1. BASIC NOTIONS 11
Remark 1.17. We need different symbols for algebras and for its underlying
modules. This will make our notation somehow heavy but, since we will sometimes
consider several algebras with the same underlying module, such a distinction is
necessary. We will, however, try to simplify the notation if no confusion may occur.
For instance, the trivial deformation of an associative algebra A = (V, μ) will be
denoted by R ⊗ A.
In the following definition, φ : V → V denotes the -linear map induced by
a continuous R-linear endomorphism φ : R ⊗ V → R ⊗ V as in diagram (1.20).
Definition 1.18. Two R-deformations (R ⊗ V, μ ) and (R ⊗ V, μ ) of an asso-
ciative algebra A with the underlying vector space V are equivalent if there exists
a continuous R-algebra isomorphism
φ : (R ⊗ V, μ )
−→ (R ⊗ V, μ )
such that φ : V → V is the identity automorphism
of V . We denote by DefA(R)
the set of equivalence classes of deformations of A with the base R.
In the important particular case when R is Artin,8 all topologies involved in
Definitions 1.16 and 1.18 are discrete, so we can omit the completions . If R
is regular local complete Noetherian, then the continuity of all maps follows from
their R-linearity, see Proposition 1.11. We can therefore avoid the topologies also
in this case and work in the realm of ‘standard’ algebra. As a matter of fact, all
base rings in this monograph will be of one of the above two types.
Let us show that the set DefA(R) behaves functorially in R. Assume that
f : R → R is a morphism of augmented rings and (R ⊗ V, μ ) an R -deformation
of A. One can easily check that μ induces an associative multiplication f!(μ) on
R ⊗ A
= R ⊗R (R ⊗ V )
which is an R -deformation of A. We will call f!(μ ) the push-forward of the
deformation μ . This construction induces a natural map (denoted by the same
f! : DefA(R ) → DefA(R ).
The sets DefA(R) therefore assemble into a covariant functor DefA(−) from the
category of complete local Noetherian rings with a given residue field, into the
category of sets. This point of view was, in the Artin case, pioneered by M. Artin
and M. Schlessinger [Sch68]. We include a brief subsection devoted to this approach
at the end of Chapter 4.
We denote by DefA(R) the set of R-deformations of an associative algebra A as
in Definition 1.16. Denote also by GA(R) the group of R-module automorphisms
φ : R ⊗ V → R ⊗ V such that φ =
, with the group structure given by the
composition. We will call GA(R) the gauge group. An automorphism φ ∈ GA(R)
acts on μ ∈ DefA(R) by φ · μ = μ , where
(1.23) μ (a, b) := φ ◦ μ
, a, b ∈ R ⊗ V.
The next proposition follows immediately from definitions.
[AM69, Theorem 8.5] that Artin local rings are precisely complete Noetherian rings
of global dimension zero.