2 1. BASIC NOTIONS

is a correspondence between augmentations of [t] and maximal ideals in [t], see

[Har77, Example 1.4.4].

For a general field , there may be more maximal ideals in [t] than augmen-

tations : [t] → . For instance, the ring [t] of polynomials with real coeﬃ-

cients contains the maximal principal ideal generated by (1 +

t2),

but the quotient

[t]/(1 + t2) is isomorphic to the field of complex numbers . This isomorphism

is induced by the ring morphism (augmentation over ) : [t] → given by

(t) = −i, where i :=

√

−1 is the imaginary unit.

Example 1.4. The truncated polynomial ring

[t]/(tn+1),

n ≥ 1, is augmented,

with the augmentation

(

0≤i≤n

aiti

)

:= a0.

It turns out that

[t]/(tn+1)

is a local Artin ring, with the unique maximal ideal

(t) and residue field – for the terminology see again [AM69, Chapter 1]. The

particular case n = 1 leads to the ring

D :=

[t]/(t2)

of dual numbers.

In the rest of this chapter, R will be an augmented ring, with the augmentation

: R → and the unit map ω : → R.

Modules over augmented rings. By an R-module we will understand a left

R-module, i.e. a module in the sense of [AM69, Chapter 2] or [ML63a, §I.1].

As usual, a vector space is a module over a field, in most cases over our basic

field . Bimodules are defined in [ML63a, §V3]. Let us formulate a couple of

useful remarks.

A unital augmented ring R is a - -bimodule (that is, left - right - bimodule),

with the bimodule structure induced by the unit map ω in the obvious manner.

Likewise, is an R-R bimodule, with the structure induced by the augmentation .

For a -vector space V and a unital augmented ring R we denote by R V the

tensor product R ⊗ V , with the left R-module action r (r ⊗ v) := r r ⊗ v, for

r , r ∈ R and v ∈ V . It is clear that R V together with the natural -linear

inclusion ι : V

∼

=

1⊗V → R V is the free R-module generated by V . This means

that, for every R-module M and a -linear map φ : V → M, there exists a unique

R-module morphism Φ : R V → M making the diagram

Φ

φ

ι

M

R V

V

❍

❍

❍

❍

❍ ❥

❄

✲

commutative. We will use both notations for R V = R ⊗ V . The advantage of

R V is that it is shorter and that it emphasizes the left R-module action, while

R ⊗ V refers directly to the tensor product structure.

Topologies and completions. To include formal deformations2 into our gen-

eral setup, it will be necessary to introduce a completed version of the free R-

module R V . To this end we recall some basic facts from Chapter 10 of [AM69],

which, along with [Lef42, Chapter II], should serve as the basic reference for this

2See

Definition 1.24 on page 14.