subsection. Some relevant notes on topologies can also be found in Section 4.1
of [CP94].
Suppose we are given a descending sequence G1 G2 G3 · · · of subgroups
of an Abelian group G. Then G has the unique linear topology with {Gi}i≥1
a fundamental system of neighborhoods of 0. The completion with respect to this
topology equals the inverse limit
G := lim
with the topology given by the fundamental system {Gi}i≥1, where
Gi := lim
Gi/(Gi Gj).
The completion G can also be described explicitly as
(1.1) G = (g1,g2,g3,...); gi G/Gi, gi = πij(gj), i j ,
where πij : G/Gj G/Gi is the canonical projection. So the completion G consists
of sequences (g1,g2,g3,...) of elements which are compatible in that gi = πij(gj),
for all i j. In this description, G appears a subspace of the Cartesian product

G/Gi of discrete spaces, with the induced topology. It is a standard fact that
the completion G is a Hausdorff space. We will often tacitly use the isomorphism
of the (discrete) quotients [AM69, Corollary 10.4]:
(1.2) G/Gi

G/Gi, i 1,
An important special case of the above situation is provided by a ring R with
a distinguished ideal a, together with an R-module M. The descending sequence
Mn :=
n 1, determines the a-adic topology of M and one can form the
completion M with respect to this topology. In particular, one can consider a local
ring R = (R, m) as a module over itself, and take its completion R with respect to
the m-adic topology. Recall that R is complete if the canonical map R R is an
Example 1.5. It is easy to prove that the completion of the ring of polynomials
[t] with respect to the (t)-adic topology equals the local ring [[t]] of formal power
series. Since the completion of any ring is complete, [[t]] is a complete ring.
Moreover, the completion of a Noetherian ring is Noetherian and [t] is Noetherian
by the Hilbert basis theorem, so [[t]] is also Noetherian [AM69, Corollary 10.27].
Example 1.6. In an Artin local ring (R, m),
= 0 for n sufficiently large.
Therefore the m-adic topology of R is discrete, so R = R and R is complete. It is
a standard fact that each Artin ring is Noetherian.
Suppose that R = (R, m) is a compete local Noetherian ring with residue field
Denote by R V = R V the m-adic completion of the R-module R V ,
(1.3) R V := lim
The linear topology of R V is given by by the fundamental system
R V = R V m R
R · · · {0}
V := lim
V, for i 0.
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