(5o3) Local rings, cohomology and Massey products.
Let k be any field and consider a local k-algebra A • Denote by ~ the maximal ideal of A and assume k = A/g};_A •
Suppose A is (essentially) of finite type over k. Then Hi(k,A;k) has a countable basis for i ~ 0. Put
Ti = Symk(Hi(k,A;k)*)A i=0,1.
0
Theorem (5.3.1) k-algebras
There exists a morphism of topological local
0 • T1 -• 0 T0 0
and an isomorphism
Proof. This follows from an obstruction calculus for lifting morph- isms in exactly the same way as one finds in [La 5] •
However we may also deduce (5.3.1) from (4.2.4).
In fact, let us consider the category of k-algebras c : A- k n
having two objects, A and k, and one nontrivial morphism n. Let c • A
--o •
be the subcategory of c consisting of the object A alone. Then the deformation functor
Def(.£/.£o) : 1 - ~
is characterized by
Def(£/.£o) (R)
R --::> R®A cr(n):> R
l
cr(n) being an R-morph- k=
w
~,
k - > A
= { y ~
A - > k
\..
,.
"'
,"";!
~
ism, making- > n k commutative
q> being
of local
a_ morphism}
r~ngs ..)
,..
Obviously this functor is prorepresented by A. On the other hand we know that the hull (the formal moduli for c relative to
.£o )
isisomorphic to
where
Ti =
Symk(A~ (.£,~)*
)" o-o
To compute the algebra cohomology A i ( c, 0 ) , we have to consider c - c
-o -
the morphism categories 1.'1or
.£o
and l'1or .£ oMor .£ :
Mor
.£o :
A - > k
/
n
Using (3a1a7) and the long exact sequence of relative cohomology we find:
(ii) An(.£,2c) is the abutment of the spectral sequence given by
. ker {Hq(k,A;A) .... Hq(k,A;k)} if p = 0
= coker{Hq(k,.A;A) .... Hq(k,A;k)] if p=1
Thus
0 otherwise
An+1(c,O ) = ~(k,A;k).
c - c -o .
(This follows from the long exact sequence
(Inclusion tO)
0 0
~
ker {Hn(k,A;A) .... Hn(k,A;k)}
$
coker{Hn-1(k,.A;.A) .... Hn-1(k,A;k))
~ W(k,A;.A)
~ An+c 1(c _, c 0 )
-o
~
ker {Hn+1(k,A;A) .... Hn+1(k,A;k)}
®
coker{Hn(k,A;.A) .... Hn(k,A;k)}
(InclusioniO) ~
0 0
In particular we find:
Ti
=
Symk(.Ai (c,O c - c )*)~-o -
~ Symk(Hi-1(k,A;k)*)~
for i=1,2o
0 0
0
Q.E.D.
It follows from the long exact sequence of algebra cohomology that
H?(k,A;k) ~ H1 (A,k;k) H1(k,A;k) ~ H2(A,k;k).
The morphism 0 :
above is obviously characterized by the homomorphism of k- vectorspaces
n 1 ,..
rr
8~ H (A,k;k)*n>o
We know that the image of 0 sits in the subspace
n
IT Sym® H 1 (A,k;k)* • Thus the first approximation of 0 is n>2
a homomorphism
which is dual to the Lie product
The
1 A 1 2
H (A,k;k) S~ H (A,k;k) --> H (A,k;k)A • later approximations
~(A,k;k)*
--> krr
n=2
are homomorphisms
n 1 ,.,
® H (A k·k)*
Sym ' '
which dualize to homomorphisms:
k n 1 2 ,.,
l i S~ H (A,k;k) -> H (A,k;k) o n=2
The k-th term of this homomorphism,
k 1 2 A
S~ H (A,k;k) --> H (A,k;k) ,
will ·be called the k!h. Massey product , and we shall see
later that, in fact, these maps are matrix Massey product in the sense of May, see (May).
The indeterminacy in the construction of 0 correspond exactly to the indeterminacy of the Massey-products.
Chapter 6. The obstruction morphism 0 •
(6.1) The obstruction morphism and Massey products.
In this section we shall go back to the general situation of (4.2) and study the morphism
0 : T1 .... T2
of (4.2.4). We shall, however, restrict to the case where
V = k,
dimkA~
(Q.,Od) <co"""'()
-
i = 1,2.By definition of the terms involved, 0 is completely determined by the restriction
Projecting onto the first r - 2 -factors we obtain homomorphisms of k-vectorspaces
Let (a1 , ••• ,aJ., •• ad } be a basis of Ad (d,Od) and let 1
1 -o -
{a1, ••• ,aj,·· ad1} be the corresponding dual basis for
A~
(d,Od)*.-o -
In the same way let (b1 , ••• ,bi,~· bd } be a basis of A~ (£,0d)
2 -o
and denote by (b1, ••• ,bi,·· bd} the dual basis.
2
Then we may write
( 1)
By construction of subspace of
*) i.e. independent of the choices made in the construction of 0 •
generated by the union of the subsets
for i=1,2, ••• , s = 2, ••• , r-1 , 1 < - j 1 < .. - H -< j S -< d1 , '1 -< j S+ -1 < •• o -~ < .i __ <d1 • Now, the dual of 0 r , is a k-linear map
r n 1 2
11. S~ Ad (£,Od) - > Ad (£,Od)
n=2 -o - -o -
which induces k-linear maps
By definition of the a~ . 's J1,.-,Jr
d2 .
(2) M (a. ® •.•• ®a. ) = I: a~ . b.
r J1 · Jr i>1 J1, ••• ,Jr ~
which is uniquely determined modulo the subspace of generated by the set
Definition (6.1.1) Mr is called the rth Massey product.
We shall sometimes use the notation for the value of Mr on a..; ® a2 ® • o o ® a~
e
· In the study of these Massey products we shall be inspired by the ideas of May and Dwyer see (May) and (Dw).
Consider the local k-algebra
Let e:i denote the image of xi in U(r) • Obviously U(r) is a k-vectorspace of finite dimension. The maximal ideal m of U(r)
is generated by and a basis for m
-
ober k is givenby the products e: • = e . .. e . • • • e: • with
I J1 J2 Js 1 < j 1 < j 2 < •• o < j < r ,
- s-
In particular U(r) is an object of 1 (see (4.2))o A morphism of local k-algebras
Ti -> U(r) i
=
1, 2is determined by the corresponding k-linear map
A~
(d,Od)* - >~
..-o
Therefore by a collection of elements of A~ (d,Od) -o
{a!) = {a! . 11 < i 1 < ••• < i < r, 1 < s < r)
,!. 1..1 ' oeo ' 1 S - S - ,.... -
Given such a collection of elements of [a!]: T1 --> U(r)
J_
be the corresponding morphism, and let [a!]
(3) T2 .Q..> T1 ~> U(r).
(a!)
1
-
denote the compositionS . 1nce 1m . 0
S
~ 2 1 ,< ')
a.2 T 1.
correspond to a collection of elements of Ad (d,Od)
-o
It is easily checked that
1:: (a1! '•". 'a1! ) = b '1 2 •
1 t ' ' - ,r
e: . . . e:. =€1 2 - -
11 1t , ,..,r i.e. the correspondence
{a.) -> {b.)
1 I
is a generalized Massey-producte Now consider the morphism
p: U(r) - U(r)/mr
of
1
a The kernel is obviously ~r which is isomorphic to k , a basis being formed by the element e:1•e2 ••• e r = e1 , ••• ,r • We know by (4.2.4) that the elements ofDef(£/~)(U(r)/mr) correspond to those morphisms *).
which composed with 0 are trivial. Therefore any collection of elements of
A1
(d,Od) such that the composition-o
(4) 2 0 1 [ai] 2
T -> T - - > U(r) -> U(r)
/£!
is trivial correspond to an element
(a.) E Def(dld )(U(r)/mr)
1. ~-o -
Since ker p2 = (mr)2 = 0 we may consider the obstruction for lifting (a.)
~ to U(r),
By construction of 0 , 0 (a. )
~
the composition 0 o [a.] =(a.)
~ ~
- -
restriction is a k-linear map
corresponds to the to
A~
(d,Od)*.-o
A2 ( d 0 ) d _, d * ... mr _,... k • -o
*) This correspondence is not one-to-one.
restriction of
In fact this
Therefore
Q(a.)
=
b 1 2 o~ ' , ••• ,r
Remark((6.1.2) This result is closely related to Theorem 2.4. of (Dw), see (La
5).
The results above may be summed up as follows:
Definition (6.1.3) Given elements collection of elements of
[a!}
-
~then a
with a'.=c.
J J j = 1, o. o ,r , is called a defining system for the Massey product (c~, ••• ,cr> if the corresponding compo- sitions of morphisms (4) are trivial.
Theorem (6.1.4) Given elements c1 , •• o,cr E
Ad
(£,0d) then there -ois a (not necessarily one-to-one) correspondence bet··.veen de·- fining systems for the Massey product (c1 ,o •• ,cr) and ele-
ments (a!) ~ € Def(d/d )(U(r)/mr) having c1 , ••• ,cr as tangents. - -o - Moreover
to U(r) •
(a!)
.!. correspond to the obstruction for lifting (a!) .!.
Bibliography.
(Dw): Dwyer, William G.: Homology, Massey product~ and maps between groups.
Journal of Pure and Applied Algebra 6 (1975) pp. 11?-190.
(La 5): Laudal, O.A.: Solvable groups and Massey products.
Arhus University, Preprint Series.
(May): May, J. Peter: Matric Massey Products.
Journal of Algebra 12 (1969) pp. 533-568.
