Algebraic Number Theory – Lecture 15
Sandro Bettin
“Only two things are infinite, the universe and human stupidity, and I’m not sure
about the former.”
– Albert Einstein
Definitions.
•A directed system is an ordered set Iin which for each pair i, j ∈Ithere exists a k∈Isuch
that i6k, j 6k.
•A projective system over a directed system Iis the set of pairs
{(Xi, fi,j)|i, j ∈I, i 6j}
where the Xiare topological spaces and the fi,j are continuous maps
fi,j :Xj−→ Xi
such that one has
fi,i =idXi,
fi,k =fi,j ◦fj,k,when i6j6k.
•The projective (or inverse) limit
X= lim
←−
i∈I
Xi
of a projective system {(Xi, fi,j )}i6j, is defined to be the subset
X={(xi)i∈I∈Y
i∈I
Xi|fi,j(xj) = xifor i6j}
of the product Qi∈IXiequipped with the product topology (i.e. the open sets of Qi∈IXiare
of the form Qi∈IAi, where Aiis open in Xiand Ai=Xifor all but a finite number of indices
i).
Remarks.
1) If the topological spaces {Xi}i∈Iare also groups then we require that the fi,j are also group
homomorphisms. It’s easy to see that the projective limit, equipped with the multiplication
induced by the componentwise multiplication, is a topological group. The analogous result
holds if the {Xi}i∈Iare rings, modules, ....
2) One can define, in a similar way, the inductive (or direct) limit, the categorical dual of the
projective limit.
Examples.
1) Let pbe a prime, I=Nand Xn=Z/pnZ. Let fm,n for n>mbe the reduction modulo pm
fm,n :Z/pnZ→Z/pmZ.
The projective limit
Zp:= lim
←−
n∈N
Z/pnZ
is the ring of the p-adic integers.
2) Let I=N,with order given by divisibility (i.e. n6miff n|m), Xn=Z/nZand fn,m for n|m
be the reduction modulo m. The pro jective limit
ˆ
Z:= lim
←
n∈N
Z/nZ
is the Pr¨ufer ring.
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