12 1. BASIC NOTIONS
Proposition 1.19. The set of equivalence classes DefA(R) of R-deformations
of an associative algebra A is the quotient
Variants. One can modify Definition 1.16 in several ways. For instance, one can
take as R an arbitrary ring augmented over and consider deformations with the
(uncompleted) R ⊗ V as the underlying space. In [Fia08], these deformations are
called global. For deformations in this sense one however loses the cohomology as
a tool and several other statements, as the invertibility of Proposition 1.21, cease
Another modification is to define an R-deformation of an associative -algebra
A as an associative R-algebra B with a -algebra isomorphism ⊗RB
→ A. There
is, however, not much to be said about R-deformations without some additional as-
sumptions on the underlying R-module M of B. In our Definition 1.16 we assumed
that it was a free complete R-module. Another assumption frequently used in alge-
braic geometry [Har77, Section III.§9] is that M is flat which, by definition, means
that the functor M ⊗R
− is left exact. One then speaks about flat deformations.
If R is a local Noetherian ring, a finitely generated R-module is flat if and only if
it is free (see Exercise 7.15, Corollary 10.16 and Corollary 10.27 of [AM69]). There-
fore, for A with a finite-dimensional underlying vector space, free deformations are
the same as the flat ones. Our restriction to free deformations includes most of
instances of algebraic deformation theory, including the (mini)versal deformations
recalled on page 16 and, of course, also the classical setup of [Ger64].
The R-linearity built in Definitions 1.16 and 1.18 implies the following lemma.
Lemma 1.20. Let (R ⊗ V, μ) be an R-deformation of A as in Definition 1.16.
Then the multiplication μ is determined by its restriction to
V ⊗V ⊂ (R ⊗ V ) ⊗R(R ⊗ V ).
Likewise, every equivalence of deformations φ : (R ⊗ V, μ ) → (R ⊗ V, μ ) is deter-
mined by its restriction to V ⊂ R ⊗ V .
Proof. To prove the first part of the lemma, we invoke the isomorphism
(R ⊗ V ) ⊗R(R ⊗ V )
R V ⊗V
of (1.17) and apply Lemma 1.9 to the case U = V ⊗V . The second part of the
proposition follows from the same lemma with U = V .
The following proposition will also be useful.
Proposition 1.21. Assume that V is a -vector space and R a local complete
Noetherian ring with residue field .9 Then every R-linear continuous morphism
φ : (R ⊗ V ) → (R ⊗ V )
of R-modules such that φ =
The proposition will follow from:
of course includes also the Artin case.