8 1. BASIC NOTIONS
satisfying appropriate coherence relations, see [ML63b]. Also the quotients by
open submodules are easy to describe; by standard linear algebra,
M ⊗S N
M ⊗S V + U ⊗S N
= M/U ⊗S N/V.
Therefore the completion of the tensor product M ⊗S N with respect to topol-
ogy (1.8) equals
(1.12) M ⊗ SN := lim
M/U ⊗S N/V.
We say that an algebraic structure is topological if all its structure operations are
There exits another topology on M ⊗S N having the universal property of
Proposition 1.12 with respect to all continuous S-bilinear maps. This topology,
however, does not have nice properties and it does not seem to play any rˆ ole in mul-
tilinear algebra, see the discussion in [BH96, §24]. There are yet some important
cases where R-bilinear continuous maps are automatically uniformly continuous.
Let us prove the following
Lemma 1.13. Let R be a local complete Noetherian ring with residue field
and V1,V2,W discrete -vector spaces. Any continuous R-bilinear map
F : (R ⊗ V1) × (R ⊗ V2) → R ⊗ W
is uniformly continuous. There is thus an one-to-one correspondence between con-
tinuous R-bilinear maps
(R ⊗ V1) × (R ⊗ V2) → R ⊗ W
and R-linear continuous maps
(R ⊗ V1) ⊗R (R ⊗ V2) → R ⊗ W.
Proof. The R-submodules mk ⊗ V1 (resp. mk ⊗ V2, resp. mk ⊗ W ), k ≥ 0,
form a basis of open neighborhoods of 0 in R ⊗ V1, (resp. R ⊗ V2, resp. R ⊗ W ).
By (1.10), the uniform continuity of F will therefore be established if we prove that,
for each k ≥ 0, there exist k1,k2 ≥ 0 such that
(1.13) F (R ⊗ V1,
⊗ V2) ⊂
⊗ W and F
⊗ V1,R ⊗ V2) ⊂
Let us prove that the first inclusion is satisfied with k1 := k. Since F is
continuous separately in each variable and mk ⊗ W is complete, it is enough to
verify the inclusion
(1.14) F (R ⊗ V1,
⊗ V2) ⊂
The R-bilinearity of F implies that
(1.15) F (R ⊗ V1,
⊗ V2) =
(R ⊗ V1,V2).
Since F (R ⊗ V1,V2) ⊂ R ⊗ W and
⊗ W ) ⊂
⊗ W , (1.15) implies (1.14)
and thus also the first inclusion of (1.13) with k1 = k. The second inclusion can be
Examples 1.14. Let R be a complete Noetherian local ring and V a discrete
k vector space. The completed R V = R ⊗ V of (1.3) is a particular instance of
the completed tensor product (1.12), with S = , M = R, and discrete N = V .