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).