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