Hatcher-Algebraic-Topology · Problem 3F.9

Cohomology

Mathematics — Graph Theory · limits-and-ext

Problem

For an abelian group

A

let

p: A \to A

be multiplication by

p

, and let

{}_p A = \ker p

pA = \mathrm{Im}\, p

, and

A_p = \mathrm{Coker}\, p

as in the proof of Proposition 3F.12. Show that the six-term exact sequences involving

\mathrm{Hom}(-, \mathbb{Z})

and

\mathrm{Ext}(-, \mathbb{Z})

associated to the short exact sequences

0 \to {}_p A \to A \to pA \to 0

and

0 \to pA \to A \to A_p \to 0

can be spliced together to yield the exact sequence

\mathrm{Hom}({}_p A, \mathbb{Z}) \to \mathrm{Ext}(A_p, \mathbb{Z}) \to \mathrm{Ext}(A, \mathbb{Z}) \xrightarrow{p} \mathrm{Ext}(A, \mathbb{Z}) \to \mathrm{Ext}({}_p A, \mathbb{Z}) \to 0

where the map labeled '

p

' is multiplication by

p

. Use this to show: (a)

\mathrm{Ext}(A, \mathbb{Z})

is divisible iff

A

is torsionfree. (b)

\mathrm{Ext}(A, \mathbb{Z})

is torsionfree if

A

is divisible, and the converse holds if

\mathrm{Hom}(A, \mathbb{Z}) = 0