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
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