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.
By (1.24),
DefA
(
[[t]]
)

=
DefA
(
[[t]]
)
/GA
(
[[t]]
)
.
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
0
: [t] defined by
0
(f) := f(0) , for f [t]. Associative [t]-algebra
structures on the (uncompleted) [t] V such that
V
: [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
GA(D)

=
u =
V
+ 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
2(A,
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
[t]/(tn+1).
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