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Hatcher-Algebraic-Topology
· Problem 3F.5
Cohomology
Mathematics — Graph Theory · limits-and-ext
Problem
Show that
E
x
t
(
A
,
Z
)
\mathrm{Ext}(A, \mathbb{Z})
Ext
(
A
,
Z
)
is isomorphic to the cokernel of
H
o
m
(
A
,
Q
)
→
H
o
m
(
A
,
Q
/
Z
)
\mathrm{Hom}(A, \mathbb{Q}) \to \mathrm{Hom}(A, \mathbb{Q}/\mathbb{Z})
Hom
(
A
,
Q
)
→
Hom
(
A
,
Q
/
Z
)
, the map induced by the quotient map
Q
→
Q
/
Z
\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}
Q
→
Q
/
Z
. Use this to get another proof that
E
x
t
(
Z
p
∞
,
Z
)
≈
Z
^
p
\mathrm{Ext}(\mathbb{Z}_{p^\infty}, \mathbb{Z}) \approx \hat{\mathbb{Z}}_p
Ext
(
Z
p
∞
,
Z
)
≈
Z
^
p
for
p
p
p
prime.
Related theory
Infinite Limits And Limits At Infinity
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