10 1. BASIC NOTIONS

Definition 1.15. An associative

R-algebra5

is a couple B = (M, μ) consisting

of an R-module M with an R-bilinear multiplication μ : M × M → M satisfying

a(bc) = (ab)c, for all a, b, c ∈ M,

where we abbreviate, as usual, μ(a, b) := ab, &c. The algebra B = (M, μ) is

commutative, if

ab = ba, for all a, b ∈ M.

It is unital, if there exists a unit e ∈ M satisfying

ae = ea, for each a ∈ M.

If M is a topological R-module, we assume that μ is

uniformly6

continuous. If

R is the basic field , we sometimes call an R-algebra simply an algebra.

The R-module M is the underlying module of the R-algebra B. A morphism

f : B → B from the associative algebra B = (M , μ ) to the associative algebra

B = (M , μ ) is a morphism f : M → M of the underlying modules commuting

with the multiplications, that is, satisfying fμ = μ (f × f).

We remind the reader that the R-bilinearity of the multiplication of B means

that, for each a , a , b , b ∈ M and r , r , s , s ∈ R,

(r a + r a )(s b + s b ) = (r s )(a b ) + (r s )(a b ) + (r s )(a b ) + (r s )(a b ).

The universal property of the tensor product implies that each R-bilinear map

μ : M × M → M gives rise to an R-module morphism (denoted by the same

symbol) μ : M ⊗R M → M. We will usually use this tensor-product notation

for structure operations of algebraic systems. The associativity of μ can then be

expressed as the equality

μ(

M

⊗R μ) = μ(μ ⊗R

M

)

of R-linear maps M ⊗R M ⊗R M → M. If M is topological and the multiplication

uniformly continuous, then μ : M ⊗R M → M is continuous in the topology (1.8).

If M is, moreover, complete, μ uniquely extends into a continuous map (denoted

again by the same symbol) μ : M ⊗ RM → M from the completed tensor product.

The central definition of this chapter reads:

Definition 1.16. Let A be an associative -algebra with the underlying vec-

tor space V , and R a local complete Noetherian ring with residue field . An

R-deformation of A is an associative continuous7 R-algebra structure on the topo-

logical R-module R V = R ⊗ V such that the map

(1.22) ⊗

V

: R ⊗ V → V

induced by the augmentation : R → is a morphism of associative -algebras. The

trivial R-deformation of A is the one given by the R-linear extension of the original

multiplication of A to R ⊗ V . Deformations in the above sense will sometimes be

called deformations with the base R or deformations over R.

5We

sometimes say also an associative algebra over R.

6See

the discussion on page 8.

7By

Lemma 1.13, such a structure is automatically uniformly continuous.