14 1. BASIC NOTIONS

m/mn+1

is maximal. Let us show that any proper ideal I ⊂ a is contained in

m/mn+1.

If I ⊂

m/mn+1,

then there exists α ∈ I represented by some x ∈ R

satisfying (x) = 0, where : R → R/m

∼

= is the augmentation map. By

multiplying with a scalar if necessary, we may clearly achieve that (x) = 1 that

is, x − 1 ∈ m. By Lemma 1.22, there exists y ∈ R such that xy = 1. Then, for

the equivalence classes in a = R/mn+1 we have 1 = [x][y] = α[y], therefore 1 ∈ I,

so I = a.

Since m/mn+1 contains all proper ideals, it is the unique maximal proper ideal,

which proves that a is local. The Artin property of a is also clear: any proper ideal

I ⊂ a is contained in m/mn+1, so Im+1 ⊂ (m/mn+1)m+1 = 0.

Let us review three most important types of deformations. As usual, for el-

ements a, b of an associative algebra A we denote by ab their product. Also the

notation := {1, 2, 3,...} of the set of natural numbers is standard.

Definition 1.24. A formal deformation is a deformation over the complete

local augmented ring [[t]].

Proposition 1.25. A formal deformation of A = (V, · ) is given by a family

(1.27) {μi : V ⊗V → V | i ∈ }

satisfying, for each k ≥ 1 and a, b, c ∈ V ,

(Dk)

μk(a, b)c + μk(ab, c) +

i+j=k

μi(μj(a, b),c) =

aμk(b, c) + μk(a, bc) +

i+j=k

μi(a, μj(b, c)).

Proof. By Lemma 1.20, the deformed multiplication μ is determined by its re-

striction to V ⊗V ⊂ ( [[t]] ⊗ V ) ⊗

[[t]]

( [[t]] ⊗ V ). Now expand μ(a, b), for a, b ∈ V ,

into a power series

μ(a, b) = μ0(a, b) + tμ1(a, b) +

t2μ2(a,

b) +

t3μ3(a,

b) + · · ·

with some -bilinear functions μi : V ⊗V → V , i ≥ 0. Clearly, μ0 must be the

original multiplication in A so, in fact,

(1.28) μ(a, b) = ab + tμ1(a, b) +

t2μ2(a,

b) +

t3μ3(a,

b) + · · · .

It is easy to see that μ is associative if and only if (Dk) is satisfied for each k ≥ 1.

Proposition 1.25 shows that the set

DefA( [[t]]) of formal deformations of A

consists of families (1.27) satisfying the infinite system (Dk), k ≥ 1, of quadratic

equations. The trivial deformation is the one with μi = 0 for each i ≥ 1. It is also

clear that

(1.29) GA

(

[[t]]

)

∼

= u =

V

+ φ1t +

φ2t2

+

φ3t3

+ · · · | φi ∈ Lin(V, V ) ,

with the group multiplication

(

V

+φ1t +

φ2t2

+ φ3

t3

+ · · · )(

V

+ φ1 t + φ2

t2

+

φ3t3

+ · · · ) :=

V

+ (φ1 + φ1 )t + (φ2 + φ1φ1 + φ2

)t2

+ (φ3 + φ2φ1 + φ1φ2 + φ3

)t3

+ · · · ,