6 1. BASIC NOTIONS

Example 1.10. Let [[t]] be the formal power series ring and U, V (discrete)

-vector spaces. Let us show that each [[t]]-linear map f : [[t]] ⊗ U → [[t]] ⊗ V

is automatically continuous.

We verify this fact by proving that, for each sequence {an}1

∞

converging to

a ∈ [[t]] ⊗ U, the sequence f(an)

∞

1

converges to f(a) in [[t]] ⊗ V . The con-

vergence {an}0

∞

→ a means that, for each n ≥ 0 there exists k ≥ 1 such that

a − ak ∈

(tn)

⊗ U. It is obvious from the description of [[t]] ⊗ U in terms of power

series with coeﬃcients in U given in Example 1.7 that the last condition in fact

says that a − ak is divisible by

tn:

a − ak =

tn

· uk

n,

for some uk

n

∈ [[t]] ⊗ U.

We conclude that f(a) − f(ak) ∈

(t)n

⊗ V , which shows that f(an)

∞

1

converges

to f(a) in the topology of [[t]] ⊗ V .

We leave as an exercise based on Theorem 11.22 of [AM69] to prove the fol-

lowing generalization of Example 1.10.

Proposition 1.11. Let R be a

regular3

local complete Noetherian ring and U,

V discrete vector spaces. Then each R-linear map f : R U → R V is continuous.

Topologized tensor products. Suppose we are given a ring S and topological

S-modules M and N. One may topologize the tensor product M ⊗S N by requiring

that the subspaces

(1.8) M ⊗S V + U ⊗S N ⊂ M ⊗S N,

where U (resp. V) are open S-submodules of M (resp. N), form a basis of open

neighborhoods of zero in M ⊗S N. This topology has a certain universal property

with respect to uniformly continuous maps which we formulate below. Let us recall

some necessary definitions.

A uniformity on a set X is a system of neighborhoods of the diagonal

Δ(X) := (x, x) | x ∈ X ⊂ X × X

satisfying suitable axioms [Kel55, Chapter 6]. A set with a uniformity is called

a uniform space. Each uniformity induces a topology on X, with a basis of open

neighborhoods of x ∈ X given by the sets

Ux := {x ∈ X | (x , x) ∈ U},

where U ∈ .

A map f : (X, ) → (Y, ) is uniformly continuous if, for each V ∈ , there

exists U ∈ such that

(x1,x2) ∈ U =⇒

(

f(x1),f(x2)

)

∈ V.

Each uniformly continuous map is continuous with respect to the induced topologies.

The cartesian product X1

× X2 of uniform spaces (Xi,

i

), i = 1, 2, has a uni-

formity

1

×

2

given by the subsets

U1 × U2 ⊂ (X1 × X1) × (X2 × X2)

∼

=

(X1 × X2) × (X1 × X2) = Δ(X1 × X2),

where Ui ∈ i. The induced topology on X1 × X2 is the product of the induced

topologies.

3See

[AM69, Theorem 11.22] for a definition of regular local rings.