l.A. ADDITIVE STRUCTURE
3
graded. An example of a filtration is the following. Let X be a CW complex, and
let X^ denote its fc-skeleton. Then
FzW{X;R)=kerlHJ{X(i-l);R) —HJ(X;R)}
defines a filtration of the cohomology module of X.
Associated to a (bounded and convergent) filtration is the bigraded module
G(M) which is defined by
G ( M ) ^ =
In general, it can be difficult to recover the giaded module M from G(M) (this is
a problem about module extensions). However, if M* is of finite type, and if R is
a field, then
•pij njri
T
[ r t
( G ( M ) ) = ©
i 9
^ - M .
yez
1.4. DIFFERENTIAL GRADED MODULES
A map / : M* Nm of degree r between graded modules is an ^-linear map which
increases degrees by r, i.e. f{Mt) C JVz+r. A differential is a map d : M* M*
of degree 1, with d2 = 0. The cohomology of (M, d) is the graded module
ff(M)=kei[Jtf^Mi+1'
An ^-linear map / : M " —• J V has bidegree (r, s) if /(M^' ) C
7V2+r^+s.
A
differential d of bidegree (r, 1 r) is a map d : M M of bidegree (r, 1 r) with
d2
= 0. The cohomology of (M, d) is the bigraded module
H ^ ( M ) = ^ ^ ^ i.
im[A^-
r
'i-
1 + r
M^]
Note that d is a differential on Tot(M), and that Tot(H**(M)) = H*(Tot(M)).
The pair (M, d) is called a differential (bijgraded module.
1.5. SPECTRAL SEQUENCES
An E2-spectral sequence is a collection of differential bigraded modules E
r
, indexed
b y r = 2,3,4,---, endowed with differentials dr of bidegree (r, 1 r), such that
E
r +
i = H(E
r
).
The spectral sequence converges if for every pair (i,j) there exists a number n = riij
such the two maps
Ei'
are trivial for all r n. The resulting module ^
E^J+1
^ E ^
2
^ E ^
3
^ is
denoted by E ^ , and one says that the spectral sequence converges to the bigraded
module E££. The spectral sequence collapses if all differentials dr vanish, and it
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