4 1. BASIC NOTIONS
Isomorphism (1.2) describes the quotients as
⊗ V )
⊗ V, for each i ≥ 0.
By (1.1), the inverse limit in (1.3) equals
(1.4) R V = (x1,x2,x3,...); xi ∈
⊗ 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
⊗ V ) ⊂
which may, in general, be a proper one. There is a natural map i : R ⊗ V → R ⊗ V
i(a) := ([a]1, [a]2, [a]3,...), a ∈ R ⊗ V,
where [a]n is, for n ≥ 1, the equivalence class of a in
⊗ V . Clearly, i(a) = 0
means that a ∈
⊗ V for each n ≥ 1. Since
⊗ 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
→ 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 +
+ · · · , 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
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