saysandare parallel,denoted bys

Denote

Moreover,(resp.)denotes the subspace ofkΓ/Igenerated by(resp.).Clearly,ask-linear spaces,

Definition 2.3Using the above notations,an admissible algebra kΓ/I is called acyclic if

It is easy to see from this definition that

(i)the fact whether the givenkΓ/Iis acyclic is independent on the choice of

(ii)if the quiver Γ is acyclic,thenkΓ/Iis acyclic;the converse is not true in general;

(iii)ifkΓ/Iis acyclic,then it is finite dimensional;the converse is not true,e.g.,kΓ/knΓ if Γ is a loop forn≥2.

Proposition 2.2Let D:kΓ→kΓ/I be a k-linear operator.

(i)If D is a differential operator,then

(a)for v∈V,

(b)for p∈E,

where the coefficients are subject to the following condition:For any path∈Q such that

(ii)Conversely,assume that the linear map D from kV⊕kE to kΓ/I satisfies(2.8)–(2.10),then D can be uniquely extended linearly to a differential operator as(2.6).

Proof(i)For a givenv∈V,sincevv=v,we have

So by the direct computation,we can get

Moreover,

So we havevD(v)v=0.That means=0.So we get(2.8).

Also,for a givenp∈E,we have

Sincet(p),h(p)∈V,by(2.8),we can easily get(2.9).

Letx,y∈V,andxy.By(2.8),

ButD(xy)=0 sincexy=0.So

For a path∈Qsuch thatsubstitutingxandyrespectively withandhwe get(2.10).

(ii)We only need to verify that the conditions of Lemma 2.2 are satisfied.Because the process is straightforward,we leave it to the readers.

Next,we apply Proposition 2.2 to display a standard basis of differential operators fromkΓ tokΓ/I,for any admissible algebrakΓ/I.

Proposition 2.3(Dif f erential operatorFor a quiverΓ=(V,E),let r∈E and s∈P with rs.Define the k-linear operator:kV⊕kE→kΓ/I satisfying

Then,the conditions of Lemma2.2are satisfied and thus,can be uniquely extended to a differential operator from kΓto kΓ/I,denoted still byfor convenience.

Proof(2.2)–(2.5)can be checked easily by the definition of

For a givens∈P,we have the corresponding inner differential operator:

Theorem 2.1LetΓ =(V,E)be a quiver and I be an ideal such that kΓ/I is an admissible algebra.Then the set

is a basis of the k-linear space of differential operators from kΓto kΓ/I,where

ProofWe only need to verify that the operators in B are linearly independent and any differential operators can be generatedk-linearly by

Step 1is linearly independentSuppose that there are∈ksuch that

Then for any givenby the definitions ofandwe have

In the last formula above,always holds.Thus,their coefficients are all zero.In particular,=0 for any∈Qwith

Thus,from(2.15),we get that

Further,for any givenr0∈E,∈Qwithr0,we have

It follows that=0 for anyr0∈E,∈Qwithr

Hence,isk-linearly independent.

Step 2is the set of k-linear generatorsLetD:kΓ →kΓ/Ibe any differential operator.Then forv∈Vandp∈E,by(2.8)–(2.10),we have

We claim thatDagrees with the differential operatordefined by the linear combination

whereandcome from(2.16)–(2.17).Any path inPis either a vertex or a product of arrows.Thus by the product rule of di ff erential operators,to showD=we only need to verify thatD(q)=for eachq=v∈Vandq=p∈E.The veri fication is straightforward,so we omit it.

We call the set B in Theorem 2.1 the standard basis of thek-linear space Di ff(kΓ,kΓ/I)generated by all di ff erential operators fromkΓ tokΓ/I.

From this theorem,we get Di ff(kΓ,kΓ/I)=whereis thek-linear space generated byfori=1,2 in(2.14).

For anyp∈E,∈is called the arrow di ff erential operator fromkΓ tokΓ/I.Letis called the space of arrow di ff erential operators.

3 H1(kΓ,kΓ/I)for an Admissible Algebra kΓ/I

Proposition 3.1Let q∈P be such that h(q)=t(q)=v0.We have

ProofNote that both sides of(3.1)arek-linearly generated by differential operators.So,by the product formula of differential operators,we only need to verify that the both sides always agree when they act on the elements ofVandE.Since the computation is direct,we omit it here.

Remark 3.1Forv∈V,it is clear thatt(v)=h(v)=v.From Proposition 3.1,we have

We callthe vertex differential operator fromkΓ tokΓ/I.Letdenote the linear space spanned bycalled the space of vertex differential operators.It is clear thatis a subspace of

Lemma 3.1Let p∈P,and then is always in the k-subspacegenerated by

ProofSupposeand then

Corollary 3.1Let q∈P be such that h(q)=t(q).Then

ProofForr∈E,rs∈P;by Lemma 3.1,supposeand it is clear thatso then we use Proposition 3.1.

Remark 3.2Forfrom Theorem 2.1 and Corollary 3.1,we know thatbut not in=DenoteThenand=0.

Denote by Inn-Dif f(kΓ,kΓ/I)the linear space consisting of inner dif f erential operators from kΓ to kΓ/I.Then,Inn-Dif f(kΓ,kΓ/I)=.Thus,we have

Since the basis of kΓ/I given in Proposition 2.1 contains the residue classes of V and E,we can see that the center of kΓ/I as a kΓ-bimodule and the center of kΓ/I as an algebra are the same,denoted by Z(kΓ/I).

Proposition 3.2Let kΓ/I be a f i nite-dimensional admissible algebra,and then

ProofBy the discussion above,dimkHH1(kΓ,kΓ/I)=Then

where the f i rst isomorphism is assured by(2.12),the second and fourth isomorphisms are trivial,and the third is because of the facts thatSoas k-linear spaces,and it follows that

IfkΓ/Iis acyclic,thenThus,we have the following corollary.

Corollary 3.2If kΓ/I is an acyclic admissible algebra(in particular,ifΓis an acyclic quiver),then

On the other hand,when Γ is a planar quiver andkΓ/Iis an acyclic admissible algebra,we can apply the approach of[10]to give a basis ofHH1(kΓ,kΓ/I).A planar quiver is a quiver with a fixed embedding into the planeThe setFof faces of a planar quiver Γ is the set of connected component of

We will need the famous Euler formula on the planar graph(see[5,9]),which states that for any finite connected planar graph(which can be thought as the underlying graph of a quiver Γ),we have

For each face of Γ,its boundary is called a primitive cycle.Letdenote the boundary of the unique unbounded facef0of Γ.Letdenote the set of primitive cycles of Γ andThen clearly,the set ΓPof primitive cycles of Γ is in bijection with the setFof the faces of Γ.So

For a facef∈F,denote bythe corresponding primitive cycle off.Suppose thatis comprised of an ordered list of arrowsp1,···,ps∈E,and define an operator fromkΓ tokΓ/Iby

where aifpiis in clockwise direction when viewed from the interior of the faceand is?otherwise.We calla face differential operator fromkΓ tokΓ/I.Letdenote the linear space spanned bycalled the space of face differential operators.

The next lemma is similar to Theorem 4.9 in[10].

Lemma 3.2LetΓbe a planar quiver with the ground field k of characteristic0,and then

(a)dim=|V|?1;

(b)dim=|F|? 1=

(c)andare linearly disjoint subspaces ofthe

Proof(a) Denote γ0= |V|.Sinceis the identity ofkΓ/I,which clearly lies in the center of kΓ/I, we have

So dim≤γ0?1.We next prove that dim≥γ0?1.We may assume thatγ0≥2.

We claim that anyγ0?1 elements of|i=1,···,γ0}are linearly independent.In fact,suppose=0,whereai∈k,which means thatis in the center ofkΓ/I.

Since Γ is connected,let the vertexvγ0be connected toviby an arrowpforiγ0.We may assume thatt(p)=viandh(p)=vγ0.We have

Soai=0.Note that Γ is connected,and we can repeat this process to getaj=0 for anyj.

(b)Let|F|=γ2.Through simple observation of the planar quiver,we can see that ifp∈Eis in the boundary,then it is at most in the boundary of two primitive cycles.Note that ifp∈Eis in the boundary of two primitive cyclesandthen the signs ofinandare opposite.Ifp∈Eis in the boundary of only one primitive cycle P,thenoccurs twice inwith an opposite sign.Thus we have

wheredenotes the primite cycle corresponding tof0as above.So dim≤ |F|?1.

We next prove that dim≥ |F|? 1.We may assume that|F|≥ 2.Suppose

wherebj∈k.Ifp∈Eis in the boundary ofandforj>0,thenSo we havebj=0.This means that ifandhave a commonp∈Ein their boundary,thenbj=0.Replacebyand repeat this process.Since the quiver is connected,we can getbj=0 for anyj>0.

(c)From[10]and Theorem 2.1,we know thatandE}arek-linearly independent sets in Di ff(kΓ)and Di ff(kΓ,kΓ/I)respectively.Based on this,andcan be linearly expressed by usingwhich is similar to the fact thatDviandcan be linearly expressed byin[10].Under this correspondence,referring to Theorem 4.9 in[10]in the same process,we obtain thatandare linearly disjoint subspaces of

By this lemma,is a basis of

Theorem 3.1LetΓbe a planar quiver and kΓ/I be an acyclic admissible algebra with the ground field k of characteristic0.Then the union set

is a basis of H1(kΓ,kΓ/I).

ProofBy the Euler formula and Lemma 3.2,we can getBecausekΓ/Iis acyclic,we haveand then

4 HH1(kΓ/I)for an Admissible Algebra kΓ/I

Lemma 4.1A differential operator of kΓ/I can naturally induce a differential operator from kΓto kΓ/I.Conversely,a differential operator D from kΓto kΓ/I satisfying D(I)=can induce a differential operator of kΓ/I.

ProofDenote bypthe canonical map fromkΓ tokΓ/I.Given a differential operatorDofkΓ/I,we claim that the compositionDpis a differential operator fromkΓ tokΓ/I.Note that the canonical map fromkΓ tokΓ/Iis an algebra homomorphism,and it can be directly verified.The converse result can be shown directly,too.

For a differential operatorDfromkΓ tokΓ/IsatisfyingD(I)=we denote bythe induced differential operator onkΓ/I.Write(I):={D|D∈Dif f(kΓ,kΓ/I),D(I)=It is clear thatfors∈P.So

Lemma 4.2(I)Dif f(kΓ/I)as k-linear spaces.

ProofThe map from F(I)to Dif f(kΓ/I)is as follows:

The proof of Lemma 4.1 assures that the map fromto Dif f(kΓ/I)is surjective.As for the injectivity,supposeD1,D2∈(I)andD1D2,so according to Lemma 2.1,there exists a

By this lemma,we can think Dif f(kΓ/I)as ak-subspace of Dif f(kΓ,kΓ/I).

From Lemma 4.2,we have

as linear spaces.This means thatHH1(kΓ/I)can be embedded into

Moreover,we have the next proposition.

Proposition 4.1Suppose that kΓ/I is a finite-dimensional admissible algebra,and then

ProofNote thatare linear spaces.By(4.2),we have pathp∈V∪Esuch thatD1(p)D2(p).Since

Corollary 4.1If kΓ/I is an acyclic admissible algebra(in particular,ifΓis an acyclic quiver),then

IfkΓ/Iis an acyclic admissible algebra,we have a standard procedure to compute dim(I).First note that for a differential operatorDfromkΓ tokΓ/I,D(I)=if and only ifD(ri)=where{r1,···,ri,···,rn}is a minimal set of generators ofI.This property follows easily from the Leibnitz rule of differential operators.SincekΓ/Iis acyclic,is finite foras given in Theorem 2.1.Suppose thatfori=1,···,n.This means that the coefficientssatisfy the system of these homogeneous linear equations.So dim(I)is equal to the dimension of the solution space of the system of homogeneous linear equations.

Now we give two examples of admissible algebras that are not monomial algebras or truncated quiver algebras,and characterize their first Hochschild cohomology.

Example 4.1Let Γ=(V,E)be the quiverandI=

In this case,So we have

Suppose that

and then we geta+b?c?d=0.Hence dim(I)=3 and dimkHH1(kΓ/I)=0.

Example 4.2Let Γ be the quiver having one vertex with two loops,or equivalently,kΓ =k.Suppose the idealI=.ThenkΓ/I=k[x,y].

In this case,=?andare the basis of Dif fk[x,y]),wherexmynmeans the multiplication ink[x,y].Sincek[x,y]is commutative,we get that

Moreover,note thatandThus we obtain the basis ofHH1(k[x,y])to be

Similarly,we can obtain the first Hochschild cohomology for

Assume thatkΓ/Iis a monomial algebra.The residue classes of paths that do not belong toIform a basis ofkΓ/I.For convenience,we also denote byQthe basis ofkΓ/IwhenkΓ/Iis a monomial algebra.

Definition 4.1A monomial algebra kΓ/I is called complete if for any parallel paths p,inΓ,p∈I implies∈I.

Proposition 4.2Suppose that kΓ/I is a complete monomial algebra with I?R2.Then the following set

is a basis ofDif f(kΓ/I),where

ProofSincekΓ/Iis complete,we have=for any∈wherepis any path inI.Then=It follows that Dif f(kΓ,kΓ/I)Dif f(kΓ/I)ask-linear spaces.Thus due to Theorem 2.1,the result follows.

Corollary 4.2Suppose that kΓ/I is an acyclic complete monomial algebra with I?R2.Then

ProofBy the proof of Proposition 4.2,dimkBy Corollary 4.1,we get the required result.

In[15],the author gave a characterization of the first Hochschild cohomology of an acyclic complete monomial algebra through a projective resolution.However,itsk-linear basis has not been constructed so far.Here,we want to reach this aim by our method.

Theorem 4.1LetΓbe a planar quiver,and kΓ/I be an acyclic complete monomial algebra with I?R2over the field k of characteristic0.Then the union set

is a basis of HH1(kΓ/I),where

ProofBy(4.1)and(I)=we haveHH1(kΓ/I)H1(kΓ,kΓ/I)in this case.So from Theorem 3.1,we can directly get this theorem.

For a truncated quiver algebrakΓ/knΓ withn≥2,we can give a standard basis of Dif f(kΓ/knΓ).kΓ/knΓ has the basis formed by the residue classes of the paths of length≤n?1,denoted also byQ.

Proposition 4.3LetΓ=(V,E)be a quiver and the field k be of characteristic0.A basis ofDif f(kΓ/knΓ)for any truncated quiver algebra kΓ/knΓwith n≥2is given by the set

where

ProofIt is clear thatfor∈Q,(knΓ)=forNote that whenris a loop of Γ,∈Dif f(kΓ,kΓ/knΓ),butMoreover,for all loopsr1,···,rsof Γ andc1,···,csnot all 0,we claim thatWithout loss of generality,we can assumec10.So we have

Then by Theorem 2.1,the union set

forms a basis of the linear space(knΓ)forI=knΓ.By Lemma 4.2,we have

Noting the map from(knΓ)to Dif f(kΓ/knΓ)in Lemma 4.2,we can see that the union setis ak-linear basis of Dif f(kΓ/knΓ).

Thus Dif f(kΓ/knΓ)=whereis thek-linear space generated byfori=1,2.

Corollary 4.3LetΓ=(V,E)be a quiver and the field k be of characteristic0.Then

ProofBy the proof of Proposition 4.3 and the definition of(I),we can see thatE,is a basis of(knΓ)forI=knΓ.By Proposition 4.3,

Then by Proposition 4.1 and the correspondence betweenDr,sandfor each pairwe get the required result.

This corollary has indeed been given as Theorem 1 in[12]and Theorem 2 in[18].The method we obtain here is different from that in[12,18].

Moreover,whenkΓ/knΓ is acyclic,we can get a basis ofHH1(kΓ/knΓ)as in Theorem 4.1.

Theorem 4.2LetΓbe a planar quiver,and kΓ/knΓfor n≥2be acyclic over the field k of characteristic0.Then the union set

is a basis of HH1(kΓ/knΓ),where

ProofSincekΓ/Iis acyclic,By Lemma 4.2,So the result can be obtained in the same way as the proof of Theorem 3.1.

AcknowledgementWe would like to thank the referees for their careful reviewing and useful comments.

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