4 1. BASIC NOTIONS

Isomorphism (1.2) describes the quotients as

R V

/(mi

⊗ V )

∼

=

R/mi

⊗ V, for each i ≥ 0.

By (1.1), the inverse limit in (1.3) equals

(1.4) R V = (x1,x2,x3,...); xi ∈

R/mi

⊗ V, πij(xj) = xi, ∀ i ≤ j

with the component-wise R-module structure. In this description, the sub-module

mi ⊗ V of R ⊗ V consists of sequences (x1,x2,x3,...) such that xj = 0 for j ≤ i.

One has, for each i ≥ 0, the inclusion

(1.5)

mi(R

⊗ V ) ⊂

mi

⊗ V

which may, in general, be a proper one. There is a natural map i : R ⊗ V → R ⊗ V

given by

i(a) := ([a]1, [a]2, [a]3,...), a ∈ R ⊗ V,

where [a]n is, for n ≥ 1, the equivalence class of a in

R/mn

⊗ V . Clearly, i(a) = 0

means that a ∈

mn

⊗ V for each n ≥ 1. Since

n≥1

(mn

⊗ V ) = ∅ (we assume that

R is complete), the map i is a monomorphism. We may use it to identify R ⊗ V

with a subspace of R ⊗ V . It is a standard fact that R ⊗ V is dense in R ⊗ V .

Finally, one has the composed inclusion of -vector spaces

ι : V → R ⊗ V → R ⊗ V

given, in the language of (1.4), by

ι(v) := (1 ⊗ v, 1 ⊗ v, 1 ⊗ v, . . .), for v ∈ V .

It is easy to show that the object V

ι

→ R V is the free complete topological R-

module generated by V – it has a universal property in the category of complete

topological R-modules similar to that of R V .

Example 1.7. The difference between R V and R V is best explained when

we take as R the power series ring [[t]] recalled in Example 1.2. The module

R V = [[t]] ⊗ V then consists of expressions

(1.6) v0 + v1t +

v2t2

+

v3t3

+ · · · , v0,v1,v2,... ∈ V,

which can be understood as power series with coeﬃcients in V . For this reason, one

sometimes denotes [[t]] ⊗ V by V [[t]]. The [t]-module R V is, up to isomorphism,

characterized by the property that it is flat, (t)-adically complete, and

R V /tR V

∼

= V.

The uncompleted R V = [[t]] ⊗ V is the subspace of [[t]] ⊗ V consisting of

expressions (1.6) such that the coeﬃcients v0,v1,v2,... span a finite-dimensional

subspace of V . In particular, for V finite-dimensional, one has a [[t]]-module

isomorphism [[t]] ⊗ V

∼

= [[t]]⊗V .

The observation made in Example 1.7 is a particular case of:

Proposition 1.8. Suppose that either V is a finite dimensional -vector space

and R a local complete Noetherian ring, or V is arbitrary and R is Artin. Then

one has an isomorphism

R V

∼

= R V

of R-modules.