CHAPTER 1

Basic notions

Augmented rings. By a ring we understand a commutative associative unital

algebra1 over a basic field , that is, the word “ring” shall mean a ring in the

sense of Chapter 1 in [AM69] which is simultaneously a -vector space and whose

structure operations are compatible with the scalar multiplication, i.e.

α(r + r ) = αr + αr , and α(r r ) = (αr )r = r (αr ),

for α ∈ , r , r ∈ R.

Definition 1.1. Let R be a ring with unit e and ω : → R the morphism

given by ω(1) := e. A morphism : R → is an augmentation of R if ω = or,

diagrammatically,

ω

R

✑

✑

✑

✑ ✑ ✸

✻

✲

A ring with an augmentation is an augmented ring.

The subspace R := Ker is called the augmentation ideal of R. Since the

quotient R/R is isomorphic to the field , the augmentation ideal is always maximal.

In this way, each augmentation determines a maximal ideal in R. Vice versa, each

maximal ideal m ⊂ R defines an augmentation R → R/m over the field R/m which

however, as we will see in Example 1.3 below, need not be isomorphic to the basic

field .

Example 1.2. The unital ring [[t]] of formal power series with coeﬃcients in

is augmented, with the augmentation : [[t]] → given by

(

i≥0

aiti

)

:= a0.

It turns out that [[t]] is a local Noetherian ring, with the unique maximal ideal (t)

and residue field , see [AM69, Chapter 1] for the terminology.

Example 1.3. Every α ∈ determines an augmentation

α

: [t] → of the

polynomial ring [t] given by

α

(f) := f(α), for f ∈ [t]. On the other hand, given

an augmentation : [t] → , take α := (t). It is clear that, for this α, = α.

There is therefore a one-to-one correspondence between augmentations of [t] and

points in the aﬃne plane .

The augmentation ideal of

α

: [t] → is the maximal ideal generated by

(t − α). If is algebraically closed, then this assignment is one-to-one, i.e. there

1See

Definition 1.15 below.

1

http://dx.doi.org/10.1090/cbms/116/01