A primer on A-infinity-algebras and their Hochschild homology

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by  Stephan Mescher

A PRIMER ON A∞-ALGEBRAS AND THEIR HOCHSCHILD HOMOLOGY
STEPHANMESCHER
Ruhr-Universita¨tBochum
Fakulta¨tfu¨rMathematik
6 Universita¨tsstraße 150
1
44801Bochum(Germany)
0
2

n ABSTRACT. Wepresentanelementaryandself-containedconstructionofA∞-algebras,
a A∞-bimodulesand theirHochschildhomologyand cohomologygroups. Inaddition,
J
wediscussthecupproductinHochschildcohomologyandthespectralsequenceofthe

5 lengthfiltrationofaHochschildchaincomplex.
1 A∞-structuresarisenaturallyinthestudyofbasedloopspacesandthegeometryof

manifolds, in particular in Lagrangian Floer theory and Morse homology. In several
]
A geometricsituations,Hochschildhomologymaybeusedtodescribehomologygroups
offreeloopspaces,seee.g.[Jon87],[Goo85]or[Sei09].
R
The objective of this article is not to introduce new material, but to give a unified
h. and coherent discussionof algebraic results from several sources. It further includes
t detailedproofsofallpresentedresults.
a
m
[

1
v CONTENTS
3
Introduction 1
6
9 Acknowledgements 2
3 1. A∞-algebrasand-bimodules 3
0
. 2. Hochschildhomologyof A∞-algebras 11
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3. Hochschildcohomologyof A∞-algebras 36
0
4. TheHochschildcupproduct 42
6
1 5. ThelengthfiltrationofaHochschildchaincomplexanditsspectralsequence 54
: References 59
v
i
X
r
a
INTRODUCTION
A∞-algebras, also known as strongly homotopy associative algebras, are generaliza-
tionsofassociativealgebras. Theyare obtainedbyformalizing andrefiningthenotion
that the product of an algebra is associative up to homotopy. They were introduced by
E-mail address:[email protected].
Date:January18,2016.
1
2 APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY
James Stasheff in [Sta63b] for the study of based loop spaces and further used in ho-
motopy theory by several authors. In the geometry of manifolds, A∞-algebras and
theirgeneralizationsto A∞-categorieswereintroducedintoMorsehomologyandFloer
homology in symplectic geometry by Kenji Fukaya in [Fuk93] (see also [Sei08b] for a
detaileddiscussion). InhistalkattheICM1994,MaximKontsevichusedA∞-structures
in the formulation of his homological version of the mirror symmetry conjecture, see
[Kon95].
The author’s aim is to give an introduction to A∞-algebras and their Hochschild (co)-
homologywhichisaselementaryandasself-containedaspossible. While A∞-algebras
are introduced and well explained in several places in the literature, e.g. in [Kel01],
[GJ90], [Tra08b] or [KS09], and while their Hochschild homology has also been de-
scribed on several occasions, e.g. in [Sei08a, Section 5] or [KS09, Section 7], the author
wasnotabletoobtainasourcewhichdiscussesthetopicscontainedinthisarticleboth
in an elementary and in a complete way, especially including all necessary sign com-
putations. Thisarticleisintendedtoprovidearemedy.
Afterpresentingthedefinitionsand somebasic constructionsof A∞-algebras and A∞-
bimodulesin Section1,wedefinetheHochschildhomologyof A∞-algebras in Section
2 and show that morphisms of A∞-bimodules induce homomorphisms between the
correspondingHochschildhomologygroups. ThelargestpartofSection2is occupied
by the proof of the latter statement and by the proof of Theorem 2.5, stating that the
differentialoftheHochschildchaincomplexisindeedadifferential.
In Section 3 we define the notion of Hochschild cohomology of A∞-algebras and pro-
vide a detailed discussion of its duality with Hochschild homology. Additionally, we
explicitly describe the case that the coefficient A∞-bimodule is given by the algebra
itself. Afterwards, we define the cup product structure of Hochschild cohomology in
Section4andshowthatitisinducedbyachainmapdefinedontheHochschildcochain
complex.
We conclude this article by discussing the spectral sequence of the length filtration of
a Hochschild chain complex in detail in Section 5. This spectral sequence has useful
convergenceproperties.Moreover,itsfirstpagehasaconcisedescriptionwhichwewill
usetoproveasimplecriterionforcertainmapstoinduceisomorphismsofHochschild
homologygroups.
ThroughoutthisarticleletRdenoteacommutativeringwithunit. Byan R-module,wealways
mean a bimodule over R. All tensor products appearing in this article refer to tensor products
of R-modules. Agraded R-moduleisalwaysassumedtobegradedoverZ.
ACKNOWLEDGEMENTS
This article is a byproduct of the author’s Ph.D. thesis in preparation, written under
the guidance of Prof. Dr. Matthias Schwarz at the University of Leipzig and partially
supportedbyascholarshipfromtheDeutscheTelekomFoundation. Theauthorthanks
MatthiasSchwarzandAlbertoAbbondandoloforvaluablediscussionson A∞-algebras
andtheirapplicationsinthegeometryofmanifolds.
APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY 3
1. A∞-ALGEBRAS AND -BIMODULES
We begin by giving an elementary definition of an A∞-algebra. Note that our sign
conventions coincide with those in [Sei08b], [Sei08a] and [GJ90], but differ from the
conventionsusedin[Tra08b]and[Tra08a].
Definition 1.1. A graded R-module A = j∈Z Aj equipped with a family of homo-
morphismsofgradedR-modules
L
µ : A⊗n → A , degµ = 2−n, foreveryn ∈ N,
n n
willbecalledan A∞-algebraover Rifthefollowingequationissatisfiedforeveryr ∈ N
andeverya ,...,a ∈ A:
1 r
∑ r+∑1−n1(−1)z1i−1µn2(a1,...,ai−1,µn1(ai,...,ai+n1−1),ai+n1,...,ar) = 0, (1.1)
n1,n2∈N i=1
n1+n2=r+1
where1
j
zj := zj(a ,...,a ) := ∑µ(x )−(j−i+1) (1.2)
i i 1 r q
q=i
for all i,j ∈ {1,2,...,r} with i ≤ j and where for every k ∈ Z we write µ(a) = k iff
a ∈ A . For a fixed r ∈ N we refer to equation (1.1) as the r-th defining equation of the
k
A∞-algebra(A,(µn)n∈N).
Example1.2. (1) Everydifferentialgradedalgebra(DGA)over R,i.e. anassociative
graded algebra over R which comes with a differential satisfying the graded
Leibniz rule, can be given the structure of an A∞-algebra. We simply put µ1
to be the differential, µ to be the algebra multiplication and µ to be zero for
2 n
every n ≥ 3. In this case, the second defining equation reduces to the graded
Leibnizrule,thethirddefiningequationisequivalenttotheassociativityofthe
multiplicationandalldefiningequationsofhigherordersimplyvanishonboth
sides.
(2) Stasheffhasshownin[Sta63a]and[Sta63b]thatthesingularchaincomplexofa
based loop space can always be equippedwith the structureof an A∞-algebra.
Themapµ isgiven(uptoinvertingthegrading)bythedifferentialofthecom-
1
plex. The map µ is given by the Pontryagin product, i.e. the map induced
2
on singular chains by the composition of loops. While it is easy to see that the
Pontryagin product is associative up to a chain homotopy, Stasheff introduced
A∞-algebrastoshowthatthisstatementcanbestronglyrefined.
Remark 1.3. It is possible to give a slightly more concise and elegant definition of an
A∞-algebraandallothernotionsdefinedinthissectionintermsoftheusualcoalgebra
structureonthetensoralgebraof A. ThismethodwasintroducedbyGetzlerandJones
in [GJ90] and furtherconsideredby Tradler in [Tra08b] and [Tra08a] and severalother
authors.
1
TheuseoftheMaltesecrossforthecoefficientsdefiningthesignsinthisarticlehappensinaccordance
with the notation of the works of Mohammed Abouzaid and Paul Seidel, e.g. [Sei08b] or [Abo10]. In
particular,theauthordistanceshimselffromanypoliticalmeaningorimplicationofthissymbol.
4 APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY
Throughouttherestofthissectionlet(A,(µn)n∈N)bean A∞-algebraover R.
Definition1.4. Let M = j∈Z Mj beagradedR-module.
(1) For a ∈ A, we puLt µ(a) := j iff a ∈ Aj and call µ(a) the index of a. We further
callkak := µ(a)−1thereducedindexofa.
(2) For m ∈ M, we put µ (m) := j iff m ∈ M and call µ (m) the index of m. We
M j M
furthercall kmk := µ (m)−1thereducedindexofm.
M
j
Inthisnotationwehavezj = ∑ka kforalliand j.
i q
q=i
Remark 1.5. Thesignsappearing in thedefiningequationsof A are bestunderstoodin
terms ofreduced indices. We consider theshifted cochain complex A[1] = j∈Z A[1]j,
givenby
L
A[1] = A ∀j ∈ Z ,
j j+1
whose differential is given by µ . In other words, the elements of A[1] are those ele-
1 j
mentsof Awhosereducedindexequals j,i.e. itholdsthat
µ (a) = kak (1.3)
A[1]
for every a ∈ A. If one considers the operation µ as defined on (A[1])⊗d instead of
d
A⊗d, then the signs in the defining equations of the A∞-algebra A will be obtained by
followingtheusualKoszulsignconvention.
ThefollowingnotionwasintroducedbyEzraGetzlerandJohnD.S.Jonesin[GJ90]and
byMartinMarklin[Mar92].
Definition 1.6. A graded R-module M equipped with a family of maps (µrM,s)r,s∈N0
where
µM : A⊗r⊗M⊗ A⊗s → M
r,s
isanR-modulehomomorphismwithdegµrM,s = 1−r−swillbecalledan A∞-bimodule
over Aifthefollowingequationholdsforeveryr,s ∈ N ,m ∈ M anda ,...,a ∈ A:
0 1 r+s
∑ ∑r1 (−1)z1i−1µrM1,s(a1,...,ai−1,µr2(ai,...,ai+r2−1),ai+r2,...,ar,m,ar+1,...,ar+s)
r1∈N0,r2∈Ni=1
r1+r2=r+1
+ ∑ ∑ (−1)z1r1µrM1,s1(a1,...,ar1,µrM2,s2(ar1+1,...,ar,m,ar+1,...,ar+s2),...,ar+s)
r1,r2∈N0 s1,s2∈N0
r1+r2=rs1+s2=s
+ ∑ ∑s1 (−1)z1r−j+1+µM(m)
s1∈N0,s2∈N j=1
s1+s2=s+1
µM (a ,...,a ,m,a ,...,a ,µ (a ,...,a ),a ,...,a ) = 0.
r,s1 1 r r+1 r+j−1 s2 r+j r+j+s2−1 r+j+s2 r+1
Werefertothisequationasthedefiningequationoftype(r,s) forthe A∞-bimodule M.
APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY 5
Notethatthedefiningequationoftype(0,0) isequivalenttothemap
µM := µM : M → M
0,0
beingadifferentialofdegree+1on M.
Remark1.7. Itfollowsfrom(1.3)thatA[1]isan A∞-bimoduleoverAwiththeoperations
definedby
µA[1] : A⊗r⊗ A[1]⊗ A⊗s → A[1],
r,s
A[1]
µ (a ,...,a ,a ,a ,...,a ) = µ (a ,...,a ,a ,a ,...,a ),
r,s 1 r 0 r+1 r+s r+s+1 1 r 0 r+1 r+s
forallr,s ∈ N .
0
Next we construct a slightly more sophisticatedexample which is taken from [Abo10,
Section 4]. Consider the tensor product A⊗ A of graded R-modules. In general, the
tensorproductoftwo A∞-algebrascannotbegiventhestructureofan A∞-algebra,see
[KS09,Section5.2]. Nevertheless,weshowthat A⊗ Awill alwaysadmit thestructure
ofan A∞-bimoduleover Aifweequipitwiththegrading
µ (b ⊗b ) = kb k+kb k,
A⊗A 1 2 1 2
i.e. ifweactuallyconsidertheproductgradingon A[1]⊗ A[1].
Definemapsµ⊗ by
r,s
µ⊗ : A⊗r⊗(A⊗ A)⊗A⊗s → A⊗ A, µ⊗ = 0 for r,s > 0,
r,s r,s
µ⊗ : A⊗r⊗(A⊗A) → A⊗ A,
r,0
a ⊗···⊗a ⊗(b ⊗b ) 7→ µ (a ,...,a ,b )⊗b for r > 0,
1 r 1 2 r+1 1 r 1 2
µ⊗ : (A⊗A)⊗A⊗s → A⊗ A,
0,s
(b1⊗b2)⊗a1⊗···⊗as 7→ (−1)kb1kb1⊗µs+1(b2,a1,...,as) fors > 0,
µ⊗ : A⊗A → A⊗ A, b ⊗b 7→ µ (b )⊗b +(−1)kb1kb ⊗µ (b ).
0,0 1 2 1 1 2 1 1 2
Theorem1.8([Abo10],Proposition4.7). (A⊗A,(µr⊗,s)r,s∈N0)isan A∞-bimoduleover A.
Proof. By definition, µ⊗ vanishes if both r > 0 and s > 0 and so does the left-hand
r,s
sideofthedefiningequationofan A∞-bimodule. Ifr = 0, s = 0,thedefiningequation
is equivalent to µ⊗ ◦µ⊗ = 0, which is clear from the definition of µ⊗ as a product
0,0 0,0 0,0
differential.
Assumethatr > 0and s = 0. Thenthethirdsumontheleft-hand sideofthedefining
equation of type (r,0) vanishes by definition, while for every a ,...,a ,b ,b ∈ A the
1 r 1 2
firstsumisgivenby:
∑ ∑r1 (−1)z1i−1µr⊗1,0(a1,...,ai−1,µr2(ai,...,ai+r2−1),ai+r2,...,ar,b1⊗b2)
r1∈N0,r2∈N i=1
r1+r2=r+1
= ∑ r∑1−1(−1)z1i−1µr1(a1,...,ai−1,µr2(ai,...,ai+r2−1),...,ar,b1)⊗b2 . (1.4)
r1,r2∈N i=1
r1+r2=r+2
6 APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY
The reader may notice the similarity of this sum to the left-hand side of the (r+1)-st
defining equation of the A∞-algebra A for the elements a1,...,ar,b1 ∈ A tensorized
with b , except that some parts of the defining equation are still missing. We will see
2
that thesepartsare containedin thesecondsumofthedefiningequation oftype(r,s)
whichwenextwritedownexplicitly.
∑ (−1)z1r1µr⊗1,0(a1,...,ar1,µr⊗2,0(ar1+1,...,ar,b1⊗b2))
r1,r2∈N0
r1+r2=r
= (−1)z1rµr⊗,0(a1,...,ar,µ0⊗,0(b1⊗b2))+µ0⊗,0(µr,0(a1,...,ar,b1⊗b2))
+ ∑ (−1)z1r1µr⊗1,0(a1,...,ar1,µr⊗2,0(ar1+1,...,ar,b1⊗b2))
r1,r2∈N
r1+r2=r
= (−1)z1rµr⊗,0(a1,...,ar,µ1(b1)⊗b2+(−1)kb1kb1⊗µ1(b2))
+µ⊗ (µ (a ,...,a ,b )⊗b )
0,0 r+1 1 r 1 2
+ ∑ (−1)z1r1µr1+1(a1,...,ar1,µr2+1(ar1+1,...,ar,b1))⊗b2
r1,r2∈N
r1+r2=r
= (−1)z1rµr+1(a1,...,ar,µ1(b1))⊗b2+(−1)z1r+kb1kµr+1(a1,...,ar,b1)⊗µ1(b2)
+µ1(µr+1(a1,...,ar,b1))⊗b2+(−1)kµr+1(a1,...,ar,b1)kµr+1(a1,...,ar,b1)⊗µ1(b2)
+ ∑ (−1)z1r1−1µr1(a1,...,ar1−1,µr2(ar1,...,ar,b1))⊗b2
r1,r2≥2
r1+r2=r+2
= ∑ (−1)z1r−1µr1(a1,...,ar1−1,µr2(ar1,...,ar,b1))⊗b2 (1.5)
r1≥2,r2∈N
+µ (µ (a ,...,a ,b ))⊗b ,
1 r+1 1 r 1 2
wherewehaveusedthefactthat
r
kµ (a ,...,a ,b )k = ∑ ka k+kb k+1 = zr +kb k+1,
r+1 1 r 1 q 1 1 1
q=1
since degµ = 1−r. Thus, we have shown that for A⊗ A, the left-hand side of
r+1
the defining equation for A∞-bimodules is given by the sum of (1.4) and (1.5), which
amountsto:
∑ r∑1−1(−1)z1i−1µr1(a1,...,ai−1,µr2(ai,...,ai+r2−1),ai+r2,...,ar,b1)⊗b2
r1,r2∈N i=1
r1+r2=r+2
+ ∑ (−1)z1r1−1µr1(a1,...,ar1−1,µr2(ar1,...,ar,b1))⊗b2
r1,r2∈N
r1+r2=r+2
= ∑ ∑r1 (−1)z1i−1µr1(a1,...,ai−1,µr2(ai,...,ai+r2−1),ai+r2,...,ar,b1)⊗b2 .
r1,r2∈N i=1
r1+r2=r+2
APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY 7
The latter sum equals the left-hand side of the A∞-equation for r+1 tensorized with
b2. Since Aisan A∞-algebra,thesumvanishes.
Weomittheremainingcase,r = 0ands > 0,sinceitisprovenanalogously. (cid:3)
The following theoremshows that the dual R-module of an A∞-bimodule can always
beequippedwiththestructureofan A∞-bimoduleoverthesame A∞-algebra.
Theorem 1.9 ([Tra08b], Lemma 3.9). Let M,(µrM,s)r,s∈N0 be an A∞-bimodule over R and
let M∗ = HomR(M,R) denote its dual R(cid:0)-bimodule. Then(cid:1) M−∗,(µr∗,s)r,s∈N0 is an A∞-
bimoduleover A,where M−∗ denotes M∗ withinvertedgrading, i.e.
(cid:0) (cid:1)
M−j := (M−∗)j := Hom (M ,R),
R −j
and where µ∗ : A⊗r⊗ M−∗⊗ A⊗s → M−∗ is for all r,s ∈ N , a ,...,a ∈ A, m ∈ M
r,s 0 1 r+s
andm∗ ∈ M−∗ definedby
µr∗,s(a1,...,ar,m∗,ar+1,...,ar+s) (m) = (−1)‡r,sm∗ µsM,r(ar+1,...,ar+s,m,a1,...,ar)
(cid:16) (cid:17)
(cid:0) (cid:1)
with
‡ := ‡ (a ,...,a ,m∗,a ,...,a ,m)
r,s r,s 1 r r+1 r+s
:= zr1· zrr++1s +µM∗(m∗)+µM(m) +µM∗(m∗)+1.
(cid:0) (cid:1)
Proof. (We repeat Tradler’s proof and add a sign computation since we use different
signconventions.)
We exhibit the operations µr∗,s r,s∈N0 to fulfill the A∞-bimodule equation. We needto
showforallr,s ∈ N ,a ,...,a ∈ A,m ∈ Mandm∗ ∈ M−∗ that
0 1 (cid:0) r(cid:1)+s
∑ ∑r1 (−1)z1i−1µr∗1,s(a1,...,µr2(ai,...,ai+r2−1),ai+r2,...,ar,m∗,ar+1,...,ar+s)(m)
r1+r2=r+1i=1
+ ∑ ∑ (−1)z1r1µr∗1,s1(a1,...,µr∗2,s2(ar1+1,...,ar,m∗,ar+1,...,ar+s2),...,ar+s)(m)
r1+r2=rs1+s2=s
+ ∑ ∑s1 (−1)z1r+j−1+µ(m∗)
s1+s2=s+1j=1
µ∗ (a ,...,a ,m∗,a ,...,a ,µ (a ,...,a ),a ,...,a )(m) =! 0.
r,s1 1 r r+1 r+j−1 s2 r+j r+j+s2−1 r+j+s2 r+1
8 APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY
whereµ(m∗) := µM∗(m∗). Bydefinitionoftheµr∗,s,theleft-handsideofthisequationis
givenby
∑ ∑r1 (−1)z1i−1+‡r1,s(a1,...,µr2(ai,...,ai+r2−1),...,ar,m∗,ar+1,...,ar+s,m)
r1+r2=r+1i=1
m∗ µM (a ,...,a ,m,a ,...,µ (a ,...,a ),...,a )
s,r1 r+1 r+s 1 r2 i i+r2−1 r
+ ∑ (cid:16) ∑ (−1)z1r1+‡r1,s1(a1,...,ar1,µ∗r2,s2(ar1+1,...,ar,m∗,ar+1,...,ar+s2),ar+s2(cid:17)+1,...,ar+s,m)
r1+r2=rs1+s2=s
µ∗ (a ,...,a ,m∗,a ,...,a ) µM (a ,...,a ,m,a ,...,a )
r2,s2 r1+1 r r+1 r+s2 s,r1 r+s2+1 r+s 1 r1
+ (cid:0)∑ ∑s1 (−1)z1r+j−1+µ(m∗)+‡r,s1(a1,...,ar,m∗(cid:1),a(cid:16)r+1,...,µs2(ar+j,...,ar+j+s2−1),...,ar+s,m) (cid:17)
s1+s2=s+1j=1
m∗ µM (a ,...,a ,µ (a ,...,a ),a ,...,a ,m,a ,...,a )
s1,r r+1 r+j−1 s2 r+j r+j+s2−1 r+j+s2 r+s 1 r
= ∑(cid:16) ∑r1 (−1)z1i−1+‡r1,s(a1,...,µr2(ai,...,ai+r2−1),...,ar,m∗,ar+1,...,ar+s,m) (cid:17)
r1+r2=r+1i=1
m∗ µM (a ,...,a ,m,a ,...,µ (a ,...,a ),...,a )
s,r1 r+1 r+s 1 r2 i i+r2−1 r
+ ∑ (cid:16) ∑ (cid:17)
r1+r2=rs1+s2=s
(−1)z1r1+‡r1,s1(a1,...,µ∗r2,s2(ar1+1,...,m∗,...,ar+s2),...,ar+s,m)+‡r2,s2(ar1+1,...,m∗,...,ar+s2,µsM1,r1(ar+s2+1,...,m,...,ar1))
m∗ µM (a ,...,a ,µM (a ,...,a ,m,a ,...,a ),a ,...,a )
s2,r2 r+1 r+s2 s1,r1 r+s2+1 r+s 1 r1 r1+1 r
+ ∑(cid:16) ∑s1 (−1)z1r+j−1+µ(m∗)+‡r,s1(a1,...,ar,m∗,ar+1,...,µs2(ar+j,...,ar+j+s2−1),...,ar+s,m) (cid:17)
s1+s2=s+1j=1
m∗ µM (a ,...,a ,µ (a ,...,a ),a ,...,a ,m,a ,...,a ) ,
s1,r r+1 r+j−1 s2 r+j r+j+s2−1 r+j+s2 r+s 1 r
(cid:16) (cid:17)
Clearlythissumvanishesforeverym∗ ∈ M−∗ ifandonlyif
r1
∑ ∑(−1)S1µsM,r1(ar+1,...,ar+s,m,a1,...,µr2(ai,...,ai+r2−1),...,ar)
r1+r2=r+1i=1
+ ∑ ∑ (−1)S2µsM2,r2(ar+1,...,µsM1,r1(ar+s2+1,...,ar+s,m,a1,...,ar1),...,ar)
r1+r2=rs1+s2=s
s1
+ ∑ ∑(−1)S3µsM1,r(ar+1,...,µs2(ar+j,...,ar+j+s2−1),...,ar+s,m,a1,...,ar) = 0,
s1+s2=s+1j=1
wherewedenotetheexponentsof(−1) in thepreviouscomputationby S , S and S ,
1 2 3
respectively.
APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY 9
Since (M,(µrM,s)r,s∈N0) is an A∞-bimodule over A, the latter equality holds true if we
canshowthatmodulo2,wehave
S ≡ zr+s +zi−1+µ (m)+k,
1 r+1 1 M
(1.6)
S ≡ zr+s2 +k, S ≡ zr+j−1+k,
2 r+1 3 r+1
for some k ∈ Z. In this case, multiplying the desired equation with (−1)k yields the
definingequationoftype(r,s)forthe A∞-bimodule M. Soitonlyremainstocheckthe
signs.
ConcerningS ,wecomputethat
1
‡ (a ,...,µ (a ,...,a ),...,a ,m∗,a ,...,a ,m)
r1,s 1 r1 i i+r1−1 r r+1 r+s
i−1 r
= ∑ ka k+kµ (a ,...,a )k+ ∑ ka k zr+s +µ(m∗)+µ (m)
q r2 i i+r2−1 q r+1 M
(cid:16)q=1 q=i+r2 (cid:17)
(cid:0) (cid:1)
+µ(m∗)+1
≡ (zr +1) zr+s +µ(m∗)+µ (m) +µ(m∗)+1
1 r+1 M
≡ zrr++1s +µ(cid:0)M(m)+µ(m∗)+‡r,s(a1,.(cid:1)..,ar,m∗,ar+1,...,ar+s,m)
andthereforebydefinitionofS :
1
S ≡ zr+s +zi−1+µ (m)+k
1 r+1 1 M 0
if we put k := µ(m∗)+‡ . This shows the first line of (1.6). Considering the sign
0 r,s
givenbyS ,wefirstcompute
2
‡ (a ,...,a ,µ∗ (a ,...,a ,m∗,a ,...,a ),a ,...,a ,m)
r1,s1 1 r1 r2,s2 r1+1 r r+1 r+s2 r+s2+1 r+s
≡ zr1 zr+s +µ µ∗ (a ,...,a ,m∗,a ,...,a ) +µ (m)
1 r+s2+1 r2,s2 r1+1 r r+1 r+s2 M
(cid:16) +µ µ∗(cid:0) (a ,...,a ,m∗,a ,...,a ) +(cid:1)1 (cid:17)
r2,s2 r1+1 r r+1 r+s2
≡ zr1 zr+s +µ(cid:0)(m∗)+µ (m)+1 +zr+s2 +µ(m∗)(cid:1)
1 r1+1 M r1+1
(cid:16) (cid:17)
≡ zr+s2 +zr1 zr+s +µ(m∗)+µ (m) +µ(m∗).
1 1 r1+1 M
(cid:16) (cid:17)
Furthermore,noticethat
‡ (a ,...,a ,m∗,a ,...,a ,µM (a ,...,a ,m,a ,...,a ))
r2,s2 r1+1 r r+1 r+s2 s1,r1 r+s2+1 r+s 1 r1
≡ zr zr+s2 +µ(m∗)+µ µM (a ,...,a ,m,a ,...,a ) +µ(m∗)+1
r1+1 r+1 M s1,r1 r+s2+1 r+s 1 r1
≡ zr (cid:16)zr+s +zr1 +µ(m∗)+(cid:16)µ (m)+1 +µ(m∗)+1 (cid:17)(cid:17)
r1+1 r+1 1 M
≡ zrr1+1(cid:0)+zr11zrr1+1+zrr1+1 zrr++1s +µ(m∗)(cid:1)+µM(m) +µ(m∗)+1.
Combiningtheselasttwocomp(cid:0)utations,thedefinitiono(cid:1)fS2 yields
S ≡zr1 +zr+s2 +zr +zr1 zr+s +µ(m∗)+µ (m) +zr1zr
2 1 1 r1+1 1 r1+1 M 1 r1+1
+zr zr+s +µ(cid:16)(m∗)+µ (m) +1 (cid:17)
r1+1 r+1 M
≡zrr++1s2 +zr1 zrr++1s(cid:0)+µ(m∗)+µM(m) +1 ≡(cid:1) zrr++1s2 +k0 ,
(cid:0) (cid:1)
10 APRIMERON A∞-ALGEBRASANDTHEIRHOCHSCHILDHOMOLOGY
implyingthefirstcongruencefromthesecondlineof(1.6). Finally,forS weconsider:
3
‡ (a ,...,a ,m∗,a ,...,µ (a ,...,a ),...,a ,m)
r,s1 1 r r+1 s2 r+j r+j+s2−1 r+s
≡ zr zr+j−1+kµ (a ,...,a )k+zr+s +µ(m∗)+µ (m) +µ(m∗)+1
1 r+1 s2 r+j r+j+s2−1 r+j+s2 M
≡ zr (cid:16)zr+s +1+µ(m∗)+µ (m) +µ(m∗)+1 ≡ zr +µ(m∗)+k . (cid:17)
1 r+1 M 1 0
Wecon(cid:0)sequentlyobtainthatS ≡ z(cid:1)r+j−1+k . Thus,wehaveshownallthreeequations
3 r+1 0
of(1.6)withk = k andthereforecompletedtheproof. (cid:3)
0
Weconcludethissectionbydefiningthenotionofmorphismsof A∞-bimodules.
Definition 1.10. Let (M,(µrM,s)r,s∈N0) and (N,(µrN,s)r,s∈N0) be A∞-bimodules over A. A
familyofmodulehomomorphisms f = (fr,s)r,s∈N0,where:
f : A⊗r⊗M⊗A⊗s → N ,
r,s
willbecalledamorphismofA∞-bimodulesfromMtoNofdegreed,denotedby f : M → N,
ofdegreed ∈ Z ifehave
deg f = d−r−s
r,s
forallr,s ∈ N andif
0
∑ ∑ (−1)d·z1r1
r1,r2∈N0 s1,s2∈N0
r1+r2=rs1+s2=s
µN (a ,...,a , f (a ,...,a ,m,a ,...,a ),a ,...,a )
r1,s1 1 r1 r2,s2 r1+1 r r+1 r+s1 r+s1+1 r+s
= ∑ ∑r1 (−1)z1i−1+d
r1,r2∈N i=1
r1+r2=r+1
f (a ,...,a ,µ (a ,...,a ),a ,...,a ,m,a ,...,a )
r1,s 1 i−1 r2 i i+r2−1 i+r2 r r+1 r+s
+ ∑ ∑ (−1)z1r1+d
r1,r2∈N0 s1,s2∈N0
r1+r2=rs1+s2=s
f (a ,...,a ,µM (a ,...,a ,m,a ,...,a ),a ,...,a )
r1,s1 1 r1 r2,s2 r1+1 r r+1 r+s2 r+s2+1 r+s
+ ∑ ∑s1 (−1)z1r+i−1+µ(m)+d
s1,s2∈N i=1
s1+s2=s+1
f (a ,...,a ,m,a ,...,a ,µ (a ,...,a ),a ,...,a )
r,s1 1 r r+1 r+i−1 s2 r+i r+i+s2−1 r+i+s2 r+s
forall a ,...,a ∈ Aandm ∈ M. Wedenote f by f : M → N.
1 r+s
Remark 1.11. The sign coefficients appearing in the defining equations of a morphism
of A∞-bimodules might differ from those used other definitions in the literature, e.g.
in [Tra08b, Section 4]. However, our choice of signs implies that a morphism of A∞-
bimodules induces a map between the corresponding Hochschild chain complexes
whichwillbethecontentofTheorem2.8inthenextsection.