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 +
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
n ≥ 1, is augmented,
with the augmentation
It turns out that
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
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
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
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
Definition 1.24 on page 14.