1. BASIC NOTIONS 9
Consider the map
V
: R V V

= V , where : R is the
augmentation. There are two open subspaces of , {0} and the whole . It is clear
that both subspaces
m V = (
V
)−1(0)
and R V = (
V
)−1(
)
are open in the topology (1.8), so (
V
) is continuous. The space V is, as each
discrete space, complete, so
V
uniquely extends into a continuous map
(1.16)
V
: R V V.
Another important particular case is S = R, M = R V1 and N = R V2 for
some discrete -vector spaces V1, V2. The completed tensor product then equals
R V1 ⊗RR V2 = lim
←−a, b
(R/ma
V1) ⊗R
(R/mb
V2).
From the obvious isomorphism
(R/ma
V1) ⊗R
(R/mb
V2)

=
R/mmax{a,b}
V1 V2
we obtain
(1.17) R V1 ⊗RR V2

= lim
←−i
R/mi
V1 V2

= R V1 ⊗V2 .
Iterating (1.17) gives a natural isomorphisms
(1.18)
k
R
R V

=
R
k
V , for each k 0.
As the last example of the completed tensor product, consider the situation
S = R, M = and N = R V . Since ⊗R (R/mn V )

=
V for each n 1,
one has
(1.19) ⊗R(R V ) = lim
←−n
⊗R
(R/mn
V )

=
lim
←−n
V

=
V.
This isomorphism induces, for each R-linear continuous φ : R V R V , the
-linear map φ : V V via the commutativity of the diagram
(1.20)
φ
⊗Rφ✲

=

=





V . V
⊗R(R V ) ⊗R(R V )
The map (1.16) then fits into the diagram
(1.21)
φ
φ
V V



V . V
R V R V
Deformations of associative algebras. We will illustrate basic notions of de-
formation theory on the particular example of associative algebras. We will see in
chapters 9 and 10 that most of the material extends to a broad class of equationally
given structures as Lie, commutative associative, Poisson, Leibniz algebras, various
bialgebras, and their diagrams. Recall
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