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 = 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.
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