Massey products for associative algebras [expository notes]

11 Pages·2011·0.319 MB·English

by  Matthew Towers

MASSEY PRODUCTS FOR ASSOCIATIVE ALGEBRAS
MATTHEWTOWERS
Abstract. Rough notes on how to define and compute Massey products in
thecohomologyofanassociativealgebra,andtheconnectionstotheBockstein
andtotheexistenceofcertainuniserialmodules.
1. Introduction
Massey products are certain higher (i.e. taking two or more arguments) prod-
ucts on the cohomology of a differential graded algebra (DGA). References include
[May69, McC01, Kra66]...While the two-fold product is (up to a sign) just the
ordinary cohomology product induced by that of the DGA, higher products are
not always defined, and don’t always have a unique value. Nevertheless they have
many applications, of which we will cover:
• the connection between a certain p-fold restricted product and the Bock-
stein homomorphism (Section 5), and
• the connection between Massey products and the existence of certain unis-
erial modules (Section 6).
ThevariousconstructionssurroundingtheMasseyproductpresentmanyoppor-
tunitiesformakingsignerrors; Ihavenodoubtmadeuseofseveralofthem. Ihave
chosen the sign conventions in the simplest way I could — this means my Massey
products may differ by sign from those defined by other authors, but since there
are almost as many sign conventions as there are papers on the Massey product
this does not significantly increase the disorder in the literature.
2. Defining the Massey triple product
A graded algebra is a algebra A equipped with a direct sum decomposition
(cid:76)
A = A such that if a ∈ A ,b ∈ A then ab ∈ A . Elements of A are
n∈Z n n m n+m n
called homogeneous of degree n.
Definition 2.1. A differential graded algebra D,d is a graded algebra D equipped
with a linear map d : D → D graded of degree +1 such that d◦d = 0 and for all
homogeneous elements x,y ∈D we have
d(xy)=d(x)y+(−1)|x|xd(y)
where |x| denotes the degree of x.
Unless stated otherwise we assume that the multiplication on D is strictly asso-
ciative. We’ll sometimes speak of non-associative DGAs, which should be assumed
to be defined as above only with the associativity condition dropped.
The cohomology H∗(D)=(cid:76)kerd/imd of an associative DGA D,d becomes a
graded associative algebra under the product induced from D.
Definition 2.2. LetD,dbeaDGAandα,β,γ behomogeneouselementsofH∗(D)
with cocycle representatives c(α),c(β),c(γ) in kerd⊂D. Suppose c(α)c(β)=d(φ)
and c(β)c(γ)=d(ψ). Then
(−1)|β|+1(φ·c(γ)+(−1)|α|+1c(α)·ψ)
1
2 MATTHEWTOWERS
is a cocycle. The Massey product (cid:104)α,β,γ(cid:105) is defined to be the set of all cohomology
classes of cocycles arising in this way (for different choices of φ and ψ).
Thereaderisentitledtobemystifiedbythe(−1)|β|+1. Whatisitdoingthere?
The reason it appears is to make this triple product a special case of the higher
products defined later on.
Theproduct(cid:104)α,β,γ(cid:105)isasubsetofH|α|+|β|+|γ|−1(D). We’llblurthedistinction
between a one-element set and its element.
Lemma 2.3. Varying the choice of representative cocycles c(α),c(β),c(γ) does not
change the Massey product (cid:104)α,β,γ(cid:105). All elements of (cid:104)α,β,γ(cid:105) lie in the same coset
of αH∗(D)+H∗(D)γ.
Proof. Direct computation. (cid:3)
Definition 2.4. Let D,d be a DGA and α be an element of H∗(D). Then the
restricted product (cid:104)α(cid:105)3 is defined in the same way as (cid:104)α,α,α(cid:105) except that we insist
that φ=ψ in the notation above.
This restricted product is a subset of the Massey product
Proposition 2.5. Suppose H∗(D) is graded commutative, in the sense that for
homogeneous x,y we have xy = (−1)|x||y|yx. Then when it exists the restricted
product contains a unique cohomology class.
This is easy to check. The hypothesis holds in many contexts: for example the
group cohomology Ext∗ (k,k), the cohomology ring Ext∗(k,k) for a Hopf algebra
kG Λ
Λ, and the Hochschild cohomology ring HH(Λ) of a k-algebra Λ are all graded
commutative.
2.1. Homotopy associative DGAs. One can define Massey products on non-
associative DGAs, so long as they are associative up to homotopy.
Definition 2.6. Let D,d be a non-associative DGA with multipliation map m :
D⊗ D →D. Then D,d is said to be homotopy associative if the maps (1⊗m)◦m
k
and (m⊗1)◦m are homotopic as maps D⊗3 →D.
What this means is that in a homotopy associative DGA D,d there’s a map
θ :D⊗3 →D, graded of degree −1 (we take d to have degree +1), such that
(1⊗m)◦m−(m⊗1)◦m=d◦θ−θ◦d⊗3
as maps D⊗3 →D.
Note that the cohomology algebra of a homotopy associative algebra is strictly
associative.
MasseyproductscanbedefinedonahomotopyassociativeDGAD,dwithmul-
tiplication m (abbreviated by ·) and homotopy θ as follows.
Definition2.7. Supposeα,β,γ arehomogeneouscohomologyelementswithcocycle
representatives c(α),c(β),c(γ) such that c(α)c(β) = d(φ) and c(β)c(γ) = d(ψ),
whereφ,ψ ∈D. ThentheMasseyproduct(cid:104)α,β,γ(cid:105)isthesetofcohomologyelements
represented by cocycles of the form
(−1)|β|+1(φ·c(γ)+(−1)|α|+1c(α)·ψ−(−1)|α|+|β|+|γ|θ(c(α)⊗c(β)⊗c(γ)))
for different choices of φ,ψ.
We recover the original definition for an associative DGA by setting θ =0. This
definition is abstracted from [Ben98b, §3.2].
MASSEY PRODUCTS FOR ASSOCIATIVE ALGEBRAS 3
3. Defining the Massey product for associative algebras
SincewewanttoworkwithMasseyproductsonthecohomologyofanassociative
algebra Λ, we must manufacture DGAs whose cohomology is the cohomology of Λ.
Fix a k-algebra Λ and a Λ-module M. If (P ,d ) is a projective resolution of M
∗ ∗
(our convention being that d :P →P and d :P →M) there’s no reason to
n n n−1 0 0
expect an interesting DGA stucture on (Hom (P ,M),δ) where δ is induced by d.
Λ ∗
We therefore consider the Yoneda cocomplex:
Hom(∗)(P ,P )=(cid:77)Hom(i)(P ,P ).
Λ ∗ ∗ Λ ∗ ∗
i∈Z
Hom(i)(P ,P ) is defined to be the set of all sequences φ of homomorphisms
Λ ∗ ∗ ∗
(φ ,φ ,φ ,...) with φ : P → P . Here P is to be interpreted as M: this
−1 0 1 i i r−i −1
must be included if we want negative degree cohomology to be independent of the
choice of resolution.
The differential on the Yoneda cocomplex is as follows: if φ has degree i then
∗
(∂φ) has components
∗
(∂φ) =φ ◦d +(−1)i+1d ◦φ :P →P .
n n−1 n n−i n n n−i−1
φ with r <−1 is to be interpreted as zero. The (−1)i+1 is there to make ∂2 =0.
r
Thedifferential∂,togetherwithmultiplicationgivenbycompositionofmaps,make
the Yoneda cocomplex into a DGA.
Thedefinitionsaresetupsothatchainhomomorphismsareexactlythecochains
intheYonedacomplex,andtwochainhomomorphismsarehomotopicexactlywhen
their difference is a boundary
There is an isomorphism of k-algebras (without unit)
H>0(Hom(∗)(P ,P ),∂)∼=(cid:77)Exti (M,M).
Λ ∗ ∗ Λ
i>0
DroppingM fromthechaincomplexwhoseendomorphismsweworkwouldallow
ustoextendtheaboveisomorphismtodegreezero. Thedisadvantageofdoingthis
is that the cohomology of the resulting endomorphism cocomplex in degree n < 0
depends on the resolution chosen: for n<0 we would get Hom (M,imd ).
Λ n
Ext0(M,M) is Hom (M,M), but zeroth degree cohomology of the Yoneda co-
Λ Λ
complex is smaller: it is isomorphic to the quotient of End (M) by the space
Λ
of homomorphisms factoring through the projective cover of M. This sort of in-
terpretation works more generally. For i > 0 we can interpret Exti (M,M) as
Λ
Hom (imd ,M) modulo those homomorphisms factoring through the inclusion
Λ i
imd (cid:44)→P (if M is semisimple and P is minimal then all such maps are zero).
i i−1 ∗
For i < 0, the degree i cohomology of the Yoneda cocomplex is isomorphic to
Hom (M,imd ) modulo the homomorphisms that factor through P (cid:16) imd
Λ −i −i −i
(even if P is a minimal resolution and M is semisimple, these can be non-zero).
∗
Both of these quotient spaces are independent of the choice of resolution.
WriteΩr forimd . Givenf :M →Ωn andg :Ωn →M representingelementsf¯
r
and g¯ of negative and postivive degree cohomology respectively (n,m>0), a map
representingtheproductg¯f¯isg◦Ωm−nf ifm−n≥0andΩn−mg◦f ifn−m>0.
Nosuchconstructionexistsforf¯g¯;theseproductsmustbetakentobezero. Thisis
anartifactofthehalf-infinitenessofourprojectiveresolutions(morecouldbedone
in the self-injective case, where there is a functor Ω−1 inverse to Ω in the stable
category).
∗
Definition 3.1. The complete cohomology ring E(cid:100)xt (M,M) is defined to be the
Λ
cohomology of the Yoneda cocomplex above.
4 MATTHEWTOWERS
ThisagreeswiththeordinaryExt-algebrainstrictlypositivedegrees. Dimension
n n+1
shifting works as follows: for n ≥ 0 we have E(cid:100)xt (Ω(M),N) = E(cid:100)xt (M,N)
Λ Λ
−n n−1
and E(cid:100)xt (Ω(M),N) = E(cid:100)xt (M,N). Note that for ordinary Ext dimension
Λ
shifting fails in degree zero: Ext0(Ω(M),N) = Hom (Ω(M),N) which is larger
Λ Λ
than Ext1(M,N) in general.
∗
Example 3.2. Let Λ=k[x]/xn, where n>2. Then E(cid:100)xt (k,k) is
Λ
k(cid:104)ξ ,ξ ,η(cid:105)/(η2,ξ ξ ,ξ ξ =1,ηξ =ξ η,ηξ =ξ η)
+ − − + + − + + − −
where ξ and ξ have degree ±2, and η has degree 1.
+ −
Remark 3.3. Complete cohomology is not in general the same thing as Tate co-
homology [Ben98a, §5.15], which is defined as follows: let G be a finite group, k a
field, and fix a projective resolution P of the trivial module k. Let P = P∗,
∗ −n−1 n
the Hopf dual, for n>0, so that there is an exact complex
···→P →P →P →P =P∗ →P =P∗ →···
2 1 0 −1 0 −2 1
called P . Then the nth Tate cohomology group of kG with coefficients in the left
±
kG-module M, for n∈Z, is the nth cohomology of Hom (P ,M). If n>0 then
kG ±
the nth Tate cohomology group is isomorphic to Extn (k,M). The zeroth Tate
kG
cohomology is Hom (k,M) modulo those maps factoring through k (cid:44)→ kG. If
kG
n < 0 then the nth Tate cohomology group is isomorphic to TorkG (M,k) where
−n−1
M is regarded as a right kG-module by m·g :=g−1m. I suppose k can be replaced
by an arbitrary self-dual module in this set-up.
Lemma 3.4. Let G be a finite group and k a field. For n < 0, E(cid:100)xtn (M,M) ∼=
kG
TorkG (DM,M) where DM is the right kG-module Hom (M,k) (not the Hopf
−n−1 k
dual which takes left modules to left modules)
Proof. Prove that for left kG-modules A,B we have
Hom (A,B)∼=A∗⊗ B
kG kG
where A∗ is the Hopf dual (a left module) and on the right hand side, A is a right
kG module via a·g :=g−1a. Apply this to the cocomplex Hom (M,P ), noting
kG n
that a cocycle M →P in this cocomplex corresponds to a kG-map M →imd
n−1 n
which represents an element of the complete cohomology. (cid:3)
Remark 3.5. Compare the definition of Gorenstein projectivity: M is Gorenstein
projective if it has a complete projective resolution , that is, an exact com-
plex (Pn,dn)n∈Z of projectives such that M ∼= imd−1 and for each projective Q,
Hom (Q,P ) is exact. See [Zha] and the references cited there.
Λ ∗
For special resolutions we can get a DGA structure directly on Hom (P ,M).
Λ ∗
Let Λ be an augmented k-algebra, so that there’s a trivial module k, and consider
the ‘bar resolution’. By bar resolution I mean the ‘standard complex’ of [CE56,
p.174] tensored over Λ with k so as to produce a projective resolution of k. If Λ is
a group algebra then up to a change of notation this is the same as the usual bar
resolution. IfP ,disthebarresolutionandf :P →k andg :P →k onedefines
∗ m n
(f ·g):P →k by
n+m
(f ·g)[x |···|x ]=(−1)mnf([x |···|x ])g([x |···|x ]).
1 n+m 1 m m+1 n+m
This makes the cobar cocomplex (Hom (P ,k),−◦d) into a DGA (it really is
Λ ∗
associative: the sign appearing in f ·(g·h) and (f ·g)·h is (−1)mn+nl+lm). The
sign (−1)mn here is to make the map s in Lemma 3.14 a homomorphism from the
cobar cocomplex to the Yoneda cocomplex on the bar resolution.
MASSEY PRODUCTS FOR ASSOCIATIVE ALGEBRAS 5
There’s a DGA structure on the cocomplex arising from the Hochschild com-
plex/standard resolution used for computing the Hochschild cohomology HH(Λ),
which works exactly as for the bar resolution.
There follow two examples, both designedto illustrate the folklore theorem that
for a group algebra kG where k has characteristic p, the p-fold restricted Massey
product is the same as the Bockstein (up to choice of sign convention). Recall that
if G is cyclic of order p>2 and k has characteristic p then
Ext∗ (k,k)∼=k{x ,y }
kG 1 2
where subscripts denote degrees, and k{S} is the free graded commutative algebra
on a set S whose elements have specified degrees. The Bockstein takes x to y .
1 2
Example 3.6. Let G=(cid:104)g(cid:105) be cyclic of order 3. We compute (cid:104)x (cid:105)3 using the DGA
1
structure on Hom (P ,k) where P is the bar resolution. Recall that
kG ∗ ∗
(cid:77) (cid:77)
P =kG[],P = kG[gi],P = kG[gi|gj]
0 1 2
i i,j
with differential d ([gi])=(gi−1)[] and d ([gi|gj])=gi[gj]−[gi+j]+[gi]. Repre-
1 2
sentative cocycles for x and y are c(x ):[gi](cid:55)→i and
1 2 1
(cid:26) 1 ¯i+¯j ≥3
c(y ):[gi|gj](cid:55)→
2 0 otherwise
where¯i,¯j are the canonical representatives of i,j in Z.
The product c(x )c(x ) is [gi|gj](cid:55)→ij, which is φ◦d where φ([gi])=i2 (because
1 1 2
j2−(i+j)2+i2 =ij mod 3). φα+αφis[gi|gj](cid:55)→ij(i+j). Thenφα+αφ+c(y )=
2
d ◦f where f([1])=0=f([g]),f([g2])=1. It follows (cid:104)x (cid:105)3 contains −y .
2 1 2
Remark 3.7. More generally, if G is a finite group and f :G→k is a homomor-
phism, so that [g] (cid:55)→ f(g) is a cocycle representing a cohomology element ξ, then
[g|h](cid:55)→f(g)f(h) is equal to d(φ) where φ([g])=(−1/2)f(g)2. The restricted triple
product (cid:104)ξ(cid:105)3 is represented by the cocycle [g|h](cid:55)→(−1/2)f(g)f(h)(f(g)+f(h)).
Here we had essentially no choice for φ so both the restricted product and the
Massey triple product are equal to −y .
2
WenowrepeatthesamecalculationusingtheYonedacocomplexoftheminimal
resolution.
Example 3.8. Identify kG with k[x]/x3. The minimal resolution P of k consists
∗
of a single copy of kG in each degree with odd differentials given by multiplication
by x and even differentials multiplication by x2. Cocycles representing x and y
1 2
are the augmentation maps on P and P . We must lift the degree one cocycle on
1 2
P to a chain map (i.e. cocycle in the Yoneda cocomplex): we take c(x ) to be
∗ 1
the map of degree one which is multiplication by x in even degrees, and 1 in odd
degrees. Then c(x )◦c(x ) is the degree two chain map given by multiplication by
1 1
x. This is ∂φ where φ is the degree one map given by 1 in even degrees and x in
odd degrees. We see that φα+αφ is a chain map lifting minus the augmentation
P →k, which is a cocycle representing y . It follows (cid:104)x (cid:105)3 contains −y .
2 2 1 2
OurdefinitionoftheMasseyproductonthecohomologyofahomotopyassocia-
tive DGA can also be used with Hopf algebras.
Example 3.9. Let Λ be a Hopf algebra and P a projective resolution of the trivial
∗
module k. The total complex P ⊗ P ,d⊗2 resolves k⊗k ∼= k, so there’s a chain
∗ k ∗
map H : P → P ⊗ P lifting the obvious isomorphism k → k⊗k. Both (H ⊗
∗ ∗ k ∗
1)◦H and (1⊗H)◦H are chain maps P⊗3 → P lifting k → k ⊗k ⊗k, they
∗ ∗
are therefore homotopic via a chain map Θ . This makes P into a homotopy
∗ ∗
6 MATTHEWTOWERS
associative coalgebra, and (Hom (P ,k),δ) into a homotopy associative DGA by
Λ ∗
defining m(f ⊗g) = (f ⊗g)◦H. Here δ is the codifferential induced by d. If θ is
the map given by composition on the right with Θ, we have
(1⊗m)◦m−(m⊗1)◦m=δ◦θ−θ◦δ⊗3
and so m is homotopy associative.
If P is the minimal resolution, the first two terms drop out and the Massey
∗
product depends only on θ — so if all Massey products are represented by zero,
then there’s a strictly coassociative comultiplication on the minimal resolution.
Remark 3.10. A module with differentiation (e.g. a projective resolution) for a
Hopf k-algebra Λ is the same thing as a module over the ring of dual numbers
Λ⊗ k[x]/x2 [CE56, p.56,IV§2]. This is still a Hopf algebra, and the comultiplica-
k
tion sends x to x⊗1+1⊗x. This explains why the differential on a tensor product
of complexes is defined the way it is.
3.1. Well-definedness. We now have several ways of defining the Massey triple
product on Ext∗(M,M). We must show they are all the same.
Λ
Lemma 3.11. The Massey product defined using the Yoneda cocomplex is inde-
pendent of the projective resolution used.
Proof. Let P and Q be projective resolutions of M. Let H and K be chain
∗ ∗
maps P → Q and Q → P of degree zero lifting the identity on M. Then
∗ ∗ ∗ ∗
HK is homotopic to id and KH is homotopic to id , so that there are elements
Q P
x and y of degree −1 in the Yoneda cocomplex such that id −d(x) = HK and
Q
id −d(y)=KH. Define a map
P
Hom∗(P ,P )→Hom∗(Q ,Q )
Λ ∗ ∗ Λ ∗ ∗
by f (cid:55)→ f¯= H ◦f ◦K. This map commutes with the differential, and induces an
isomorphism of the cohomology of the two Yoneda cocomplexes (observe that, if
φ is a cocycle, then KHφKH differs from φ by a coboundary). We need to show
∗
that this isomorphism ‘commutes with’ the Massey triple product. Let α,β,γ be
cohomology elements represented by cocycles c(α),c(β),c(γ) in Hom∗(P ,P ), so
Λ ∗ ∗
that we need
(cid:104)c(α),c(β),c(γ)(cid:105)=(cid:104)c(α),c(β),c(γ)(cid:105).
There’s a lot of abuse of notation here, hopefully the meaning is clear. This can be
proved by direct computation, though it is quite a mess. The crucial identity is
a¯¯b−ab=−ad(y)b
for a,b∈Hom∗(P ,P ). (cid:3)
Λ ∗ ∗
Remark3.12. Computingisthewrongmethodofproofhere—thehigherproducts
would be even worse. How should it be done?
Therearethreeotherthingstocheck: thattheHopfalgebradefinitionisresolution-
independent, that it agrees with the Yoneda definition, and that the bar resolution
definition is the same as the Yoneda definition.
Lemma 3.13. Let σ :D,d→E,e be an injective morphism of DGAs inducing an
injective map σ¯ :H(D)→H(E) on cohomology. Then if α,β,γ ∈H(D) are such
that their Massey triple product exists, we have
σ¯(cid:104)α,β,γ(cid:105)=(cid:104)σ¯(α),σ¯(β),σ¯(γ)(cid:105)
MASSEY PRODUCTS FOR ASSOCIATIVE ALGEBRAS 7
Proof. ⊆ is clear. For the other inclusion, choose representatives c(α),c(β),c(γ) of
α,β,γ so that σ¯(α) can be represented by σ(c(α)) etc. If
σ(c(α))σ(c(β))=e(x)
then e(x)∈imσ, so e(x)=σ(y) for a unique y ∈D. Since σ¯(y)=0 in H(E) and
σ¯ is an isomorphism in cohomology, y = d(z) for some z ∈ D. This argument is
enough to give the reverse inclusion. (cid:3)
Lemma 3.14. Let Λ be an augmented algebra. Then the Massey product structure
on Ext∗(k,k) is independent of whether it is calculated using the cobar cocomplex
Λ
or the Yoneda cocomplex.
Proof. Let C,d be the DGA arising from the cobar cocomplex and Y,δ the Yoneda
cocomplex on the bar resolution. Define a map s:C →Y by
s(f)([x |···|x ])=(−1)|f|(r+1)f([x |···|x ])[x |···|x ]
1 r r−n+1 r 1 r
ThisisamonomorphismofDGAs(thatis,sismultiplicativeandδ◦s=s◦d)which
induces an isomorphism on cohomology. To see that it is injective let t : Y → C
be the linear map sending φ to ∂ ◦φ : B → k (B ,∂ is the bar resolution);
∗ 0 0 |φ| ∗
then t◦s=id . Note that t isn’t multiplicative, though it does commute with the
C
differentials. The result now follows from Lemma 3.13. (cid:3)
4. Higher Massey products
Let D,d be a DGA, and for homogeneous c define c˜ to be (−1)|c|+1c. Let
α ,...,α be homogeneous cohomology elements; we wish to define the Massey
1 m−1
product (cid:104)α ,...,α (cid:105). For this we need the existence of a defining system [c ]
1 m−1 ij
for α ,...,α , that is, a collection of elements c of D, for m ≥ j > i ≥ 1 and
1 m−1 ij
(i,j)(cid:54)=(1,m) such that
• c is a cocycle representing α , and
i,i+1 i
• d(c )=(cid:80)j−1 c˜ c for all i<j except i=1,j =m.
ij k=i+1 ik kj
The value of this defining system is
m−1
(cid:88)
c˜ c ,
1k km
k=2
which is a cocycle (use |c |+1=|c |+|c | for any i<k <j).
ij ik kj
The defining system can be thought of as a m×m strictly upper triangular
matrix over D with 1,m entry equal to zero. Define the rth superdiagonal to be
the sequence of entries of the form c . The entries in the rth superdiagonal
i,i+r
witness that the (r+1)-fold sub Massey products — that is, Massey products of
the form (cid:104)α ,α ,...,α (cid:105) — contain zero. This is clearly a necessary condition
i i+1 i+r
for the existence of a defining system for the α s.
i
Definition4.1. TheMasseyproduct(cid:104)α ,...,α (cid:105)isthesetofcohomologyclasses
1 m−1
represented by the values of all possible defining systems for α ,...,α .
1 m−1
TheMasseyproductisasubsetofH(cid:80)|αi|+3−m(D)(because|c1k|=|c12|+···+
|c |+2−k and |c |=|c |+···+|c |+k+1−m). When m=3 it
k−1,k km k,k+1 m−1,m
differs from the usual product on H∗(D) by a sign, and when m=4 it is the same
as the triple product defined earlier.
The formulas involved have obvious similarities to matrix multiplication. Take
a strictly upper triangular matrix C =[c ] of elements of D. The condition for it
ij
to be a defining system is, up to signs, that C˜C =d(C) except possibly in the top
right hand corner, where a cocycle representing an element of the Massey product
8 MATTHEWTOWERS
appears. This is noted by May in [May69]. Our Massey products are the special
case of his matric Massey products when each of his matrices V is 1×1.
i
There is a general version of our definition of the restricted triple product:
Definition 4.2. The restricted product (cid:104)α(cid:105)n is defined as the set of cohomology
elements represented by the values of all defining systems for (cid:104)α,...,α(cid:105) (where α
appears n times) in which all superdiagonals are constant.
5. Massey products and the Bockstein
Reduction mod p, written γ :ZG→F G, is a ring homomorphism. The associ-
p
ated change of rings map
ExtnFpG(FpG⊗ZGZ∼=Fp,Fp)→ExtnZG(Z,k)
is an isomorphism, even though F G isn’t a flat ZG-module, because if P is a ZG
p ∗
projective resolution of Z then FpG⊗ZG P∗ is a FpG projective resolution of Fp.
This means we can identify Ext∗ (F ,F ) with Ext∗ (Z,F ).
FpG p p ZG p
There’s also a change of rings map Extn (F ,F )→Extn (F ,F ), but it isn’t
FpG p p ZG p p
onto: inparticularindegreeoneitdoesn’thittheelementrepresentedbytheshort
exact sequence
(1) 0→F →Z/p2 →F →0.
p p
Applying HomZG(Z,−) gives a long exact sequence in which the connecting
homomorphism is the Yoneda splice with (1).
Definition 5.1. The Bockstein β : Extn (F ,F ) → Extn+1(F ,F ) is the map
FpG p p FpG p p
obtained by identifying Ext∗ (F ,F ) with Ext∗ (Z,k) then multiplying by (1).
FpG p p ZG
Remark 5.2. The Bockstein, like any connecting homomorphism, commutes with
change of rings maps — in particular the usual restriction and inflation maps in
group cohomology.
The Bockstein is very easy to describe in degree one: the degree two cocycle in
Example 3.6 is a typical example of a cocycle representing such a Bockstein. If
f :G→F is a group homomorphism then it gives rise to a degree one cocycle on
p
the bar resolution f([g])=f(g) (all degree one cocycles arise this way).
Lemma 5.3. The Bockstein of the degree one cohomology element arising from a
homomorphism f :G→F can be represented by
p
(cid:26)
1 f(g)+f(h)≥p
[g|h](cid:55)→
0 otherwise
wherethebardenotesthecanonicalrepresentativeofanelementofF in{0,1,...,p−
p
1}⊂Z.
In [Kra66] Kraines proves by an explicit calculation that in characteristic p,
the p-fold restricted product of a degree one cohomology element is equal to its
Bockstein. Let the degree one cohomology element arise from a homomorphism
f :G→F . On the first superdiagonal Kraines has [g](cid:55)→f(g), and on the nth he
p
puts
(cid:18) (cid:19)
f(g)
[g](cid:55)→
n
If the binomial coefficient doesn’t make sense because the “numerator” is smaller
than the “denominator” then it is to be interpreted as zero. In any case it is
to be regarded as an element of F in the obvious way. Up to cocycles, that is
p
MASSEY PRODUCTS FOR ASSOCIATIVE ALGEBRAS 9
group homomorphisms, and signs Kraines’ (1/2)f(g)(f(g) − 1) agrees with our
(−1/2)f(g)2. The value of this defining system is
p (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:26)
(cid:88) f(g) f(h) f(g)+f(h) 1 f(g)+f(h)≥p
[g|h](cid:55)→ = =
r p−r p 0 otherwise
r=0
using the binomial identity
(cid:18) (cid:19) n (cid:18) (cid:19)(cid:18) (cid:19)
i+j (cid:88) i j
= .
n r n−r
r=0
This represents the Bockstein of the cohomology element represented by f.
Remark 5.4. Is there a more conceptual proof that the Bockstein and the p-fold
restrictedproductofadegreeoneelementareequal? Whataboutrestrictedproducts
in higher degrees — these correspond to the Bockstein composed with a certain
Steenrod pth power operation. Why?
6. Massey products and uniserial modules
I learned about the connection from [BC87, p.59].
Fix simple modules S for some algebra Λ and elements α ∈Ext1(S ,S ). A
i i Λ i+1 i
uniserial module is said to be of type (S ,...,S ;α ,...,α ) if its composition
1 n 1 n−1
seriesisS ,...,S (startingwiththesocle),andeachsubquotientmodulewithtop
1 n
S and socle S fits into a short exact sequence representing α .
i+1 i i
LetS,T,U besimplemodulesforsomealgebraΛ,andletη ∈Ext1(S,T),η ∈
ST Λ TU
Ext1(T,U). Then there exists a uniserial module M of type (U,T,S;η ,η ) if
Λ TU ST
and only if η η =0∈Ext2(S,U).
TU ST Λ
What about larger modules? Then Massey products come in to play. We have
the following generalization of the above result:
Let Λ be an algebra with simple modules S ,...,S and corresponding repre-
1 n
sentations ρ : Λ → End (S ). If we wish to define a module of length two with
i k i
composition factors S ,S , the representation should be
1 2
(cid:18) (cid:19)
ρ (λ) ρ (λ)
λ(cid:55)→ 1 21
0 ρ (λ)
2
where ρ ∈ Hom (Λ,Hom (S ,S )). The condition on ρ for this to be a rep-
21 k k 2 1 21
resentation is that it should be a derivation when hom (S ,S ) is regarded as a
k 2 1
Λ bimodule in the obvious way, in other words, that ρ is a 1-cocycle from the
21
Hochschild complex for Λ to Hom (S ,S ).
k 2 1
This suggests looking at the Hochschild cocomplex C∗(Λ,E) = Hom (Λ⊗∗,E)
k
where E =End (⊕S ), which is a DGA in the usual bar resolution way. The coho-
k i
mology of this cocomplex computes the Hochschild cohomology Ext∗ (Λ,E). The
Λe
Masseyproductsinthefollowingpropositionaretobecomputedinthiscocomplex.
Proposition 6.1. If M is a module with composition factors T ,...,T , where
1 N
each T is identified with one of the S , and corresponding representation
i j
 ρ ρ ρ ... ... ρ 
1 21 31 n1
 0 ρ2 ρ32 ... ... ρn2 
 . 
 . 
 0 0 ρ3 ... . 
(2) ρ= ... ... ... 
 
 .. 
 . ... 0 ρn−1 ρn,n−1 
0 0 ... ... ... ρ
n
10 MATTHEWTOWERS
then the matrix
 0 ρ ρ ... ρ 0 
21 31 n−1,1
 0 0 ρ32 ... ... ρn2 
 . 
 . 
 0 0 0 ... . 
(3)  ... ... ... 
 
 .. 
 . ... 0 0 ρn,n−1 
0 0 ... ... ... 0
is a defining system for the Massey product (cid:104)ρ ,ρ ,...,ρ (cid:105) whose value is
21 32 n,n−1
the coboundary ±∂ρ . Conversely given such a defining system over C∗(Λ,E) of
n1
whose value is a coboundary ∂ρ , we can define a module M using the matrix (2).
n1
Proof. Compare ρ(ab) and ρ(a)ρ(b). (cid:3)
The resulting module is uniserial if all of the ρ are non-zero.
i+1,i
Lemma 6.2. The module defined by (2) is uniserial if each ρ (cid:54)=0
i+1,i
Proof. Induct on n, the case n = 2 is straightforward. For the inductive step, it
is enough to prove that socM is T , the submodule corresponding to the top-left
1
block of the matrix. Suppose s ∈ socM. Consider the image of s in M/T , which
1
is uniserial and which has socle (T +T )/T by induction. It follows s lies in the
2 1 1
submoduleofM correspondingtothetopleft2×2block. Applyingthen=2case
to this submodule we get s∈T . (cid:3)
1
In particular, under the hypothesis of the lemma, the module is indecomposable
of composition length n.
NotethatExt∗ (Λ,M⊗M∗)∼=Ext∗ (Λ,End (M))∼=Ext∗(M,M)asalgebras,
Λe Λe k Λ
so our Massey products are really happening in Ext∗((cid:76)S ,(cid:76)S ), which is the
Λ i i
“matrix ring” [Ext∗(S ,S )] .
Λ i j ij
7. A algebras
∞
TheindeterminacyofMasseyproductsisaseriousdrawback. ThetheoryofA
∞
algebrasallowsustointroducehigherproductsonthecohomologyofanassociative
algebrawithnoindeterminacy,whichgeneralisetheMasseyproductwhenitexists.
The downside is that they are slightly harder to define.
An A algebra is a Z-graded vector space A = A equipped with a map m :
∞ ∗ n
A⊗n →A for each n≥1, satisfying
(cid:88)
(−1)r+stm ◦(id⊗r⊗m ⊗id⊗t)
r+t+1 s
r+s+t=n
In particular: m m =0, so that A,m is a differential complex.
1 1 1
m ◦m =m ◦(m ⊗1+1⊗m ):A⊗2 →A
1 2 2 1 1
so that m is a derivation with respect to the multiplication given by m .
1 2
m ◦(1⊗m +m ⊗1)=m ◦m +m ◦(m ⊗1⊗1+1⊗m ⊗1+1⊗1⊗m ):A⊗3 →A
2 2 2 1 3 3 1 1 1
so m gives a homotopy associative multiplication on A,m .
2 1
An ordinary DGA is an A algebra with higher maps zero. This is the key
∞
to defining the higher operations on the cohomology of an associative algebra: we
begin with the ordinary DGA arising from the Yoneda cocomplex of a projective
resolution, give it such a ‘trivial’ A structure, then apply the following result:
∞