1. BASIC NOTIONS 15
where φiφj denotes the standard composition of linear maps. Observe that, by
Proposition 1.21, each u as in (1.29) indeed induces an invertible
φ : [[t]] ⊗ V → [[t]] ⊗ V.
It is the quotient of an infinite dimensional aﬃne quadratic algebraic variety, modulo
an action of a pro-unipotent group. From the point of view of singularity theory,
this is the worst situation.
Expansion (1.28) exhibits μ as a one-dimensional family, depending on the
parameter t, of associative products whose value at t = 0 is the original undeformed
multiplication. Its ‘formality’ means that no kind of convergence is required, so
the series (1.28) has only a ‘formal’ meaning. In [Fia08], all deformations with
a complete local base are called formal.
Let [t] be the polynomial ring as in Example 1.3, with the augmentation
: [t] → defined by
(f) := f(0) ∈ , for f ∈ [t]. Associative [t]-algebra
structures on the (uncompleted) [t] ⊗ V such that ⊗
: [t] ⊗ V → V is
a morphism of associative algebras are examples of global deformations of A = (V, · )
in the sense of [FP02]. It is easy to verify that these deformations are precisely
finite expressions (1.28).
Definition 1.26. An infinitesimal deformation, sometimes also called a first
order deformation, of an algebra A is a deformation over the local Antin ring
D := [t]/(t2) of dual numbers.
Notice that in [Fia08], all deformations over a local base (R, m) with m2 = 0 are
called infinitesimal. We leave the proof of the following version of Proposition 1.25
as an exercise.
Proposition 1.27. An infinitesimal deformation of A = (V, · ) is given by
a linear map μ1 : V ⊗ V → V fulfilling
(1.30) aμ1(b, c) − μ1(ab, c) + μ1(a, bc) − μ1(a, b)c = 0
for each a, b, c ∈ V .
Therefore DefA(D) consists of linear maps μ1 : V ⊗V → V satisfying (1.30).
It is easy to see that
+ φ1t | φ1 ∈ Lin(V, V )
Lin(V, V ),
with the abelian group structure of point-wise addition of linear maps, and the
action on μ1 ∈ DefA(D) given by
(1.31) φ1(μ1)(a, b) := μ1(a, b) + φ1(ab) − φ1(a)b − aφ1(b).
Not very surprisingly, the set
(1.32) DefA(D) = DefA(D)/GA(D)
of isomorphism classes of infinitesimal deformations of A is a vector space that
equals the second Hochschild cohomology group HH
A) recalled in the next
chapter, see Theorem 2.3.
Definition 1.28. Let n ≥ 1. An n-deformation of an algebra A is a deforma-
tion over the local Artin ring