16 1. BASIC NOTIONS

1-deformations are infinitesimal (= first-order) deformations of Definition 1.26.

We have the following version of Proposition 1.25 which generalizes Proposition 1.27.

Proposition 1.29. An n-deformation of A = (V, · ) is given by a family

{μi : V ⊗V → V | 1 ≤ i ≤ n}

of -linear maps satisfying condition (Dk) of Theorem 1.25 for 1 ≤ k ≤ n.

The proof is obvious, as well as the description

GA

(

[t]/(tn+1)

)

∼

=

u =

V

+ φ1t +

φ2t2

+ · · · +

φntn

| φ1,...,φn ∈ Lin(V, V )

of the gauge group. We leave both as an exercise.

(Mini)versal deformations. A deformation ω of an associative algebra A with

a base S would be universal, if for any other deformation μ with the base R there

exists a unique ring morphism f : S → R such that the push-forward f!(ω) of ω

along f is equivalent to μ. Unfortunately, as a consequence of the fact that the

category of algebraic varieties is not closed under quotients, universal deformations

seldom exist, the uniqueness of f is too much to ask. Under some mild assumptions,

there however exist miniversal deformations. Recall that a deformation ω of an

algebra A with the base S is miniversal, if

(i) for any deformation μ of A with the base R there exists a, not necessarily

unique, ring morphism f : S → R such that

f!(ω) is equivalent to μ, and

(ii) if the maximal ideal of R satisfies m2 = 0, then f is unique.

Deformations satisfying (i) only are called versal. The existence of miniversal defor-

mations for a large class of algebras was proved in [FP02] and the citations therein;

see also [SS84].

Deformations in algebraic geometry. Let us mention briefly how deformations

are treated in algebraic geometry. Since we are not going to follow this direction

in the sequel, we will be very telegraphic here, referring to [Har10, Ser06] for

terminology and details.

For a point α of a scheme Y , let (Oα, mα) denote the local ring of α and

k(α) := Oα/mα its residue field. The inclusion {α} → Y induces a morphism

Spec k(α) → Y . Let E be another scheme and p : E → B a proper flat morphism.

The fiber of p over α ∈ Y is the pull-back

(1.33)

✲

❄ ❄

✲

Y .

Spec k(α)

E

p

Fα

It turns out that Fα is a scheme over k(α) whose underlying topological space

equals

p−1(b)

⊂ E.

Let X be a scheme over k(b) isomorphic to Fb, via a fixed isomorphism which

is considered to be a part of the structure. The morphism p : E → Y can be viewed

as a family of deformations of the scheme X parametrized by the points of Y . The

flatness and properness of the morphism p : E → Y guarantee that the fibers vary

in a ‘controlled’ way, and that the above concept is invariant under the base change.

An R-deformation B of an associative -algebra A as in Definition 1.16 should

then ‘ideologically’ be the same as a morphism Spec B → Spec R of spectra whose