1. BASIC NOTIONS 5
Proof. For V finite dimensional and R complete, the proposition follows from
[AM69, Proposition 10.13]. If R is Artin,
mi
= 0 for i sufficiently large, so the
inverse limit in (1.3) stabilizes. The topology of R V is therefore discrete and
R V

= R V as (discrete) topological R-modules.
Lemma 1.9. Let U and V be (discrete) vector spaces, and R = (R, m) a local
complete Noetherian ring with residue field . Then there is a natural one-to-one
correspondence
(1.7) Φ : Lin
(
U, R V
)

=
←→ LinR
c
(
R U , R V
)
: Ψ,
where Lin(−, −) denotes, as usual, the space of -linear maps and LinR(−,
c
−) the
space of continuous R-linear maps. Moreover, (1.7) restricts, for each k 0, to
the isomorphism
Φk : Lin
(
U,
mk
V
)

=
←→ LinR
c
(
R U ,
mk
V
)
: Ψk.
Proof. The lemma, of course, follows from the universal property of R U in
the category of complete topological R-modules, but we include a direct proof here.
Let us define first the correspondences Φ and Ψ.
For a -linear map φ : U R V denote by
˜
φ : R U R V its R-linear
extension given by
˜(r
φ u) := rφ(u), for u U and r R. Clearly,
˜(mi
φ U) =
miφ(U)

mi(R
V ),
so, by (1.5),
˜(mi
φ U) mi V . The map
˜
φ therefore induces a map
R U
mi U
−→
R V
mi V
of the quotients which, combined with the isomorphisms
R U
mi U

=
R/mi
U and
R V
mi V

=
R/mi
V
gives a map
˜
φ
i
:
R/mi
U
R/mi
V . Define finally, in the notation (1.4) for
elements of the inverse limits,
Φ(φ)(x1,x2,x3,...) :=
(˜1(x1),
φ
˜
φ 2(x2),
˜
φ 3(x3),...).
It is easy to verify that Φ(φ) : R U→ R V is a well-defined continuous map.
The definition of the inverse correspondence Ψ is even simpler. Given an R-
linear map f : R U→ R V , Ψ(f) : U R V is the composition
Ψ(f) : U
ι
R U
f
−→ R V .
It is clear that Ψ◦Φ = . It therefore remains to prove that Ψ is a monomorphism.
So assume that Ψ(f) = 0 for a continuous f : R U R V and prove that
then f = 0. Since R V is dense in R V , it is enough to show that f(r v) = 0
for r R and v V . But this is obvious, since
f(r v) = rf(1 v) = rf(ι(v)) = rΨ(f)(v) = 0.
We leave the proof that the isomorphisms Φ resp. Ψ restrict to Φk resp. Ψk as an
exercise.
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