In mathematics, the Bockstein spectral sequence is a

spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...

relating the homology with mod ''p'' coefficients and the homology reduced mod ''p''. It is named after

Meyer Bockstein Meyer Bockstein (also Меер Феликсович Бокштейн or Meer Feliksovich Bokshtein or Bokstein) (4 October 1913 to 2 May 1990) was a topologist from Moscow who introduced the Bockstein homomorphism. The Bockstein spectral sequenc ...

Definition

Let ''C'' be a chain complex of

torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only e ...

s and ''p'' a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. Then we have the exact sequence: :

0 \longrightarrow C \overset\longrightarrow C \overset \longrightarrow C \otimes \Z/p \longrightarrow 0.

Taking integral homology ''H'', we get the

exact couple In mathematics, an exact couple, due to , is a general source of spectral sequences. It is common especially in algebraic topology; for example, Serre spectral sequence can be constructed by first constructing an exact couple. For the definition o ...

of "doubly graded" abelian groups: :

H_*(C) \overset \longrightarrow H_*(C) \overset \longrightarrow H_*(C \otimes \Z/p) \overset \longrightarrow.

where the grading goes:

H_*(C)_ = H_(C)

and the same for

H_*(C \otimes \Z/p),\deg i = (1, -1), \deg j = (0, 0), \deg k = (-1, 0).

This gives the first page of the spectral sequence: we take

E_^1 = H_(C \otimes \Z/p)

with the differential

^1 d = j \circ k

. The

derived couple In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they ha ...

of the above exact couple then gives the second page and so forth. Explicitly, we have

D^r = p^  H_*(C)

that fits into the exact couple: :

D^r \overset\longrightarrow D^r \overset \longrightarrow E^r \overset\longrightarrow

where

^r j = (\text p) \circ p^

and

\deg (^r j) = (-(r-1), r - 1)

(the degrees of ''i'', ''k'' are the same as before). Now, taking

D_n^r \otimes -

of :

0 \longrightarrow \Z \overset\longrightarrow \Z \longrightarrow \Z/p \longrightarrow 0,

we get: :

0 \longrightarrow \operatorname_1^(D_n^r, \Z/p) \longrightarrow D_n^r \overset\longrightarrow D_n^r \longrightarrow D_n^r \otimes \Z/p \longrightarrow 0

. This tells the kernel and cokernel of

D^r_n \overset\longrightarrow D^r_n

. Expanding the exact couple into a long exact sequence, we get: for any ''r'', :

0 \longrightarrow (p^ H_n(C)) \otimes \Z/p \longrightarrow E^r_ \longrightarrow \operatorname(p^ H_(C), \Z/p) \longrightarrow 0

. When

r = 1

, this is the same thing as the

universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...

for homology. Assume the abelian group

H_*(C)

is finitely generated; in particular, only finitely many cyclic modules of the form

\Z/p^s

can appear as a direct summand of

H_*(C)

. Letting

r \to \infty

we thus see

E^\infty

is isomorphic to

(\text H_*(C)) \otimes \Z/p

References

* * J. P. May
A primer on spectral sequences
Spectral sequences {{topology-stub