Higher Koszul algebras and A-inﬁnity algebras · 2017-02-06 · 336 J.-W. He, D.-M. Lu / Journal...

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Journal of Algebra 293 (2005) 335–362

www.elsevier.com/locate/jalgebr

Higher Koszul algebras andA-infinity algebras

Ji-Wei Hea,b, Di-Ming Lu c,∗

a Institute of Mathematics, Fudan University, Shanghai 200433, Chinab Department of Mathematics, Shaoxing College of Arts and Science, Shaoxing 312000, China

c Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received 23 September 2004

Available online 11 July 2005

Communicated by J.T. Stafford

Abstract

We study a class ofA∞-algebras, named(2,p)-algebras, which is related to the class ofp-homo-geneous algebras, especially to the class ofp-Koszul algebras. A general method to construct(2,p)-algebras is given. Koszul dual of a connected graded algebra is defined in terms ofA∞-algebra. It isproved that ap-homogeneous algebraA is p-Koszul if and only if the Koszul dualE(A) is a reduced(2,p)-algebra and generated byE1(A). The(2,p)-algebra structure of the Koszul dualE(A) of ap-Koszul algebraA is described explicitly. A necessary and sufficient condition for ap-homoge-neous algebra to be ap-Koszul algebra is also given when the higher multiplications on the Kodual are ignored. 2005 Elsevier Inc. All rights reserved.

Keywords:A∞-algebra; Koszul algebra; Connected graded algebra

0. Introduction

Throughout we work over a base fieldk with characteristic 0, all vector spaces alinear maps are overk.

* Corresponding author.E-mail addresses:[email protected] (J.-W. He), [email protected] (D.-M. Lu).

0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2005.05.025

336 J.-W. He, D.-M. Lu / Journal of Algebra 293 (2005) 335–362

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A Koszul algebrais a connected gradedk-algebraA = k ⊕ A1 ⊕ A2 ⊕ · · · with nicehomological properties. The concept was introduced by Priddy in [18] where it was ca homogeneous Koszul algebra. The algebra structure of Koszul dual Ext∗

A(kA, kA) of aKoszul algebraA is worked out explicitly [11,18]. The concept of higher Koszul algbra (p-Koszul algebra) is a generalization of that of Koszul algebra. Forp � 2, ap-Koszulalgebrais ap-homogeneous algebraA such that the trivialA-modulekA admits a pure resolution [3,21]. Whenp = 2, a 2-Koszul algebra is a usual Koszul algebra. Some impograded algebras arep-Koszul algebras [3]. Some significant applications ofp-Koszul alge-bras were found in algebraic topology, algebraic geometry, quantum group, and Lie a([1–5,7,8,11,15,16,19,20], etc.). For example, some Artin–Schelter regular algebrasglobal dimension 3 arep-Koszul algebras which are fundamental in non-commutativejective geometry.

Let A be a p-Koszul algebra, the Koszul dual Ext∗A(kA, kA) of A will be denoted

by E(A). Compared with (usual) Koszul algebras, the algebra structure onE(A) of ap-Koszul algebra forp � 3 is not very clear. Our aim of this paper is to describe thegebra structure onE(A), and to discuss the properties ofp-Koszul algebras in terms oA∞-algebras. For this, we introduce the concept of(2,p)-algebras.

A (2,p)-algebraE is anA∞-algebra with two non-trivial multiplicationsm2 andmp.Roughly speakingE is a graded associative algebra such that some compatibility ctions betweenm2 andmp are required. For anyp � 3, we provide a method to construa (2,p)-algebraE(A;p) naturally from a given positively graded associative algebraA,which suggested a new approach of studying graded objects. Keller has shown in [1a quadratic algebra is Koszul if and only if the higher multiplications of its Koszulare trivial. However, the Koszul dual of ap-Koszul algebraA admits non-trivial highermultiplications whenp � 3. As we will see, the Koszul dualE(A) is a (2,p)-algebra.Moreover, the(2,p)-algebra structure of the Koszul dualE(A) will be given explicitly.We also give a criterion for ap-homogeneous algebra to be ap-Koszul algebra in terms oA∞-algebras.

1. Preliminaries

In what follows, unadorned⊗ means⊗k and Hom means Homk .A graded(associative) algebrais aZ-graded vector spaceA = ⊕

n∈ZAn with graded

linear mapsmA : A ⊗ A → A andηA : k → A of degrees 0 such thatmA ◦ (mA ⊗ 1A) =mA ◦ (1A ⊗ mA) andmA ◦ (1A ⊗ ηA) = mA ◦ (ηA ⊗ 1A) = 1A. Let A andB be gradedalgebras, amorphism of graded algebrasf : A → B is a graded linear map of degreesuch thatf ◦ mA = mB ◦ (f ⊗ f ) andf ◦ ηA = ηB . An augmented graded algebrais agraded algebraA = ⊕

n∈ZAn such that there is a graded algebra morphismεA : A → k

with εA ◦ ηA = 1k . A morphism of augmented graded algebrasf : A → A′ is a gradedalgebra morphism such thatεA = εA′ ◦ f . An augmented positively graded algebraA =⊕

n�0 An with A0 = k is called aconnected graded algebra. A connected graded algebA is said to belocally finite if dim(An) < ∞ for all n � 1, and is said to be generateddegree 1 ifAn = mA(Ai ⊗ Aj) for all i + j = n, i, j � 1 andn � 2.

J.-W. He, D.-M. Lu / Journal of Algebra 293 (2005) 335–362 337

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A differential graded algebra(or DGA, for short) is a graded algebraA together with adifferentialdA : A → A of degree 1 such that (1)dA ◦ dA = 0, and (2)dA(ab) = dA(a)b +(−1)|a|adA(b) for homogeneous elementsa, b ∈ A, where |a| is the degree ofa. Anaugmented DGAis a DGAA such thatA is augmented as a graded algebra andεA ◦dA = 0.A morphism of(augmented) DGAs f :A → A′ is a morphism of (augmented) gradalgebras such thatf ◦ dA = dA′ ◦ f . Due to the degree of the differentiald of a DGA is 1,we usually write a DGA asA = ⊕

n∈ZAn, and denoteHA the cohomology ofA.

Let A be a graded algebra. Aleft graded moduleoverA is aZ-graded vector spaceM =⊕n∈Z

Mn with a graded linear mapµ :A⊗M → M of degree 0 such thatµ◦ (ηA ⊗1M) =1M andµ ◦ (mA ⊗ 1M) = µ ◦ (1A ⊗ µ). Right gradedA-modules and graded bi-modulare defined similarly. WhenA is a DGA, aleft differential graded moduleover A is agradedA-moduleM with a differentialdM :M → M of degree 1 such thatdM ◦ dM = 0and dM(a · m) = dA(a) · m + (−1)|a|a · dM(m) for homogeneous elementsa ∈ A andm ∈ M , where|a| is the degree ofa and “·” is the A-module action. Right differentiagraded modules and differential graded bi-modules are defined similarly.

Let A be a graded algebra. The category of right gradedA-modules (or left gradeA-modules) will be denoted by GrMod-A (or A-GrMod). For M,N ∈ GrMod-A, letHomA(M,N) be the set of gradedA-module morphisms of degree 0 and let Homd

A(M,N)

be the set of gradedA-module morphisms of degreed . Further, let

HomA(M,N) =⊕d∈Z

HomdA(M,N),

where Hom0A(M,N) = HomA(M,N). Let ExtiA(M,N) be the derived functors o

HomA(M,N). SinceM andN are graded modules, ExtiA(M,N) is a graded abelian grou

and we write the component of degreej by ExtiA(M,N)j . Hence

Ext∗A(M,N) =⊕i∈Z

ExtiA(M,N)

is a bi-graded abelian group with(i, j)th component ExtiA(M,N)j . The second grading oExt∗A(M,N) is induced by the gradings ofM andN .

Let U be a vector space. ThenU∗ is the dual space Hom(U, k). Let U and V befinite-dimensional vector spaces. As we know,V ∗ ⊗ U∗ ∼= (U ⊗ V )∗. For conveniencethe isomorphismV ∗ ⊗ U∗ → (U ⊗ V )∗ is defined by(g ⊗ f )(u ⊗ v) = f (u)g(v).

Let V be a finite-dimensional vector space, and letT (V ) = k ⊕ V ⊕ V ⊗2 ⊕ · · ·, withthe usual gradingT (V ) is a graded algebra. For a given integerp � 2, ap-homogeneoualgebrais a graded algebraA = T (V )/(R), in whichR is a subspace ofV ⊗p. Clearly, ap-homogeneous algebra is a connected graded algebra. LetA be ap-homogeneous algebrwe recall that thehomogeneous dualof A is defined byA! = T (V ∗)/(R⊥) whereR⊥ ⊆(V ⊗p)∗ ∼= (V ∗)⊗p is the orthogonal complement ofR.

For the sake of convenience, we introduce a functionp :N → N by

p(n) ={

pm if n = 2m,

pm + 1 if n = 2m + 1.


n

-r

Definition 1.1 [3]. A p-homogeneous algebraA is (right)p-Koszul if the trivialA-modulekA admits a linear projective resolution

· · · → Pn → Pn−1 → ·· · → P1 → P0 → kA → 0, (1.1)

with Pn generated in degreep(n). Whenp = 2, A is called aKoszul algebra.

A left p-Koszul algebra is defined similarly. It is shown in [3,21] that ap-homogeneousalgebraA is left p-Koszul if and only if it is rightp-Koszul. Henceforth we say that aalgebra isp-Koszul without reference to “left” or “right.”

Example 1.2. Let A be an AS-regular algebra over the fieldk of global dimension 3 generated in degree 1. ThenA is generated by two elements with relations of degree 3 oA

is generated by three elements with relations of degree 2 [1]. IfA is generated by twoelements, then the trivialA-modulekA has a minimal projective resolution [1]:

0→ A[−4] → A[−3] ⊕ A[−3] → A[−1] ⊕ A[−1] → A → k → 0.

HenceA is 3-Koszul in this case. IfA is generated by three elements, thenkA has a minimalprojective resolution [1]:

0→ A[−3] → A[−2] ⊕ A[−2] ⊕ A[−2] → A[−1] ⊕ A[−1] ⊕ A[−1] → A → k → 0.

HenceA is Koszul in this case.

Definition 1.3. A Z-graded vector spaceE = ⊕n∈Z

En is called anA∞-algebra if it isequipped with a family of multiplications{mn}n�1, wheremn : E⊗n → E is a gradedlinear map of degree 2− n for n � 1, satisfying theStasheff identities(see [13]):

SI(n)∑

(−1)i+jkml(1⊗i ⊗ mj ⊗ 1⊗k) = 0

for all n � 1, where the sum runs over all decompositionsn = i + j + k, (i, k � 0 andj � 1), andl = i + 1+ k.

The multiplications{mn}n�3 are called thehigher multiplicationsof E.

Definition 1.4. Let E and E′ be A∞-algebras. Amorphism ofA∞-algebras(or A∞-morphism)f :E → E′ is a family of graded linear maps

fn :E⊗n → E′

of degree 1− n satisfying themorphism identities(see [13]):

MI(n)∑

(−1)i+jkfl(1⊗i ⊗ mj ⊗ 1⊗k) =

∑(−1)εmr(fi ⊗ · · · ⊗ fir )
1


by

].

.

d

is

nt in

mentedfined

for all n � 1, where the first sum runs over all decompositionsn = i + j + k with j � 1,i, k � 0, where putl = i + 1 + k, and the second sum runs over all 1� r � n and alldecompositionsn = i1 + · · ·+ ir with all is � 1; the sign on the right-hand side is givenε = (r − 1)(i1 − 1) + (r − 2)(i2 − 1) + · · · + (ir−1 − 1).

An A∞-morphismf is strict if fi = 0 for all i �= 1.f is said to be aquasi-isomorphismif f1 is a quasi-isomorphism.

An augmentedA∞-algebraE is anA∞-algebra satisfying the following conditions:

(1) There is a strictA∞-morphismηE : k → E such thatmn(1⊗i ⊗ ηE ⊗ 1⊗j ) = 0 for alln �= 2 andi + j = n − 1, andm2(1⊗ ηE) = m2(ηE ⊗ 1) = 1E .

(2) There is a strictA∞-morphismεE :E → k such thatεE ◦ ηE = 1.

An augmentedA∞-algebra defined here is a strict unitalA∞-algebra as defined in [12,13Augmented differential graded algebras are examples of augmentedA∞-algebras.

If E andE′ are augmentedA∞-algebras, anA∞-morphismf :E → E′ is called anaugmentedA∞-morphismif

(1) εE′ ◦ f = εE , and(2) f1 ◦ ηE = ηE′ , andfn(1⊗i ⊗ ηE ⊗ 1⊗j ) = 0 for all n � 2.

Two augmentedA∞-algebrasE andE′ are said to bequasi-isomorphicas augmentedA∞-algebras if there is an augmentedA∞-morphismf :E → E′ that is a quasi-isomorphism

The following theorem is given in [9,12,13].

Theorem 1.5. LetA be an augmented DGA, and letE = HA. Then there is an augmenteA∞-structure{mi} on E such thatm1 = 0, m2 is induced by the multiplicationmA of A,andE is quasi-isomorphic toA as augmentedA∞-algebras.

In this paper, we are mainly interested in a connected graded algebraA and its EXT-algebra Ext∗A(kA, kA). As we have seen, Ext∗

A(kA, kA) is a bigraded object. Hence itnecessary to introduce an extra grading for anA∞-algebra. AbigradedA∞-algebra is aZ × Z-graded vector spaceE = ⊕

i,j∈ZEi

j with multiplicationsmn (n � 1) satisfying theStasheff Identities SI(n) as in Definition 1.3, where the degree of a nonzero elemeEi

j is (i, j) and the degree ofmn is (2 − n,0). In other words, eachmn must preservethe second grading. The second grading is called theAdams gradingin [12,13]. Hencea bigradedA∞-algebra is also called anA∞-algebra with an Adams grading(see [12]).Naturally,k can be viewed as a bigradedA∞-algebra that is concentrated in degree(0,0).For convenience, we writeEi = ⊕

j∈ZEi

j . HenceEi is aZ-graded object. Amorphismf

of bigradedA∞-algebras satisfies the same Morphism Identities MI(n), but eachfn mustpreserve the second grading. Similarly, we can define augmented bigradedA∞-algebras,(augmented) bigraded algebras, augmented differential bigraded algebras (or augDBGA, for short) and differential bigraded modules. Morphisms of objects can be desimilarly. For example, ifA = k ⊕ A1 ⊕ A2 ⊕ · · · is a connected graded algebra, thenA


nts of

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-

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ri-.2).

can be regarded as an augmented bigraded algebra with first degree of the elemeA

always being zero.Note that in an augmented DBGA, both the multiplicationm and the differentiald

preserve the second grading. Comparing with Theorem 1.5, we have the following the

Theorem 1.5′. Let A be an augmented DBGA, and letE = HA. Then there is an augmented bigradedA∞-algebra structure{mi} onE such thatm1 = 0, m2 is induced bymA,andE is quasi-isomorphic toA as augmented bigradedA∞-algebras.

For more properties ofA∞-algebras andA∞-morphisms, we refer to papers [9,12,13

2. p-Koszul algebras and Koszul dual

Let A = k ⊕ A1 ⊕ A2 ⊕ · · · be a locally finite connected graded algebra. Endowedthe Yoneda product, Ext∗

A(kA, kA) is a bigraded algebra, called theKoszul dualof A anddenoted byE(A). The(i, j)-component ofE(A) is denoted byEi

j (A) = ExtiA(kA, kA)j .

Also, writeEi(A) = ExtiA(kA, kA) = ⊕j∈Z

Eij (A) for all i � 0. In this section, we discus

some relations between ap-Koszul algebra and its Koszul dual.Let A = T (V )/(R) be ap-homogeneous algebra. Recall that the minimal projec

resolution ofkA begins with

· · · → R ⊗ A → V ⊗ A → A → kA → 0. (2.1)

Hence the first three components of the Koszul dualE(A) areE0(A) = k, E1(A) = V ∗,andE2(A) = R∗. In particular,E1(A) is concentrated inE1−1(A) andE2(A) is concen-trated inE2−p(A). Recall that ap-Koszul algebraA is ap-homogeneous algebra such ththe trivialA-modulekA admits a linear resolution

· · · → Pn → Pn−1 → ·· · → P1 → P0 → kA → 0, (2.2)

wherePn is generated in degreep(n). The following lemma gives an equivalent definitiof p-Koszul algebras which is characterized by the Koszul dual.

Lemma 2.1 [2,21]. Let A be ap-homogeneous algebra. ThenA is a p-Koszul algebra ifand only ifEn(A) is concentrated in degree(n,−p(n)).

For a Koszul algebra, there is a more precise description on the Koszul dual.

Theorem 2.2 [7,11]. Let A be a quadratic algebra. ThenA is Koszul if and only if itsKoszul dualE(A) is generated byE1(A).

If A is ap-homogeneous algebra withp � 3, it is impossible forE(A) to be generatedby E1(A) sinceE1(A) · E1(A) = 0 [21]. However we will show that there is a similar cterion for ap-homogeneous algebra to be ap-Koszul algebra (see Theorems 2.5 and 6


t.

e

First we have to compute the product ofE(A) for ap-Koszul algebraA. In order to avoidambiguity, we usef ∗ g to denote the product ofE(A) and usef · g to denote the producof A!. Fori, j � 1, if p(i)+p(j) �= p(i + j), thenEi(A)∗Ej(A) = 0 by [21, Lemma 2.1]

Let A = T (V )/(R) be ap-Koszul algebra(p � 3). Note that the trivial modulekA hasa minimal projective resolution (see [3,21])

· · · δn+1−−−→ Kp(n) ⊗ Aδn−→ Kp(n−1) ⊗ A

δn−1−−−→ · · · δ1−→ Kp(0) ⊗ A ε−→ kA → 0,

where

Kp(n) =⋂

i+j+p=p(n)i,j�0

V ⊗i ⊗ R ⊗ V ⊗j for n � 2,

Kp(1) = K1 = V, and Kp(0) = K0 = k.

The differentialδ is given as follows. Forx1, . . . , xp(n) ∈ V , anda ∈ A,

δn(x1, . . . , xp(n), a) ={

x1 ⊗ · · · ⊗ xp(n)−p+1 ⊗ xp(n)−p+2 · · · · ·xp(n) · a if n is even,x1 ⊗ · · · ⊗ xp(n)−1 ⊗ xp(n) · a, if n is odd,

wherexp(n)−p+2 · · · · ·xp(n) · a andxp(n) · a are products of elements inA.Moreover, we haveA!

p(n)∼= K∗

p(n) (see [21], or combine results of [3,5]) forn � 0.

HenceEn(A) ∼= A!p(n) for n � 0.

Forϕ ∈ A!p(n)

∼= En(A) andψ ∈ A!p(m)

∼= Em(A), we viewϕ andψ as linear maps from

Kp(n) to k andKp(m) to k, respectively. Letϕ :Kp(n) ⊗ A → kA andψ :Kp(m) ⊗ A → kA

be the rightA-module morphism induced byϕ and ψ , respectively. Now consider thfollowing diagram:

· · · δn+m+1Kp(n+m) ⊗ A

ϕm

δn+m · · · δn+1Kp(n) ⊗ A

ϕ0ϕ

δn · · ·

· · · δm+1Kp(m) ⊗ A

δm

ψ

· · · δ1Kp(0) ⊗ A

εkA 0

kA

whereϕi for i = 0,1, . . . ,m is defined as follows.(1) Letn be even. Forx1, . . . , xp(n+i) ∈ V , anda ∈ A, the rightA-module morphismϕi

is given as

ϕi(x1, . . . , xp(n+i), a) = ϕ(x1, . . . , xp(n))xp(n)+1 ⊗ · · · ⊗ xp(n+i) ⊗ a.

Clearly,δi ◦ ϕi = ϕi−1 ◦ δn+i . Now for x1, . . . , xp(n+m) ∈ V , anda ∈ A,


t the

milar

ψ ◦ ϕm(x1, . . . , xp(n+m), a) = ψ(ϕ(x1, . . . , xp(n))xp(n)+1, . . . , xp(n+m), a

)= ϕ(x1, . . . , xp(n))ψ(xp(n)+1, . . . , xp(n+m))a

= (ψ · ϕ)(x1, . . . , xp(n+m))a. (2.3)

Hence, whenn is even, we have

ψ ∗ ϕ = ψ · ϕ (2.4)

for ϕ ∈ En(A) = A!p(n) andψ ∈ Em(A) = A!

p(m), where the “·” is the product ofA!.(2) Let n be odd. Forx1, . . . , xp(n+i) ∈ V , anda ∈ A, the rightA-module morphismϕi

is given as follows:

(a) if i is odd, defineϕi to be

ϕi(x1, . . . , xp(n+i), a) = ϕ(x1, . . . , xp(n))xp(n)+1 ⊗ · · · ⊗ xp(n+i)−p+2

⊗ (xp(n+i)−p+3 . . . xp(n+i)a),

where (xp(n+i)−p+3 . . . xp(n+i)a) is the product of the elementsxp(n+i)−p+3, . . . ,

xp(n+i), a in A;(b) if i is even, defineϕi to be

ϕi(x1, . . . , xp(n+i), a) = ϕ(x1, . . . , xp(n))xp(n)+1 ⊗ · · · ⊗ xp(n+i) ⊗ a.

It is not hard to checkδi ◦ϕi = ϕi−1 ◦ δn+i . If ψ ∗ϕ �= 0 thenp(n)+ p(m) = p(n+m).Sincen is odd,m must be even. By the computations similar to (2.3), we get thaidentity (2.4) also holds whenn is odd.

Let E = ⊕i�0,j∈Z

E ij be the bigraded vector space withE i

−p(i) = A!p(i) andE i

j = 0 if

j �= −p(i). HenceE i is concentrated in degree(i,−p(i)). Define a multiplication onE asfollows: for ψ ∈ E i andϕ ∈ Ej ,

ψ ∗ ϕ ={

ψ · ϕ at least one ofi andj is even,0 otherwise,

where “·” is the product ofA!. Then it is not hard to check thatE is a bigraded algebra.By discussions and settings above, we have the following proposition.

Proposition 2.3. Let A be ap-Koszul algebra,p � 3, and E(A) its Koszul dual. ThenE(A) ∼= E as bigraded algebras.

Remark. After we finished the work we found that Berger and Marconnet also got a siresult in [6].

Lemma 2.4. Let A = T (V )/(R) be ap-homogeneous algebra andE(A) = ⊕i�0 Ei(A)

be its Koszul dual. ThenEi (A) = 0 for all i � 0 andj < p(i).
−j


ssr

tor

at

Proof. Let

· · · → P2m+1∂2m+1−−−→ P2m

∂2m−−→ P2m−1 → ·· · → P1∂1−→ P0

ε−→ kA → 0

be a minimal resolution ofkA. We prove inductively thatPi is generated in degrees not lethanp(i) for all i � 0. SinceP0 ∼= A, P1 ∼= V ⊗A, andP2 ∼= R⊗A, the statement holds foPi wheni � 2. If the statement holds forP2m, it is also holds forP2m+1 by minimality of∂2m. Now suppose thatP2m is generated in degrees not less thanmp. We have to show thaP2m+2 is generated in degrees not less than(m + 1)p. Obviously, there is a graded vectspaceW = Wmp ⊕Wmp+1 ⊕· · · such thatP2m

∼= W ⊗A. In what follows, we identifyP2m

with W ⊗ A. Specifically,P2m,mp = Wmp ⊗ k andP2m,mp+j = Wmp ⊗ V ⊗j ⊕ Wmp+1 ⊗V ⊗j−1 ⊕ · · · ⊕ Wmp+j for 0� j � p − 1. LetK = Kmp+1 ⊕ Kmp+2 ⊕ · · · ⊕ Kmp+p−1 ⊕· · · = ker(∂2m). By minimality of ∂2m, we getKmp+j ⊆ Wmp ⊗V ⊗j ⊕· · ·⊕Wmp+j−1 ⊗V

for 1 � j � p − 1. LetU = Ump+1 ⊕ Ump+2 ⊕ · · · ⊕ Ump+p−1 ⊕ · · · = K/KJ , whereJ

is the graded Jacobson radical ofA. We may considerU as a graded subspace ofK . Inparticularly,

Ump+j ⊆ Kmp+j ⊆ Wmp ⊗ V ⊗j ⊕ Wmp+1 ⊗ V ⊗j−1 ⊕ · · · ⊕ Wmp+j−1 ⊗ V

= (Wmp ⊗ V ⊗j−1 ⊕ Wmp+1 ⊗ V ⊗j−2 ⊕ · · · ⊕ Wmp+j−1

) ⊗ V

for 1 � j � p − 1. Let Qj = Wmp ⊗ V ⊗j−1 ⊕ · · · ⊕ Wmp+j−1. Now Ump+j ⊆ Qj ⊗ V

for 1� j � p − 1. By minimality, we haveP2m+1 ∼= U ⊗A. Also, we identifyP2m+1 withU ⊗ A. More precisely,

P2m+1,mp+1 = Ump+1 ⊗ k,

and

P2m+1,mp+j+1 = Ump+1 ⊗ V ⊗j ⊕ Ump+2 ⊗ V ⊗j−1 ⊕ · · · ⊕ Ump+j+1 ⊗ k

for 0 � j � p − 2. The differential∂2m+1 is defined as follows: fora ∈ A andx ∈ U ,∂2m+1(x ⊗ a) = x · a, wherex · a is the rightA-module action inP2m. Let us denoteK ′ = K ′

mp+2 ⊕ · · · ⊕ K ′mp+p−1 ⊕ · · · = ker(∂2m+1). What remains to be shown is th

K ′mp+j = 0 for 2� j � p − 1. Forj = 2, we haveK ′

mp+2 ⊆ Ump+1 ⊗ V ⊆ Q1 ⊗ V ⊗ V .By the definition of∂2m+1, it follows K ′

mp+2 = 0. For 2< j � p − 1,

K ′mp+j ⊆ Ump+1 ⊗ V ⊗j−1 ⊕ · · · ⊕ Ump+j−1 ⊗ V

⊆ Q1 ⊗ V ⊗ V ⊗j−1 ⊕ · · · ⊕ Qj−1 ⊗ V ⊗ V.

By the definition of∂2m+1 and the choice ofUmp+i , 1� i � p − 1, we get thatK ′mp+j = 0

for 2� j � p−1. By minimality of the projective resolution, we get the desired result.�Now we can prove the criterion theorem for ap-homogeneous algebra to be ap-Koszul

algebra.


t

nce

hat

e

fsinceThis

on fromszul

s

Theorem 2.5. LetA be ap-homogeneous algebra, andE(A) its Koszul dual. ThenA is ap-Koszul algebra if and only ifE(A) is generated byE1(A) andE2(A).

Proof. Whenp = 2, this is Theorem 2.2 since in this caseE2(A) is generated byE1(A).Hence it is sufficient to prove the statement forp � 3. Suppose thatA is p-Koszul. Propo-sition 2.3 says that the productEi(A) ∗ Ej(A) is the same asA!

p(i) · A!p(j) when at leas

one ofi or j is even. ClearlyA! is generated byA!1 as a connected graded algebra. He

A!p(n) = A!

p(i) ·A!p(j) whenp(n) = p(i)+ p(j). Hence forn � 3, En(A) = Ei(A) ∗Ej(A)

with i + j = n andp(n) = p(i) + p(j). Conversely, suppose thatE(A) is generated byE1(A) andE2(A). It is known thatE0(A) = k, E1(A) = V ∗, andE2(A) = R∗. SinceE(A) is a bigraded algebra andR ⊆ V ⊗p, we haveE0(A) = E0

0(A), E1(A) = E1−1(A),and E2(A) = E2−p(A). Since the product onE(A) preserves the bidegree, we get t

E1−1(A) ∗ E1−1(A) ⊆ E2−2(A). By Lemma 2.4,E2−2(A) = 0 since 2< p(2) = p. Now

E3(A) = E1−1(A) ∗ E2−p(A) ⊕ E2−p(A) ∗ E1−1(A)

is concentrated in degree(3,−(p + 1)). Inductively we proveEn(A) is concentrated indegree(n,−p(n)). Suppose that fors < n, it is true thatEs(A) is concentrated in degre(s,−p(s)). Let i + j = n. For i, j < n, if i = 2s + 1 andj = 2t + 1, then

Ei−(sp+1)(A) ∗ E

j

−(tp+1)(A) ⊆ E2(s+t)+2−(p(s+t)+2)(A).

But (s + t)p + 2 < p(2(s + t) + 2) = (s + t + 1)p, we getEi(A) ∗ Ej(A) = 0 byLemma 2.4. Otherwise at least one ofi or j is even, thenp(n) = p(i) + p(j). Hencefor n � 3

En(A) =∑

i,j�1i+j=n

Ei(A) ∗ Ej(A) =∑

i,j�1, i+j=np(n)=p(i)+p(j)

Ei(A) ∗ Ej(A).

By the inductive hypothesisEi(A) is concentrated in degree(i,−p(i)) andEj(A) is con-centrated in degree(j,−p(j)). HenceEn(A) is concentrated in degree(i + j,−p(i) −p(j)) = (n,−p(n)). By Lemma 2.1, the statement follows.�

3. The A∞-version of Koszul dual

In this section, we revisit the Koszul dual of a connected graded algebra in terms oA∞-algebras. The use ofA∞-algebras in graded ring theory is a completely new approachthe concept ofA∞-algebra was introduced to non-commutative algebra very recently.method has advantages for higher Koszul algebras because one can get informatithe non-trivial higher multiplications on its Koszul dual. We make a definition of Kodual of a connected graded algebra in terms ofA∞-algebras.

Let D = ⊕n�0 Dn be an augmented DGA withD0 = k. By Theorem 1.5, there i

an augmentedA∞-structure onHD which is quasi-isomorphic toD asA∞-algebras. In


is

s

of

.

fact theA∞-structure onHD can be constructed explicitly. The following paragraphessentially a copy from [12].

Let H = HD = ⊕n�0 Hn, let B = ⊕

n�0 Bn andZ = ⊕n�0 Zn be the coboundarie

and cocycles ofD, respectively. Forn � 1, there are subspacesLn andH̄ n of Dn such thatZn = Bn ⊕ H̄ n andDn = Bn ⊕ H̄ n ⊕ Ln. Of course, there are many different choicesH̄ n andLn. In what follows, we identifyHn with H̄ n. Let Pr :D → H be the projectionto H . Now we define a linear mapQ :D → D of degree−1 with the following propertiesFor n � 1, Qn :Dn → Dn−1 is defined asQn = 0 when restricted toHn ⊕ Ln, andQn =(dn−1|Ln−1)−1 when restricted toBn. Define a sequence of linear mapsλn : D⊗n → D ofdegree 2− n as follows. There is no mapλ1, but we formally setQλ1 = −idD , andλ2 isthe multiplication ofD. Forn � 3, λn is defined by the recursive formula

λn =∑

s+t=ns,t�1

(−1)s+1λ2[Qλs ⊗ Qλt ].

In what follows, we useλn to denote both the mapD⊗n → D and its restriction to(HD)⊗n.

Lemma 3.1 [12,17]. Let {λn} be defined as above. Letm1 = 0 and mn = Pr ◦λn : (HD)⊗n → HD for n � 2. Then(HD, {mn}) is an augmentedA∞-algebra and itis quasi-isomorphic toD as augmentedA∞-algebras.

Remark. If D is an augmented DBGA, then(HD, {mn}) is an augmented bigradedA∞-algebra and(HD, {mn}) is quasi-isomorphic toD as augmented bigradedA∞-algebras.

Let A be a locally finite connected graded algebra. LetI = A1 ⊕ A2 ⊕ · · ·. SinceA is agraded associative algebra, there is a natural graded projective resolution

· · · → I⊗n ⊗ Adn−→ I⊗n−1 ⊗ A

dn−1−−−→ · · · d2−→ I ⊗ Ad1−→ A ε−→ kA → 0, (3.1)

called thebar resolution(see [14], where the algebra is non-graded) ofkA, where thedifferentialdn : I⊗n ⊗ A → I⊗n−1 ⊗ A is given by

dn(a1 ⊗ · · · ⊗ an ⊗ a) =n−1∑i=1

(−1)i+1a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an ⊗ a

+ (−1)n+1a1 ⊗ · · · ⊗ an−1 ⊗ ana. (3.2)

Let P denote the following complex of graded rightA-modules:

· · · → I⊗n ⊗ Adn−→ I⊗n−1 ⊗ A

dn−1−−−→ · · · d2−→ I ⊗ Ad1−→ A → 0. (3.3)

If we takeA as a bigraded algebra with the first degree of the elements ofA being alwayszero, thenP can be regarded as a differential bigraded rightA-module with differential


nts

d

e

d given by (3.2). The bigrading ofP is given as follows. For homogeneous elemea1, . . . , an ∈ I anda ∈ A,

bideg(a1 ⊗ · · · ⊗ an ⊗ a) = (−n, |a1| + · · · + |an| + |a|).Hence the degree of the differentiald of P is (1,0).

Let I ∗ = Hom(I, k). ThenT (I ∗) = k ⊕ I ∗ ⊕ I ∗ ⊗ I ∗ ⊕ · · · is an augmented bigradealgebra with bigrading given as follows. For homogeneous elementsf1, . . . , fn ∈ I ∗,

bideg(f1 ⊗ · · · ⊗ fn) = (n, |f1| + · · · + |fn|

).

Define a bigraded morphism∂ : T (I ∗) → T (I ∗) of degree(1,0) by

∂(f1 ⊗ · · · ⊗ fn)(a1 ⊗ · · · ⊗ an+1)

= −(−1)n(f1 ⊗ · · · ⊗ fn)

(n∑

i=1

(−1)i+1a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an+1

)

= −(−1)nn∑

i=1

(−1)i+1fn(a1)fn−1(a2) . . . fn−i+1(aiai+1) . . . f1(an+1).

It is not hard to see that∂ ◦ ∂ = 0 and(T (I ∗), ∂) is an augmented DBGA.Applying HomA(−, kA) on (3.3), we get a complex

· · · ← HomA(I⊗n ⊗ A,kA)d∗n←− · · · d∗

2←− HomA(I ⊗ A,kA)

d∗1←− HomA(A,kA) ← 0, (3.4)

where forf ∈ HomA(I⊗n ⊗A,kA), d∗(f ) = −(−1)nf ◦ d . By abuse of notations, we usHomA(PA, kA) to denote the complex (3.4). Since HomA(I⊗n ⊗A,kA) ∼= Hom(I⊗n, k) ∼=I ∗⊗n, it is easy to see that there is a natural isomorphism of complexes

ξ : HomA(PA, kA) → T (I ∗).

Forf ∈ T (I ∗), define

f ⇀ (a1 ⊗ · · · ⊗ an ⊗ a) =n∑

i=0

〈f,a1 ⊗ · · · ⊗ ai〉ai+1 ⊗ · · · ⊗ an ⊗ a,

for all a1, . . . , an ∈ I and a ∈ A. It is easy to check that “⇀” defines a left bigradedT (I ∗)-module action onP . We next show that the left action “⇀” is compatible withthe differentiald of P .

Forf1, . . . , fs ∈ I ∗, a1, . . . , an ∈ I anda ∈ A,


i-

y

d(f1 ⊗ · · · ⊗ fs ⇀ (a1 ⊗ · · · ⊗ an ⊗ a)

)= d

(fs(a1)fs−1(a2) . . . f1(as)as+1 ⊗ · · · ⊗ an ⊗ a

)=

n−s∑t=1

(−1)t+1fs(a1)fs−1(a2) . . . f1(as)as+1 ⊗ · · · ⊗ as+t as+t+1 ⊗ · · · ⊗ an ⊗ a,

∂(f1 ⊗ · · · ⊗ fs) ⇀ a1 ⊗ · · · ⊗ an ⊗ a + (−1)sf1 ⊗ · · · ⊗ fs ⇀ d(a1 ⊗ · · · ⊗ an ⊗ a)

= −(−1)ss∑

t=1

(−1)t+1fs(a1)fs−1(a2) . . . fs−t+1(atat+1) . . . f1(as+1)as+2

⊗ · · · ⊗ an ⊗ a

+ (−1)ss∑

t+1

(−1)t+1fs(a1)fs−1(a2) . . . fs−t+1(atat+1) . . . f1(as+1)as+2

⊗ · · · ⊗ an ⊗ a

+ (−1)sn−s∑t=1

(−1)s+t+1fs(a1)fs−1(a2) . . . f1(as)as+1 ⊗ · · · ⊗ as+t as+t+1

⊗ · · · ⊗ an ⊗ a

=n−s∑t=1

(−1)t+1fs(a1)fs−1(a2) . . . f1(as)as+1 ⊗ · · · ⊗ as+t as+t+1 ⊗ · · · ⊗ an ⊗ a.

HenceP is a differential bigraded leftT (I ∗)-module. The leftT (I ∗)-module action onPis compatible with the rightA-module action. HenceP is a differential bigradedT (I ∗)-A-bimodule. Thus there is a natural DBGA morphism

χ :T (I ∗) → EndA(PA),

where EndA(PA) is the DBGA consisting of bigraded rightA-module morphisms fromPA

to itself. The differentialδ of the DBGA EndA(PA) is given as follows. Forf ∈ EndA(PA)

whose degree is(n,m), thenδ(f ) = d ◦ f − (−1)nf ◦ d .

Lemma 3.2. The DBGA morphismχ is a quasi-isomorphism.

Proof. Since (3.1) is a graded projective resolution ofkA, there is a natural quasisomorphismζ : EndA(PA) → HomA(PA, k). One can check that the composition

T (I ∗) χ−→ EndA(PA)ζ−→ HomA(PA, k)

ξ−→ T (I ∗)

is the identity. Henceχ is a quasi-isomorphism.�Since (3.1) is a graded projective resolution ofkA, Ext∗A(kA, kA) = H(EndA(PA)).

By Lemma 3.2, Ext∗ (kA, kA) ∼= H(T (I ∗)). SinceT (I ∗) is an augmented DBGA, b
A


ithd.

on-

reen

ties

u,

Theorem 1.5′ there is an augmented bigradedA∞-algebra structure on Ext∗A(kA, kA)

with m1 = 0 and m2 is induced by the multiplication ofT (I ∗). Of course, the in-duced augmented bigradedA∞-algebra structure on Ext∗

A(kA, kA) is unique up toquasi-isomorphism; that is, there may be twoA∞-algebra structures{mi} and {m′

i} onExt∗A(kA, kA) induced fromT (I ∗), but they are quasi-isomorphic.

Definition 3.3. Let A be a locally finite connected graded algebra. TheKoszul dualof A isthe augmented bigradedA∞-algebra (up to quasi-isomorphism)

E(A) := Ext∗A(kA, kA)

induced from the augmented DBGA(T (I ∗), ∂) as in Theorem 1.5′.If we need to point out a specificA∞-algebra structure, we shall say theKoszul dual

E(A) of A with A∞-structure{mi} induced fromT (I ∗), or theA∞-structure{mi} inducedfromT (I ∗) of the Koszul dualE(A) of A.

Remark. By definition the Yoneda product on Ext∗A(kA, kA) is the multiplication of

the cohomology algebraH(EndA(PA)). By Lemma 3.2, the DBGA EndA(PA) is quasi-isomorphic to the DBGAT (I ∗). Hence the Koszul dual in Definition 3.3 coincides wthe usual definition of Koszul dual (Section 2) if the higher multiplications are ignore

4. (2,p)-algebras

We will focus on a special class ofA∞-algebras in this section, those with only one ntrivial higher multiplication. Such anA∞-algebra will be called a(2,p)-algebra. Given apositively graded algebraA = ⊕

n�0 An and any integerp � 3, we provide a method foconstructing a(2,p)-algebra fromA. We will see that there are deep relations betw(2,p)-algebras andp-homogeneous algebras.

The positive integerp � 3 is assumed in this section.

Definition 4.1. Let E = ⊕n∈Z

En be anA∞-algebra. IfE has only two non-trivial multi-plicationsm2 andmp, then(E,m2,mp) is called a(2,p)-algebra. If E is an augmentedA∞-algebra, then(E,m2,mp) is called anaugmented(2,p)-algebra.

Since a(2,p)-algebra has only two non-trivial multiplications, the Stasheff identiare automatically satisfied except for the following three cases:

SI(3) m2(m2 ⊗ 1) = m2(1⊗ m2),SI(2p − 1)

∑i+j=p−1(−1)i+pjmp(1⊗i ⊗ mp ⊗ 1⊗j ) = 0,

SI(p + 1)∑

i+j=p−1(−1)imp(1⊗i ⊗ m2 ⊗ 1⊗j ) = m2(1⊗ mp) − (−1)pm2(mp ⊗ 1).

The identity SI(3) says that(E,m2) is a graded algebra.What motivate us to define(2,p)-algebra is the following example given by L

Palmieri, Wu, and Zhang in [13].


dif-

Example 4.2. Let E be the graded algebrak[x1, x2]/(x21) with |x1| = 1 and |x2| = 2.

Define a(2,p)-algebra structure onE as follows.For s � 0, set

xs ={

xs/22 if s is even,

x1x(s−1)/22 if s is odd.

Then{xs}s�0 is a basis of the graded vector spaceE. For i1, . . . , ip � 0, define

mp(xi1, . . . , xip ) ={

xj if all is are odd,0 otherwise,

wherej = 2−p+∑s is . The multiplicationm2 is the product of the algebrak[x1, x2]/(x2).

It is direct to check that(E,m2,mp) is a(2,p)-algebra.

Definition 4.3. Let E andE′ be two(2,p)-algebras.E andE′ are said to beisomorphicifthere is a strictA∞-morphismf from E to E′ such thatf is bijective.

Let g = {gn} :E → E′ be anA∞-morphism that is a quasi-isomorphism. Since theferentials ofE andE′ are zero, it follows thatg1 : (E,m2) → (E′,m2) is an isomorphismof graded algebras.

Definition 4.4. An augmented(2,p)-algebra(E,m2,mp) is called areduced(2,p)-algebraif the following conditions are satisfied:

(i) E = k ⊕ E1 ⊕ E2 ⊕ · · ·;(ii) m2(E

2t1+1 ⊗ E2t2+1) = 0 for all t1, t2 � 0;(iii) mp(Ei1 ⊗ · · · ⊗ Eip) = 0 unless all ofi1, . . . , ip are odd.

A reduced(2,p)-algebraE is said to begenerated byE1 if for all n � 2,

En =∑

i+j=ni,j�1

m2(Ei ⊗ Ej) +

∑mp(Ei1 ⊗ · · · ⊗ Eip),

where the sum in the second sigma runs all decompositionsi1 + · · · + ip + 2 − p = n,(i1, . . . , ip � 1).

The(2,p)-algebra constructed in Example 4.2 is a reduced(2,p)-algebra.

Theorem 4.5. Let A = A0 ⊕ A1 ⊕ A2 ⊕ · · · be a positively graded algebra. LetE =E0 ⊕ E1 ⊕ E2 ⊕ · · · be a graded vector space such thatEn = Ap(n) for all n � 0. De-fine multiplications onE by:


that

nt

(i) for a ∈ En = Ap(n) andb ∈ Em = Ap(m),

m2(a, b) ={

a · b, at least one ofn andm is even,0, otherwise;

(ii) for a1 ∈ En1, a2 ∈ En2, . . . , ap ∈ Enp ,

mp(a1, a2, . . . , ap) ={

a1 · a2 · · · · · ap all of n1, n2, . . . , np are odd,0 otherwise;

where“ ·” is the product ofA.

Then(E,m2,mp) is a (2,p)-algebra. Moreover, ifA0 = k, thenE is a reduced(2,p)-algebra.

Notation. The(2,p)-algebra constructed as above is denoted byE(A;p).

Proof. We only prove the theorem whenp is odd. The proof is similar whenp is even.Note that the Koszul sign conventions are always assumed. It is not hard to seeE

is a graded associative algebra, that is,m2(1⊗ m2) = m2(m2 ⊗ 1). We need to check

(a)∑

i+j=p−1(−1)i+jpmp(1⊗i ⊗ mp ⊗ 1⊗j ) = 0;

(b)∑

i+j=p−1(−1)imp(1⊗i ⊗ m2 ⊗ 1⊗j ) = m2(1⊗ mp) − (−1)pm2(mp ⊗ 1).

Since im(mp) ⊆ ⊕t�1 E2t , (a) is satisfied. When applied to a homogeneous elemex,

the only cases that (b) might be failed are:

(1) x = a ⊗ b1 ⊗ · · · ⊗ bp,(2) x = b1 ⊗ · · · ⊗ bp ⊗ a,(3) x = b1 ⊗ · · · ⊗ bt ⊗ a ⊗ bt+1 ⊗ · · · ⊗ bp for 1� t � p − 1,(4) x = b1 ⊗ · · · ⊗ bp+1,

wherea is in an even component ofE andb1, . . . , bp+1 are in odd components ofE.For case (1),

∑i+j=p−1

(−1)imp(1⊗i ⊗ m2 ⊗ 1⊗j )(x) = mp

(m2(a, b1), b2, . . . , bp

)

= mp

((a · b1), b2, . . . , bp

) = (a · b1) · b2 · · ·bp,

and

(m2(1⊗ mp) − (−1)pm2(mp ⊗ 1))(x) = m2(a, b1 · b2 . . . bp) = a · (b1 · b2 . . . bp).


By the associativity ofA,

∑i+j=p−1

(−1)imp(1⊗i ⊗ m2 ⊗ 1⊗j )(x) = (m2(1⊗ mp) − (−1)pm2(mp ⊗ 1)

)(x).

Case (2) is similar to case (1). For case (3),

∑i+j=p−1

(−1)imp

(1⊗i ⊗ m2 ⊗ 1⊗j

)(x)

= (−1)t−1mp

(b1, . . . , bt−1,m2(bt , a), bt+1, . . . , bp

)+ (−1)tmp

(b1, . . . , bt ,m2(a, bt+1), bt+2, . . . , bp

)= (−1)t−1(b1 . . . bt−1 · (bt · a) · bt+1 . . . bp

)+ (−1)t

(b1 . . . bt · (a · bt+1) · bt+2 . . . bp

) = 0,

and

(m2(1⊗ mp) − (−1)pm2(mp ⊗ 1)

)(x) = 0.

For case (4),

∑i+j=p−1

(−1)imp(1⊗i ⊗ m2 ⊗ 1⊗j )(x) = 0,

and

(m2(1⊗ mp) − (−1)pm2(mp ⊗ 1)

)(x)

= (−1)p|b1|b1 · (b2 . . . bp+1) − (−1)p(b1 . . . bp) · bp+1 = 0.

Hence (b) holds. Thus(E,m2,mp) is a(2,p)-algebra.Clearly,E is a reduced(2,p)-algebra providedA0 = k. �

Remark. Similar to the settings before Proposition 2.3, the(2,p)-algebraE(A;p) can beregarded as a bigradedA∞-algebra. In fact, setEn

−p(n)(A;p) = Ap(n) andEnj (A;p) = 0

for j �= −p(n). ThenE(A;p) is a bigradedA∞-algebra withEn(A;p) concentrated indegree(n,−p(n)).

Proposition 4.6. Let(E,m2,mp) be a reduced(2,p)-algebra. If(E,m2,mp) is generatedbyE1, then the graded associative algebra(E,m2) is generated byE1 andE2. Moreover,E2t+1 = m2(E

1 ⊗ E2t ) for all t � 1.


ssible

t

Proof. Clearly, we haveE3 = m2(E1 ⊗ E2 + E2 ⊗ E1). We claim thatm2(E

2 ⊗ E1) ⊆m2(E

1 ⊗ E2). It is sufficient to show that for all homogeneous elementsx ∈ E2 andy ∈ E1, m2(x, y) ∈ m2(E

1 ⊗ E2). Since(E,m2,mp) is generated byE1, we getE2 =mp((E1)⊗p). Without loss of generality, we may assumex = mp(x1, . . . , xp), wherexi ∈ E1 for 1� i � p. By the Stasheff identity SI(p + 1),

m2(x, y) = m2(mp(x1, . . . , xp), y

) = m2(x1,mp(x2, . . . , xp, y)

) ∈ m2(E1 ⊗ E2).

Next, we want to show thatE4 = m2(E2 ⊗ E2). By the hypotheses,

E4 = m2(E2 ⊗ E2) +

∑mp(Ei1 ⊗ · · · ⊗ Eip),

where 1� i1, . . . , ip � 3 are odd and the sum runs over all possible cases. For all poi1, . . . , ip , we showmp(Ei1 ⊗· · ·⊗Eip) ⊆ m2(E

2 ⊗E2). Since the degree ofmp is 2−p,one ofi1, . . . , ip must be 3 and the rest are 1. Ifip = 3, then by SI(p + 1),

mp(E1 ⊗ · · · ⊗ E1 ⊗ E3) = mp

(E1 ⊗ · · · ⊗ E1 ⊗ m2(E

1 ⊗ E2))

⊆ m2(mp(E1 ⊗ · · · ⊗ E1) ⊗ E2) = m2(E

2 ⊗ E2).

If it = 3, 0< t < p, by SI(p + 1), we have

mp

((E1)⊗t−1 ⊗ E3 ⊗ (E1)⊗p−t

) = mp

((E1)⊗t−1 ⊗ m2(E

1 ⊗ E2) ⊗ (E1)⊗p−t)

⊆ mp

((E1)⊗t ⊗ E3 ⊗ (E1)⊗p−t−1)

⊆ · · · ⊆ mp

((E1)⊗p−1 ⊗ E3) ⊆ m2(E

2 ⊗ E2).

Now suppose that for 3< t < n,

Et =∑

i1,i2�1, i1+i2=t

m2(Ei1 ⊗ Ei2) and Et = m2(E

1 ⊗ Et−1)

whent is odd.If n is odd, then

En =∑

i1,i2�1, i1+i2=n

m2(Ei1 ⊗ Ei2)

since the image ofmp lies in the even components ofE. It suffices to show tham2(E

i1 ⊗ Ei2) ⊆ m2(E1 ⊗ En−1). But m2(E

i1 ⊗ Ei2) ⊆ m2(m2(E1 ⊗ Ei1−1) ⊗ Ei2) if

i1 > 1 is odd. By the Stasheff identity SI(3), we havem2(Ei1 ⊗ Ei2) ⊆ m2(E

1 ⊗ En−1).Similarly we see thatm2(E

i1 ⊗ Ei2) ⊆ m2(E1 ⊗ En−1) if i1 is even andi2 is odd.

If n is even, then

En =∑

i ,i �1, i +i =n

m2(Ei1 ⊗ Ei2) +

∑j +···+j =n

mp(Ej1 ⊗ · · · ⊗ Ejp),

1 2 1 2 1 p


s,

e-

thece

thats

wherej1, . . . , jp � 1 are odd. Sincen > 3, at least one of a given sequencej1, . . . , jp isnot less than 3. Without loss of generality, we assumejp � 3. By the inductive hypotheseEjp = m2(E

1 ⊗ Ejp−1). Now

mp(Ej1 ⊗ · · · ⊗ Ejp) = mp

(Ej1 ⊗ · · · ⊗ Ejp−1 ⊗ m2(E

1 ⊗ Ejp−1))

⊆ m2(mp(Ej1 ⊗ · · · ⊗ Ejp−1 ⊗ E1) ⊗ Ejp−1)

⊆ m2(En−jp+1 ⊗ Ejp−1)

by SI(p + 1). This completes the proof.�Next we want to show that the higher multiplicationmp of a reduced(2,p)-algebra

(E,m2,mp) which is generated byE1 is determined bym2 andmp acting on(E1)⊗p.We need the following lemma.

Lemma 4.7. Let (E,m2,mp) be a reduced(2,p)-algebra. If the graded associative algbra (E,m2) is generated byE1 andE2, andE2t+1 = m2(E

1 ⊗E2t ) for all t � 1, then thehigher multiplicationmp is determined bym2 andmp|(E1)⊗p .

Proof. Suppose thatm2 andmp|(E1)⊗p are fixed. We prove the lemma by induction ondegrees of elements. Leta1, . . . , ap−1 ∈ E1 andb ∈ E3 be homogeneous elements. SinE2t+1 = m2(E

1 ⊗ E2t ) for t � 1, it is no harm to assume thatb = m2(x, y) for somex ∈ E1 andy ∈ E2. Then forp − 2� i � 0,

mp(a1, . . . , ai, b, ai+1, . . . , ap−1) = mp

(a1, . . . , ai,m2(x, y), ai+1, . . . , ap−1

)= mp

(a1, . . . , ai, x,m2(y, ai+1), . . . , ap−1

).

The degree ofc = m2(y, ai+1) is 3. Continuing the foregoing procedure, we get

mp(a1, . . . , ai, b, ai+1, . . . , ap−1) = mp

(a′

1, . . . , a′p−1, z

)for somez ∈ E3 and a′

i ∈ E1. Without loss of generality, letz = m2(x1, x2) for somex1 ∈ E1 andx2 ∈ E2. Then

mp(a1, . . . , ai, b, ai+1, . . . , ap−1) = mp

(a′

1, . . . , a′p−1,m2(x1, x2)

)= m2

(mp(a′

1, . . . , a′p−1, x1), x2

).

Hencemp(a1, . . . , ai, b, ai+1, . . . , ap−1) is determined bym2 andmp|(E1)⊗p . Now sup-pose thatmp(Ei1 ⊗ · · · ⊗ Eip) is determined bym2 andmp|(E1)⊗p for i1 + · · · + ip � n

(n > p + 2). Let a1, . . . , ap be homogeneous elements with odd degrees such|a1| + · · · + |ap| = n + 2. Then there is an element in{ai | 1 � i � p} whose degree i


ed

d-

nalge-

not less than 3. Without loss of generality, we assume|ap| � 3 andap = x · y = m2(x, y)

for somex ∈ E1 andy ∈ E|ap |−1. By the Stasheff identity SI(p + 1), we have

mp(a1, . . . , ap) = mp

(a1, . . . , ap−1,m2(x, y)

) = m2(mp(a1, . . . , ap−1, x), y

).

Now |a1| + · · · + |ap−1| + |x| � n. By the inductive hypothesis,mp(a1, . . . , ap) is deter-mined bym2 andmp|(E1)⊗p . �Proposition 4.8. Let (E,m2,mp) be a reduced(2,p)-algebra. IfE is generated byE1,then the higher multiplicationmp is determined bym2 andmp|(E1)⊗p .

Proof. This is a direct result of Lemma 4.7 and Proposition 4.6.�If (E,m2,mp) and(E,m2,m

′p) are two reduced(2,p)-algebra structures on a grad

vector spaceE = k ⊕ E1 ⊕ E2 ⊕ · · · generated byE1 such that the underlying gradeassociative algebra structures are the same, and ifmp = m′

p on (E1)⊗p, then by Proposition 4.8,mp = m′

p. By this observation, we have the following proposition.

Proposition 4.9. Let (E,m2,mp) and (E′,m′2,m

′p) be reduced(2,p)-algebras. Iff =

{fi} :E → E′ is a quasi-isomorphism ofA∞-algebras andE is generated byE1, theng = f1 :E → E′ is an isomorphism of(2,p)-algebras.

Proof. Since (E,m2,mp) is generated byE1, then the graded algebra(E,m2) isgenerated byE1 and E2 by Proposition 4.6. Sincef is a quasi-isomorphism, thef1 : (E,m2) → (E′,m′

2) is an isomorphism of graded algebras. Hence the graded

bra(E′,m′2) is generated byE′1 andE′2. Fory1, . . . , yp ∈ E1, by MI(p) we have

f1 ◦ mp(y1, . . . , yp) = m′p

(f1(y1), . . . , f1(yp)

). (4.1)

Hence we have a commutative diagram

(E1)⊗pf

⊗p1

mp

(E′1)⊗p

m′p

E2f1

E′2.

Since the(2,p)-algebraE is generated byE1, we haveE2 = mp((E1)⊗p). Hencemp|(E1)⊗p is surjective. Sincef1 andf

⊗p

1 are isomorphisms, we get thatE′2 = m′p(E′⊗p

).

Hence the(2,p)-algebra(E′,m′2,m

′p) is generated byE′1.

The bijection f1 induces a(2,p)-algebra structure(m′′2,m

′′p) on E′ as follows.

For x, y ∈ E′, define m′′2(x, y) = f1m2(f

−11 (x), f −1

1 (y)). For x1, . . . , xp ∈ E′, definem′′

p(x1, . . . , xp) = f1mp(f −1(x1), . . . , f−1(xp)). Clearly, we have(E′,m′′,m′′

p) is a
1 1 2


e

9 are

Itytions

ussed

s

ap

(2,p)-algebra. Moreover,f1 : (E,m2,mp) → (E,m′′2,m

′′p) is an isomorphism. Sinc

f1 : (E,m2) → (E′,m′2) is an isomorphism of graded algebra, for allx, y ∈ E′ then

m′′2(x, y) = f1m2

(f −1

1 (x), f −11 (y)

) = m′2

(f1f

−11 (x), f1f

−11 (y)

) = m′2(x, y).

For allx1, . . . , xp ∈ E′1, by (4.1) we have

m′′p(x1, . . . , xp) = f1mp

(f −1

1 (x1), . . . , f−11 (xp)

) = m′p

(f1f

−11 (x1), . . . , f1f

−11 (xp)

)= m′

p(x1, . . . , xp).

Hencem′2 = m′′

2 andm′p|

(E′1)⊗p = m′′p|

(E′1)⊗p . Thus the(2,p)-algebra structures(m′2,m

′p)

and(m′′2,m

′′p) onE′ are the same. Henceg = f1 is an isomorphism. �

Remark. If the (2,p)-algebras involved are bigraded, then Propositions 4.8 and 4.also true.

Let V be a finite-dimensional vector space, and letR be a subspace ofV ⊗p. ThenA =T (V )/(R) and its homogeneous dualA! = T (V ∗)/(R⊥) arep-homogeneous algebras.is easy to see that the reduced(2,p)-algebrasE(A;p) and E(A!;p) are generated bE1(A;p) andE1(A!;p), respectively. In the rest of this section, we discuss some relabetweenA andE(A;p) (or E1(A!;p)). More will be done in Section 6.

Example 4.10. Let A be thep-homogeneous algebrak[x]/(xp). ThenE(A;p) = k ⊕ kx.This (2,p)-algebra is nothing but a vector space. Hence there is nothing to be discaboutE(A;p). Let y be the dual basis of the 1-dimensional spacekx. ThenA! is the freealgebrak[y]. Now E(A!;p) = k ⊕ ky ⊕ kyp ⊕ kyp+1 ⊕ · · · ⊕ kynp ⊕ kynp+1 ⊕ · · · . The(2,p)-algebra structure onE(A!;p) is as follows:

m2(ypn1+1, ypn2+1) = 0,

m2(ypn1, ypn2+1) = m2(y

pn2+1, ypn1) = y(n1+n2)p+1,

m2(ypn1, ypn2) = y(n1+n2)p, for all n1, n2 � 0;

mp(ypn1+1, . . . , ypnp+1) = yp(n1+···+np)+2, for all n1, . . . , np � 0;mp(others) = 0.

One can check that the(2,p)-algebraE(A!;p) is isomorphic to the(2,p)-algebra obtainedin Example 4.2.

Lemma 4.11. LetA = T (V )/(R) be ap-homogeneous algebra. Then thep-homogeneoualgebra is determined by the restriction of the higher multiplicationmp on (E1(A;p))⊗p.

Clearly,E1(A;p) = V . By the definition ofmp, one can see that the kernel of the mmp :V ⊗p → E2(A;p) equalsR. Hence the structure ofA is determined.


s.

asi-

ative

rading

he

Theorem 4.12. Let A = T (V )/(R) andB = T (U)/(S) be twop-homogeneous algebraThenA andB are isomorphic as graded algebras if and only if the(2,p)-algebrasE(A;p)

andE(B;p) are quasi-isomorphic asA∞-algebras.

Proof. The “only if” part of the theorem is trivial. Now suppose that there is a quisomorphismf = {fn} :E(A;p) → E(B;p). SinceE(A;p) and E(B;p) are reduced(2,p)-algebras generated in degree 1, we get thatg = f1 :E(A;p) → E(B;p) is an iso-morphism of(2,p)-algebras by Proposition 4.9. In particular, we have a commutdiagram

V ⊗p∼=

mp

U⊗p

mp

E2(A;p)∼=

E2(B;p).

Applying Lemma 4.11, we get the desired result.�

5. Computation on higher multiplications

In this section,A is a locally finite connected graded algebra generated byA1, and(T (I ∗), ∂) is the DBGA constructed in Section 3. Suppose thatR = ⊕

n�2 Rn ⊂ T (A1) isa minimal graded vector space of the relations ofA. ThenA = T (A1)/(R). The minimalprojective resolution ofkA begins with

· · · → R ⊗ A → A1 ⊗ A → A → kA → 0.

Hence Ext1A(kA, kA) = A∗1 and Ext2A(kA, kA) = R∗.

For convenience, we ignore the second grading ofT (I ∗) andE(A) for a moment. Thisis no harm since all the morphisms involved in this section preserve the second gnaturally.

We know thatT (I ∗) is a DGA and the Koszul dualE(A) of A is induced byT (I ∗).We use Lemma 3.1 to compute the higher multiplications ofE(A). To do this, we needto construct analogue of the mapsλ andQ constructed in Section 3. Essentially all of tmaterial in this section is taken from [12].

SinceA is generated byA1, the multiplicationAn−1⊗A1 → An is surjective. Forn � 2,let the linear mapsξn :An → An−1 ⊗ A1 be a split injection such that the composition

Anξn−→ An−1 ⊗ A1 → An

is the identity. Letθn be the composition

Anξn−→ An−1 ⊗ A1

ξn−1⊗1−−−−→ An−2 ⊗ A1 ⊗ A1ξn−2⊗1⊗2−−−−−−→ · · · ξ2⊗1⊗n−1−−−−−−→ A⊗n.
1


o-

on

p

d a

SinceR is a minimal graded vector space of the relations ofA, there is an injectionηn :Rn → An−1 ⊗ A1. We can viewRn as a subspace ofAn−1 ⊗ A1 via this injection.

Let T (I) = k ⊕ I ⊕ I⊗2 ⊕ · · ·. For homogeneous elementsa1, . . . , as ∈ I , let the bide-gree ofa1 ⊗ · · · ⊗ as be (−s, |a1| + · · · + |as |). Define a differentialδ :T (I) → T (I) ofdegree(1,0) as follows. Fors � 1, the restriction ofδ on I⊗s → I⊗s−1 is written asδ−s .Thenδ−1(I ) = 0, and

δ−s(a1, . . . , as) =s−1∑i=1

(−1)i+1a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ as

for s � 2. Clearly,(T (I ), δ) is a complex. It is not hard to see that the complex(T (I ∗), ∂) isthe dual of(T (I ), δ); that is,∂(f1, . . . , fs) = δ∗(f1, . . . , fs) = −(−1)s(f1 ⊗ · · · ⊗ fs) ◦ δ.

So, we writeδ−sn to be the restriction ofδ−s on the subspace

⊕Ai1 ⊗ · · · ⊗ Ais , where

the direct sum runs over all decompositionsn = i1 + · · · + is (i1, . . . , is � 1).

Lemma 5.1. LetW−2n = ⊕

i+j=n Ai ⊗ Aj . Then there is a decomposition

W−2n = im

(δ−3n

) ⊕ Rn ⊕ ξn(An).

Proof. It is clear thatW−2n = ker(δ−2

n ) ⊕ ξn(An). SinceR∗ ∼= Ext2A(k, k) ∼= H 2(T (I ∗)),there is a decomposition ker(δ−2

n ) = im(δ−3n ) ⊕ Rn. �

Let T 2 = I ∗ ⊗ I ∗ andT 2−n = ⊕i+j=n A∗

i ⊗ A∗j . By Lemma 5.1, we have a decomp

sitionT 2−n = (ξn(An))∗ ⊕ R∗

n ⊕ (im(δ−3n ))∗. This decomposition is just the decompositi

T 2−n = B2−n ⊕ H 2−n ⊕ L2−n in Section 3. Hence we can defineQ2−n :T 2−n → T 1−n = A∗n as

the compositionA∗1 ⊗ A∗

n−1∼= (An−1 ⊗ A1)

∗ ξ∗n−→ A∗

n for all n � 2. Then we get a maQ2 :T 2 → T 1 = I ∗.

Theorem 5.2. Let A be a locally finite connected graded algebra generated byA1. LetR = ⊕

n�2 Rn ⊂ T (A1) be a minimal graded vector space of the relations ofA. Then thereis an augmentedA∞-algebra structure{mn} induced fromT (I ∗) of the Koszul dualE(A)

such thatm1 = 0, m2 is the Yoneda product, and the higher multiplicationmn restricted to(E1(A))⊗n = (A∗

1)⊗n is the composition

(A∗

1

)⊗n ∼= (A⊗n

1

)∗ (θn−1⊗1)∗−−−−−−→ (An−1 ⊗ A1)∗ η∗

n−→ R∗n.

Remark. The proof of this theorem is a modification of that in [12]. Keller also statesimilar result in [10] without proof.

Proof. Let Pr2 be the projection in Merkulov’s construction restricted toT 2. Write Pr2−n =Pr2|T 2−n

. Then Pr2−n|A∗1⊗A∗

n−1is the composition

A∗ ⊗ A∗ ∼= (An−1 ⊗ A1)∗ η∗

n−→ R∗n
1 n−1


nt

.-

l

he-5.2

and Pr2−n(others) = 0. By Lemma 3.1, there is an augmentedA∞-algebra structure(E(A), {m′

n}) with m′1 = 0 and forn � 2,

m′n = Pr2−n ◦ λn = Pr2−n ◦

∑i+j=ni,j�1

(−1)i+1λ2(Qλi ⊗ Qλj ),

which is quasi-isomorphic toT (I ∗) as augmentedA∞-algebras. Since the homotopyQ2

is defined fromA∗1 ⊗ A∗

n−1 to A∗n for all n � 2, by the definition of Pr2−n we have

m′n = −Pr2−n ◦ λ2(1⊗ Qλn−1)

= (−1)2Pr2−n ◦ λ2(1⊗ Qλ2(1⊗ λn−2)

)...

= (−1)nPr2n−1 ◦ λ2(1⊗ Qλ2

(1⊗ Qλ2

(. . .Qλ2(1⊗ Qλ2)

)))= (−1)nPr2n−1 ◦ λ2 ◦ (1⊗ Qλ2) ◦ (

1⊗2 ⊗ Qλ2) ◦ · · · ◦ (

1⊗n−2 ⊗ Qλ2)

when restricted to(A∗1)

⊗n. Note thatλ2 is the multiplication ofT (I ∗). Consider the fol-lowing commutative diagram:

(A∗1)

⊗n

�

1⊗n−2⊗Qλ2(A∗

1)⊗n−2 ⊗ A∗

2

�

1⊗n−3⊗Qλ2 · · · 1⊗Qλ2A∗

1 ⊗ A∗n−1

�

Pr2−nR∗

n

�

(A⊗n1 )∗

(ξ2⊗1⊗n−2)∗(A2 ⊗ A⊗n−2

1 )∗(ξ3⊗1⊗n−3)∗ · · · (ξn−1⊗1)∗

(An−1 ⊗ A1)∗ η∗

nR∗

n.

Setmn = (−1)nm′n. We see that when restricted to(A∗

1)⊗n, mn is the composition

(A∗

1

)⊗n ∼= (A⊗n

1

)∗ (θn−1⊗1)∗−−−−−−→ (An−1 ⊗ A1)∗ η∗

n−→ R∗n.

Similar to [13, Lemma 4.2], let us setf1(x) = (−1)|x|x for homogeneous elemex ∈ E(A), and letf = {fi} with fi = 0 for i �= 1. Thenf is a strictA∞-morphism(E(A), {mn}) → (E(A), {m′

n}). Clearlyf1 is bijective, hencef is a quasi-isomorphismHence(E(A), {mn}) is quasi-isomorphic toT (I ∗) as augmentedA∞-algebras. By the construction ofλ2, m2 is induced by the multiplication ofT (I ∗), hencem2 is just the Yonedaproduct by Lemma 3.2. Hence{mn} is an augmentedA∞-structure of the Koszul duaE(A) of A by Definition 3.3. �Remark. SinceT (I ∗) is an augmented DBGA and all the morphisms in the proof of Torem 5.2 preserve the second grading, theA∞-algebra structure obtained in Theoremis an augmented bigradedA∞-algebra by the remark of Lemma 3.1.


eraded

f a

d

1.

6. (2,p)-algebras and Koszul dual of p-Koszul algebras

Let A be a locally finite-connected graded algebra. As we know, Koszul dualE(A) isdefined up to quasi-isomorphism. In this section, we prove that ifA is ap-Koszul algebrathen eachA∞-structure of the Koszul dualE(A) is a (2,p)-algebra. Moreover, all th(2,p)-algebra structures are isomorphic. We then give a criterion for a connected galgebra to be ap-Koszul algebra in terms ofA∞-algebra.

Lemma 6.1. LetA be ap-Koszul algebra(p � 3), and letE = E(A) = k ⊕E1 ⊕E2 ⊕· · ·be the Koszul dual ofA with an augmented bigradedA∞-structure{mi} induced fromT (I ∗). Then(E(A), {mi}) is a reduced(2,p)-algebra.

Proof. Since A is a p-Koszul algebra,E2m(A) is concentrated in degree−mp andE2m+1(A) is concentrated in degree−(mp + 1) by Lemma 2.1. Keep in mind thatmn

preserves the second degree, we have

mn(E2m1+s1 ⊗ · · · ⊗ E2mn+sn) ⊂ E

2(m1+···+mn)+s1+···+sn+2−n−(p(m1+···+mn)+s1+···+sn) ,

wheresi = 0 or 1 (1� i � n), so 0� s1 + · · · + sn � n, we have 2− n � s1 + · · · + sn +2− n � 2. Hence the only possible cases formn �= 0 are

(a) s1 + · · · + sn + 2− n = 0 ands1 + · · · + sn = 0,(b) s1 + · · · + sn + 2− n = 1 ands1 + · · · + sn = 1,(c) s1 + · · · + sn + 2− n = 2 ands1 + · · · + sn = p.

The cases (a) and (b) implyn = 2. Notice thatp � 3, we have

m2(E

2m1+1−(m1p+1) ⊗ E

2m2+1−(m2p+1)

) ⊂ E2(m1+m2+1)−(p(m1+m2)+2) = 0.

The case (c) impliesn = p; that is,si = 1 for all 1� i � p. HenceE(A) is a reduced(2,p)-algebra. �

By Lemma 6.1, each augmented bigradedA∞-algebra structure of the Koszul dual op-Koszul algebra is a reduced(2,p)-algebra.

Theorem 6.2. Let A be a locally finite connected graded algebra generated byA1. LetE = E(A) = k ⊕ E1 ⊕ E2 ⊕ · · · be the Koszul dual ofA with an augmented bigradeA∞-structure{mi} induced fromT (I ∗). For p � 3, A is ap-Koszul algebra if and only if

(i) (E(A), {mi}) is a reduced(2,p)-algebra, and(ii) the reduced(2,p)-algebraE(A) is generated byE1.

Proof. Let us suppose thatA is a p-Koszul algebra. Then (i) follows from Lemma 6.Next we prove (ii). It is no harm to assume that theA∞-structure{mi} is just the


ul-

5,

get

a.a

ee-

ulti-

en

A∞-structure constructed in Theorem 5.2. Suppose thatA = T (V )/(R) with R ⊂ V ⊗p.By (i), (E(A), {mi}) is a reduced(2,p)-algebra, that is,mi = 0 for i �= 2,p. Since(E(A),m2) is the usual Yoneda algebra, we get that the graded algebra(E(A),m2) isgenerated byE1(A) andE2(A) by Theorem 2.5. By the construction of the higher mtiplication mp in Theorem 5.2, we haveE2(A) = R∗ = mp((A∗

1)⊗p) = mp((E1(A))⊗p).

Hence the reduced(2,p)-algebra(E(A),m2,mp) is generated byE1(A).Now suppose thatE(A) has the properties as stated in the theorem. SinceA is a con-

nected graded algebra generated byA1, we assume thatA = T (A1)/(R) with R ⊂ T (A1)

being a minimal graded vector space of the relations ofA. As we have seen in SectionE1(A) = A∗

1 andE2(A) = R∗. HenceE1(A) is concentrated inE1−1(A). By the hypothe-ses,R∗ = E2(A) = mp(E1 ⊗ · · · ⊗ E1). Sincemp preserves the second grading, wethat mp(E1 ⊗ · · · ⊗ E1) is concentrated in degree(2,−p). HenceR∗ is concentrated indegree−p, that is,R is concentrated in degreep. HenceA is ap-homogeneous algebrSince the reduced(2,p)-algebra(E(A),m2,mp) is generated byE1, the graded algebr(E(A),m2) is generated byE1 andE2 by Proposition 4.6. Since(E(A),m2) coincideswith the usual Koszul dual by the remark of Definition 3.3, we get thatA is a p-Koszulalgebra by Theorem 2.5.�

Let A be ap-Koszul algebra(p � 3). Since any two(2,p)-algebra structures of thKoszul dual ofA are quasi-isomorphic asA∞-algebras, then they are isomorphic by Thorem 6.2 and Proposition 4.9. Hence we have

Proposition 6.3. The(2,p)-algebra structure of the Koszul dualE(A) of a p-Koszul al-gebraA (p � 3) is unique up to isomorphism.

If A is a Koszul algebra, Keller showed in [10] that there is no non-trivial higher mplication on the Koszul dual ofA. Comparing Theorems 2.2 and 6.2, we get

Theorem 6.4. For all p � 2, if A is p-Koszul algebra, thenE(A) is generated byE1(A)

as an augmentedA∞-algebra.

For ap-Koszul algebraA, the multiplicationm2 of the (2,p)-algebraE(A) has beengiven in Proposition 2.3. The higher multiplicationmp is explicitly given in the followingtheorem.

Theorem 6.5. LetA be ap-Koszul algebra,p � 3, andA! be its homogeneous dual. ThEn(A) = A!

p(n) and the(2,p)-algebra structure onE(A) is: for f1 ∈ Ei(A), f2 ∈ Ej(A),

m2(f1, f2) ={

f1 · f2 if at least one ofi andj is even,0 otherwise,

and forf1 ∈ Ei1(A), . . . , fp ∈ Eip(A),


itiondnis

l com-

Math.

.310070.Conf.

ath.

ijing

lec-

bra, in:1183,

4)

l, QC,

mp(f1, f2, . . . , fp) ={

f1 · f2 · · · · · fp if i1, . . . , ip are all odd,0 otherwise,

where“ ·” is the product of the graded algebraA!.

Proof. Let E = ⊕n�0 A!

p(n) be the bigraded algebra given in Section 2. By Propos2.3,E(A) = E as a bigraded algebra. By Theorem 4.5,(m2,mp) given above is a reduce(2,p)-algebra structure. Clearlymp|(E1(A))⊗p coincides with the multiplication derived iTheorem 5.2. By Theorem 6.2 and Proposition 4.8, the(2,p)-algebra obtained as abovethe Koszul dual ofA. �

Combining Theorems 6.2, 6.5 and 4.5, we have

Theorem 6.6. Let A be ap-homogeneous algebra,p � 3. ThenA is a p-Koszul algebraif and only ifE(A) = E(A!;p) as augmentedA∞-algebras.

Acknowledgments

We thank J.J. Zhang for several helpful conversations and the referee for usefuments and valuable suggestions.

References

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