1. BASIC NOTIONS 3

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

←−i

G/Gi,

with the topology given by the fundamental system {Gi}i≥1, where

Gi := lim

←−j

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

∞

i=1

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 :=

anM,

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

isomorphism.

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),

mn

= 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

←−i

R/mi

⊗ V.

The linear topology of R V is given by by the fundamental system

R V = R ⊗ V ⊃ m ⊗ R ⊃

m2

⊗ R ⊃

m3

⊗ R ⊃ · · · ⊃ {0}

where

mi

⊗ V := lim

←−j

mi/mj

⊗ V, for i ≥ 0.