§18. Complements on extensions of algebras
This section assembles a number of functorial constructions on rings which will be used repeatedly in §§19 and 20; it contains no non-trivial result.
18.1. Inverse images of augmented rings
(18.1.1) Given a ring (not necessarily commutative), the category of -rings has as objects the pairs formed by a ring and a ring homomorphism , and as morphisms (also called -homomorphisms) the ring homomorphisms such that, if and are the homomorphisms (called structural) defining the -ring structure on and respectively, the diagram
u
B ───────→ C
↖ ↗
ρ σ
(18.1.1.1) ↖ ↗
A
is commutative. The kernel of is a two-sided ideal which is an -bimodule (for ).
If is a ring homomorphism, then for every -ring , the pair is an -ring, and if is an -homomorphism from the -ring
to the -ring , it is also an -homomorphism from the -ring to the -ring ; one defines in this way a canonical functor from the category of -rings to that of -rings.
(18.1.2) Let , , be three -rings, and , two -homomorphisms. Recall that
one calls fibre product of and over (for the -homomorphisms and ), and denotes by , the sub-ring of the product ring consisting of the pairs such that ; the
restrictions , of the projections and to are still called the
canonical projections; the -ring structure on is defined by the homomorphism (where and are the structural homomorphisms), which does send
into by virtue of (18.1.1). The characteristic property of is that, for every pair of
-homomorphisms , such that , there exists a unique -homomorphism
such that , . One may also say that is the
projective limit of the projective system formed by , , and the -homomorphisms , , in the category
of -rings .
(18.1.3) Let be the two-sided ideal of , kernel of ; it is immediate that the kernel
of is the ideal formed by the elements (x, 0) with ; the
restriction of is thus an isomorphism of rings without unit element,
and also an isomorphism of -bimodules. Likewise, if is the kernel of , the kernel
of is the ideal of elements (0, y) with , and the restriction of is an isomorphism (in the same sense). Finally, it is clear that the kernel
of the -homomorphism of into is , so that one has the commutative diagram
0 0 0
│ │ │
↓ ↓ ↓
i_2
𝔍' ⊕ 𝔎' ───→ 𝔎' ──────→ 𝔎
│ │ │
(18.1.3.1) │ ↓ j ↓
│ q p_1
0 ──→𝔍' ───────→ G ───────→ F ──→ 0
│ │ │
i_1│ p_1│ │ g
↓ ↓ ↓
0 ──→𝔍 ────────→ E ───────→ B ──→ 0
f
The definitions and results of (18.1.2) and (18.1.3) extend at once to the fibre product of an arbitrary family
of -rings defined by a family of -homomorphisms .
We leave the formulation of these results to the reader.
(18.1.4) In agreement with the terminology of (M, VIII), we shall call an -ring augmented over an
-ring equipped with a surjective -homomorphism (called the augmentation of ), ; the kernel
of is called the augmentation ideal. One says that the augmented -ring is trivial if there
exists an -homomorphism of -rings which is a right inverse of the augmentation (in
other words ). The exact sequence of -bimodules
is then split; in other words, one can identify the -bimodule with , and the multiplication in is then given by
(b, z)(b', z') = (bb', bz' + zb' + zz'),
being considered as a -bimodule by means of .
(18.1.5) With the notations of (18.1.2), suppose that the -homomorphism is surjective, in other words that
it makes an augmented -ring over . Then it is clear that is also surjective, in other words
defines on a structure of augmented -ring over , which one calls the inverse image by of the
augmented ring .
Proposition (18.1.6).
For the inverse image by of the augmented -ring to be a trivial augmented -ring, it is necessary and sufficient that there exist an -homomorphism making commutative the diagram
F
↙ │
u │ g
↙ ↓
E ──→ B
f
The condition is evidently necessary, taking , where is an -homomorphism right
inverse of . Conversely, if there exists an -homomorphism satisfying the condition of the statement, the
existence of the right inverse of follows from the universal property of the fibre product (18.1.2)
applied to the -homomorphisms and .
This result entails in particular that if is a trivial augmented ring, so are all its inverse images.
(18.1.7) Let us resume the situation described in (18.1.2) and (18.1.3); we evidently have on a
structure of -bimodule, coming from its structure of -bimodule and from the ring homomorphism .
Since is bijective, we have furthermore an injection , which
is a homomorphism of -bimodules. We shall see conversely that, when one further assumes that and are
surjective, in other words that and are augmented -rings over , the datum of such a homomorphism
allows one to reconstitute the augmented ring (over ) from the augmented ring (over ).
More precisely, let us give ourselves an -ring augmented over and an -ring augmented over , the augmentation ideals being denoted by and respectively:
0
│
↓
𝔎
θ │
↓
(18.1.7.1) 0 ──→ 𝔍' ──→ G ────→ F ──→ 0
h
│
↓ g
B
│
↓
0
The homomorphism defines on all ideals of (and in particular on ) a structure of -bimodule. Let
be a homomorphism of -bimodules making the diagram (18.1.7.1) commutative; this
implies that is injective and that ; since
is a two-sided ideal of , and since , the composite homomorphism factors as , where is surjective; furthermore, the image
of in is the kernel of (that of being
), and the restriction to of the canonical homomorphism is injective. One can therefore form the fibre product , and since the two
-homomorphisms and are such that , they define a unique
-homomorphism by the universal property of (18.1.2); we shall see that is bijective. It
suffices to prove this when is considered as a homomorphism of -bimodules; one notes then that is compatible
with the finite filtrations on and formed respectively by and , and by
and ; furthermore, one has seen (18.1.3) that is the identity and that is bijective, hence itself is bijective
(Bourbaki, Alg. comm., chap. III, §2, n° 8, cor. 3 of th. 1).
18.2. Extensions of a ring by a bimodule
(18.2.1) Let be an augmented -ring over , the augmentation, the
augmentation ideal. If one has , then is not only an -bimodule but also a
-bimodule, since is isomorphic to ; more precisely, every is of the form with , and if , are in one has (resp. ), so that the value of
xz' (resp. z'x) does not depend on the element , and can be written bz' (resp. z'b), which
defines the -bimodule structure under consideration. Conversely, if is equipped with a structure of
-bimodule such that and for and , it is clear that
.
(18.2.2) One calls an -extension of an -ring by a -bimodule an exact sequence of homomorphisms of -bimodules
0 → L ─→ E ─→ B → 0
j f
where is an -ring, an -homomorphism of rings, and one has, for and ,
j(f(x) z) = x j(z), j(z f(x)) = j(z) x,
whence it follows (18.2.1) that is a two-sided ideal of square zero of . By abuse of language one also says
that is an extension of by . One says that two -extensions , of by are -equivalent
if there exists an isomorphism of -rings (also called an -equivalence of
-extensions) making commutative the diagram
E
↗ │ ↘
j │u f
↗ ↓ ↘
(18.2.2.1) 0 → L B → 0
↘ ↑ ↗
j' │ f'
↘ │ ↗
E'
(18.2.3) One says that an -extension of by a -bimodule is -trivial if is a trivial
augmented -ring (over ) (18.1.4). One defines on the product -bimodule a structure of -ring
by setting , as one verifies at once, and it is immediate that the canonical maps and define an -extension of by which is -trivial, the canonical map
being an -homomorphism right inverse of . One says that this extension is the trivial type
extension of by and denotes it by ; it is immediate that every -trivial -extension of by
is -equivalent to .
One will note that every extension of the -ring itself by an -bimodule is necessarily -trivial.
(18.2.4) Given two -extensions , , a morphism of the first to the second is by definition a triple of homomorphisms of -bimodules such that the diagram
0 ──→ L ──→ E ──→ B ──→ 0
j f
(18.2.4.1) w │ u │ v │
↓ ↓ ↓
0 ──→ L' ─→ E' ─→ B' ─→ 0
j' f'
is commutative, and being -homomorphisms of rings and being such that and for and (in other words, the pair constitutes a di-homomorphism of the
-bimodule into the -bimodule ); it is clear that if (u', v', w') is a morphism from
to an -extension , then is still a morphism, which
justifies the terminology.
The consideration of the two commutative squares of the diagram (18.2.4.1) will lead us to two operations on
extensions of -rings.
(18.2.5) In the first place, consider an -extension of by
0 → L' ─→ E' ─→ B' → 0
j' f'
and an -homomorphism of rings , and let be the inverse image by of the
augmented -ring (18.1.5), so that one has a commutative diagram
0 ──→ L_0 ──→ F ──→ B ──→ 0
p_1 p_2 │
i ↓ ↓ v
0 ──→ L' ──→ E' ──→ B' ──→ 0
j' f'
whose rows are exact, and being the canonical homomorphisms; one has seen (18.1.3) that is
bijective, and it also follows from the definition (18.1.2) that , so that one can consider as an
-extension of by , which one calls the inverse image by of the extension of by
( being naturally considered as -bimodule by means of the ring homomorphism ). The functorial
character of the fibre product with respect to each of the factors shows furthermore that if one has a morphism between two extensions of
0 ──→ L'_1 ──→ E'_1 ──→ B' ──→ 0
g' 1_{B'} │
h │ │ ↓
↓ ↓
0 ──→ L'_2 ──→ E'_2 ──→ B' ──→ 0
one deduces from it a morphism inverse image by
0 ──→ L'_1 ──→ E'_1 ×_{B'} B ──→ B ──→ 0
h │ g' ×_{B'} 1_B │ 1_B │
↓ ↓ ↓
0 ──→ L'_2 ──→ E'_2 ×_{B'} B ──→ B ──→ 0
In particular, if and are -equivalent -extensions of by , their inverse images by are -equivalent -extensions of by .
The definition of the fibre product shows that when one has a morphism (18.2.4.1) of -extensions, it factors
through the inverse image of by ; more precisely, there exists a unique -homomorphism making commutative the diagram
0 ──→ L ──→ E ──→ B ──→ 0
w_0│ u_0│ 1_B │
↓ ↓ ↓
0 ──→ L_0 ─→ F ──→ B ──→ 0
p_1 p_2 │
i ↓ ↓ v
0 ──→ L' ──→ E' ──→ B' ──→ 0
j' f'
where is the restriction of to and , .
(18.2.6) Let us study in particular the inverse images of extensions by surjective -homomorphisms. Consider a
surjective -homomorphism , an -extension of by a -bimodule , and the two-sided ideal
of , kernel of (which can be considered as -bimodule by means of the augmentation );
one has seen (18.1.7) that every homomorphism of -bimodules making commutative the
diagram
𝔎
θ │
↓
F ──→ B
determines an extension of by whose inverse image by is equivalent to , and that every -extension of by having this latter property is obtained in this way (up to -equivalence). Furthermore, for two homomorphisms , of -bimodules from into to give two -equivalent -extensions of by , it is necessary and sufficient that there exist an -equivalence of the -extension onto itself such that
; this follows at once from what was seen in (18.2.5) and from the definition of the
canonical bijection of onto the fibre product (18.1.7).
(18.2.7) Consider now the left square of (18.2.4.1), and let us first recall the notion of amalgamated sum in
the category of -bimodules: given three -bimodules , , and two -homomorphisms ,
, the amalgamated sum is the inductive limit of the inductive system formed by , ,
and the -homomorphisms , in the category of -bimodules . One defines this
-bimodule as the quotient of the product by the sub--bimodule image of under the homomorphism
. Its characteristic property is that, for every pair of homomorphisms of -bimodules
, such that , there exists a unique homomorphism
such that and , where
and are the canonical maps.
(18.2.8) Consider now an -extension of by a -bimodule :
and on the other hand let be a homomorphism of -bimodules. Let be the amalgamated sum -bimodule ; let us show how one can endow this -bimodule with a structure of -ring and define an -extension
For this, note that is endowed with a structure of -bimodule by means of the augmentation homomorphism ; one can therefore form the trivial type -extension (18.2.3). Consider then the map from into ; this is a homomorphism of -bimodules ( being considered as -bimodule
by means of the canonical homomorphism ). Indeed, for and , one has ; now by definition of the -bimodule structure on
, and and ; one verifies likewise that is a homomorphism of left
-modules. One can then apply to the commutative diagram
0
│
↓
L
θ │
↓ j
0 ──→ L' ──→ G ──→ E ──→ 0
│
↓ f
B
│
↓
0
the result of (18.1.7). As by definition, our assertion is an immediate consequence of (18.1.7).
One says that the -extension of by is deduced from by means of the homomorphism . The functorial character of the amalgamated sum in each of its summands shows furthermore that if one has a morphism of extensions
0 ──→ L ──→ E_1 ──→ B_1 ──→ 0
1_L│ g │ h │
↓ ↓ ↓
0 ──→ L ──→ E_2 ──→ B_2 ──→ 0
one deduces from it canonically a morphism of extensions
0 ──→ L' ──→ E_1 ⊕_L L' ──→ B_1 ──→ 0
1_{L'}│ g ⊕ 1_{L'}│ h │
↓ ↓ ↓
0 ──→ L' ──→ E_2 ⊕_L L' ──→ B_2 ──→ 0
In particular, if E_1 and E_2 are -equivalent -extensions of by , the extensions of by which
one deduces from them by means of are -equivalent.
When one has a morphism (18.2.4.1) of -extensions, it factors through the -extension of by deduced
from by means of the homomorphism ( being considered as -bimodule by means of the homomorphism
): indeed, the definition of the amalgamated sum shows that there exists a unique -homomorphism
of -bimodules, making commutative the diagram
0 ──→ L ──→ E ──→ B ──→ 0
j_1│ u_0│ 1_B │
↓ ↓ ↓
0 ──→ L' ──→ H ──→ B ──→ 0
j_0 f_0
1_{L'}│ u_0│ v │
↓ ↓ ↓
0 ──→ L' ──→ E' ──→ B' ──→ 0
j' f'
with , , and being the canonical homomorphisms; one verifies immediately that is also a ring homomorphism.
Let us finally note the functorial properties relative to trivial extensions:
Proposition (18.2.9).
Let , be two -rings, a -bimodule, a -bimodule, an -homomorphism of rings, a homomorphism of -bimodules such that is a di-homomorphism of bimodules. Then there exists a unique -homomorphism of rings making commutative the diagrams
D_B(L) ──→ D_{B'}(L') D_B(L) ──→ D_{B'}(L')
u u
↑ ↑ ↑
│ │ │
B ───→ B' L ───→ L'
v w
where the vertical arrows are the canonical injections.
Indeed, can only be the map , and it remains to verify that this is an -homomorphism
of rings, which results trivially from the definition (18.2.3). One notes that also makes commutative the diagram
D_B(L) ──→ D_{B'}(L')
u
│ │
↓ ↓
B ───→ B'
v
where this time the vertical arrows are the augmentations.
Proposition (18.2.10).
Let be an -ring, a -bimodule, an -extension of by . One defines a bijective map of the set of homomorphisms of -rings of into (in other words, the set of -homomorphisms right inverse of the augmentation ) onto the set of -equivalences of onto by making correspond to every the -equivalence ; the inverse map makes correspond to every the -homomorphism .
This results at once from the definitions.
18.3. The group of classes of -extensions
(18.3.1) Consider a fixed -ring and a fixed -bimodule ; then the relation " and are -equivalent" between -extensions , of by is an equivalence relation, and for this relation one can speak of the set of classes of -equivalent -extensions of by . To see this, it suffices to remark that if, for every , is an element of whose image in is , every is written in a unique way in the form , where , and one can write , , where and are elements of , the maps and from to having to satisfy conditions expressing that is an -ring, which it is pointless to write here. Every -extension of by is therefore -equivalent to an -extension of which is the underlying set, from which one draws our conclusion at once.
(18.3.2) The -ring being fixed, let us provisionally denote by the set of classes of -extensions of
by . For every -homomorphism of -bimodules, one defines canonically a map by making correspond to the class of an -extension of by the class of the -extension deduced from it by means of , by virtue of (18.2.8). If is a second homomorphism
of -bimodules, one has in addition
(18.3.2.1) T(w' ∘ w) = T(w') ∘ T(w).
Indeed, one knows that there exists a canonical isomorphism of -bimodules
E ⊕_L L'' ⥲ (E ⊕_L L') ⊕_{L'} L''
by virtue of the general properties of inductive limits (cf. for example (I, 3.3.9)), and it is immediate to verify
that this is indeed an -equivalence of -extensions, whence
the relation (18.3.2.1). If denotes the category of -bimodules, one sees that one has thus
defined a covariant functor .
(18.3.3) Consider now a family of -bimodules and their product ; the projections define maps
whence a canonical map
(18.3.3.1) ∏_α T(pr_α) : T(L) → ∏_{α ∈ I} T(L_α).
We shall see that this map is bijective. Indeed, for every , let be an -extension of
by , and let be the fibre product of the over (18.1.3); it is immediate that is
an -extension of by and that if one replaces each by an
-equivalent -extension , the fibre product of the over is -equivalent to
. One has thus defined a map , and it is clear that this map is inverse
to (18.3.3.1), whence our assertion. One will canonically identify with by
the map (18.3.3.1). One further verifies immediately that if is a second family of
-bimodules and, for each , is a homomorphism of -bimodules,
then, setting , is identified with
when one makes the preceding identification.
(18.3.4) This being so, for a -bimodule , the addition is a homomorphism of -bimodules, and the same holds for the symmetry of the additive law of . One deduces from this a composition law
T(s) : T(L) × T(L) → T(L)
on by virtue of (18.3.3), and this law is a commutative group law of which is the symmetry, as follows
from the definition of a group object by means of commutative diagrams . We shall denote by
the commutative group thus defined and we shall say that it is the group of classes of -extensions
of by .
(18.3.5) Let us denote by the category whose objects are the triples where is a ring, an -ring and a -bimodule; the morphisms of this category are the triples where and are two ring homomorphisms making commutative the left square of the diagram
A ────→ B L
u │ v │ w │
↑ ↑ ↓
A' ───→ B' L'
where the horizontal arrows are the structural homomorphisms; finally is a homomorphism of commutative groups such that and
whatever and (in other words, is a homomorphism of -bimodules when one endows with the -bimodule structure defined by ). The composition of morphisms is defined by , which is justified at once. We propose to show that
(18.3.5.1) (A, B, L) ↦ Exan_A(B, L)
is a covariant functor from the category to the category Ab of commutative groups. It is thus a matter
of, for every triple as above, defining a homomorphism of commutative groups
(u, v, w)_* : Exan_A(B, L) → Exan_{A'}(B', L').
By virtue of the definition of morphisms in , one can write
(u, v, w) = (1_{A'}, 1_{B'}, w) ∘ (1_{A'}, v, 1_L) ∘ (u, 1_B, 1_L)
where, in the first factor, is endowed with its -bimodule structure defined by ; we shall therefore first define when two of the homomorphisms , , are reduced to the identity.
(18.3.6) We shall take first for the map
(18.3.6.1) w_* : Exan_A(B, L) → Exan_A(B, L')
denoted in (18.3.2); it is immediate to verify that this is a group homomorphism, this property expressing
itself by the commutativity of diagrams, transformed by from analogous diagrams for and .
The map is the map
(18.3.6.2) v^* : Exan_A(B, L) → Exan_A(B', L)
defined in the following way: if is an -extension of by , one has seen that is an
-extension of by (18.2.5), and that if one replaces by an -equivalent -extension , is -equivalent to ; the image by of the class of is the class of . One verifies at once that if is a homomorphism of -bimodules, the diagram
Exan_A(B, L) ──→ Exan_A(B', L)
v^*
w_* │ │ w_*
↓ ↓
(18.3.6.3)
Exan_A(B, L') ──→ Exan_A(B', L')
v^*
is commutative, and being considered as -bimodules by means of in the right-hand column. Replacing and respectively by and , and by the addition in , one concludes that is indeed a group homomorphism.
Finally, the map is the map
(18.3.6.4) u^* : Exan_A(B, L) → Exan_{A'}(B, L)
obtained by making correspond to an -extension of by the ring considered as -ring by means of
(18.1.1), which is evidently an -extension of by , being also considered as -ring by means of ;
it is clear that an -equivalence is also an -equivalence, whence the map (18.3.6.4), which, for every
homomorphism of -bimodules, still makes commutative the diagram
Exan_A(B, L) ──→ Exan_{A'}(B, L)
u^*
w_* │ │ w_*
↓ ↓
(18.3.6.5)
Exan_A(B, L') ─→ Exan_{A'}(B, L')
u^*
from which one concludes as above that is a group homomorphism.
This being so, one sets , and one verifies easily, on account of the commutativity of the diagrams (18.3.6.3) and (18.3.6.5),
that one has , which completes the
proof that (18.3.5.1) is a functor.
The existence of the group homomorphism (18.3.6.1) shows in particular that if is a trivial -extension of
by , the extension of by defined in (18.2.8) is also trivial, which one moreover
verifies without difficulty in a direct fashion.
On the other hand, for every element of the centre of , the homothety is an endomorphism of the -bimodule , hence is an endomorphism of the commutative group , and by functoriality, these endomorphisms define on a canonical structure of -module.
(18.3.7) Let , be two rings, a homomorphism, an -ring and a -bimodule. The kernel of the group homomorphism
u^* : Exan_A(B, L) → Exan_{A'}(B, L)
is formed by definition of the classes of -extensions of by which are -trivial when one considers them as -extensions by means of . One denotes this kernel by the notation when this does not lead to confusion.
If is a ring, and if one has a commutative diagram of -homomorphisms of -rings
B' ──→ B
(18.3.7.1) ↑ ↑
│ │
A' ──→ A
one deduces canonically homomorphisms
(18.3.7.2) Exan_{A/Λ}(B, L) → Exan_{A'/Λ}(B, L) → Exan_{A'/Λ}(B', L)
which come from the commutativity of the diagram
Exan_Λ(B, L) ──→ Exan_{A'}(B, L) ──→ Exan_{A'}(B', L)
│ │ │
↓ ↓ ↓
Exan_A(B, L) ──→ Exan_A(B, L) ──→ Exan_A(B', L)
where the arrows are deduced from those of (18.3.7.1) by functoriality.
Proposition (18.3.8).
Let be a ring, a two-sided ideal of , the quotient ring; is then canonically endowed with a structure of -bimodule. For every -bimodule , let be the additive group of homomorphisms of -bimodules from to . One then defines a canonical isomorphism of commutative groups
(18.3.8.1) η_L : Hom_C(𝔍 / 𝔍^2, L) ⥲ Exan_B(C, L)
by making correspond to every -homomorphism (which is a fortiori a -homomorphism) the class of the extension deduced from the extension of by by means of ; the inverse isomorphism makes correspond to the class of a -extension of by the homomorphism such that the composite is the restriction to of the structural homomorphism .
Let , which is a -extension of by . For every -extension of by , the structural homomorphism is such that the composite is the canonical homomorphism . As the image of by is null, is contained in the kernel of , that is , and as is of square zero, one has ; hence factors as , and if is the restriction of to , one has a commutative diagram
0 ──→ 𝔍 / 𝔍^2 ──→ F ──→ C ──→ 0
w │ u │ 1_C │
↓ ↓ ↓
0 ──→ L ──→ E ──→ C ──→ 0
j p
in other words is a morphism of extensions (18.2.4). One deduces from it a morphism of extensions
0 ──→ L ──→ E' ──→ C ──→ 0
1_L│ u' │ 1_C │
↓ ↓ ↓
0 ──→ L ──→ E ──→ C ──→ 0
where (18.2.8); it is a matter of proving that is a
-equivalence, which will establish that is bijective. By virtue of the construction made in (18.2.8),
everything comes down to proving (taking account of (18.1.7)) that the inverse image of the extension of
by the canonical homomorphism is -trivial, which is evident since factors as
(18.1.6).
It remains to see that is a group homomorphism; now, for every -homomorphism , it is immediate that the diagram
η_L
Hom_C(𝔍 / 𝔍^2, L) ─⥲ Exan_B(C, L)
│ │
Hom(1, h)│ h_*│
↓ ↓
Hom_C(𝔍 / 𝔍^2, L') ─⥲ Exan_B(C, L')
η_{L'}
is commutative. It suffices to apply this remark to the homomorphism defining the addition to conclude.
18.4. Extensions of algebras
(18.4.1) Let be a commutative ring. The category of -algebras is then a full subcategory of that of -rings, characterized by the fact that the structural homomorphisms are central.
If is an -algebra and a -bimodule, it is immediate that the trivial type -extension is an
-algebra. Likewise, in the construction of (18.2.5) (resp. of (18.2.8)), if , and (resp. and
) are -algebras, the same holds for (resp. for ). Finally, it is clear that if
is an -algebra and an -extension of by a -bimodule which is an -algebra, every -extension
of by , -equivalent to , is also an -algebra. One then deduces at once from the definition (18.3.4)
that the classes of equivalent -extensions of by which are -algebras form a subgroup, denoted , of . Let be the full subcategory of the category defined in
(18.3.5), whose objects are such that is commutative and an -algebra. Then what precedes shows
that
(A, B, L) ↦ Exal_A(B, L)
is a covariant functor from to Ab. The results of (18.3.7) are unchanged when one replaces Exan
by Exal everywhere.
(18.4.2) Suppose always the ring commutative. The remarks of (18.4.1) remain valid when one replaces "algebra"
by "commutative algebra" and "bimodule" by "module". If is a commutative -algebra and a -module,
the classes of equivalent -extensions of by which are commutative -algebras (or, what comes to the same thing, commutative -rings) form a subgroup of , denoted . If is the full subcategory of formed by the triples where and are commutative and is a -module,
(A, B, L) ↦ Exalcom_A(B, L)
is still a covariant functor from to Ab. One can also in (18.3.7) replace Exan by Exalcom
everywhere. Finally, if in (18.3.8) one supposes that is a commutative ring and that is a -module, the same
reasoning gives a canonical isomorphism
(18.4.2.1) Hom_C(𝔍 / 𝔍^2, L) ⥲ Exalcom_B(C, L)
where the first member is the group of homomorphisms of -module.
(18.4.3) Let be a commutative ring. An important case of extensions of -algebras is formed by the
-algebras , extensions of an -algebra by a -bimodule , such that , as an -module, is a trivial
extension of the -module by the -module ; in other words, the exact sequence of -modules is split. One then says that is an extension of Hochschild of by . This will always be the
case when is a projective -module, and in particular when is a commutative field. As an -module, one can
identify with , the multiplication in being given by with
. If one writes that this multiplication defines on a structure of -algebra, one finds
(M, XIV, 2) that must be an -bilinear map of into , such that
(18.4.3.1) f(xy, z) + f(x, y) z = x f(y, z) + f(x, yz)
in other words, is a 2-cocycle on with values in , in the sense of Hochschild; for the extension to be -trivial, it is necessary and sufficient that one have
(18.4.3.2) f(x, y) = x g(y) − g(xy) + g(x) y
where is an -linear map of into , in other words must be a 2-coboundary in the sense of Hochschild. One deduces at once that the classes of Hochschild extensions of by form a subgroup of , isomorphic to the Hochschild cohomology group .
If is a commutative -algebra, and a -module, the commutative Hochschild extensions of by correspond to the symmetric 2-cocycles, that is to say those such that . The classes of commutative Hochschild extensions of by therefore form a subgroup of , isomorphic to the subgroup of image of the group of symmetric 2-cocycles, which we shall denote .
(18.4.4) Let us limit ourselves to the case where and are commutative and a -module, and recall in
this case the equivalent definition of the Hochschild cohomology groups (M, IX, 6). One considers a
complex of -modules, where ( times), the -module structure being defined
by ; the boundary, of degree , , is defined by
d_n(x_1 ⊗ x_2 ⊗ ⋯ ⊗ x_{n+1}) = (x_1 x_2) ⊗ x_3 ⊗ ⋯ ⊗ x_{n+1}
− x_1 ⊗ (x_2 x_3) ⊗ ⋯ ⊗ x_{n+1} + ⋯
+ (−1)^{n−1} x_1 ⊗ x_2 ⊗ ⋯ ⊗ (x_n x_{n+1})
+ (−1)^n (x_{n+1} x_1) ⊗ x_2 ⊗ ⋯ ⊗ x_n
which is indeed -linear since is commutative.
A 2-cocycle of this complex with values in is a -linear map of into such that
; but as , the
cocycle is determined by the -bilinear map of
into , and on writing the preceding condition for , one recovers for the condition (18.4.3.1). Likewise, a
2-coboundary will be a map of the form , where is linear, and here again is determined by the -linear map
of into ; one then obtains , which gives (18.4.3.2)
again. One proceeds likewise for every , and one thus sees that one has
(18.4.4.1) H^i_A(B, L) = H^i(Hom_B(P_•, L)).
(18.4.5) Under the conditions of (18.4.4), one can interpret in the same way the group
(18.4.3). For this, let us modify the complex in degree 3, by considering a new complex
P'_• : P'_3 ──→ P_2 ──→ P_1;
d'_3 d_2
we shall take , coinciding with on P_3, and
being given on by
d'_3(x ⊗ y ⊗ z) = x ⊗ y ⊗ z − x ⊗ z ⊗ y.
The relation follows from the commutativity of . With the notations introduced above, a 2-cocycle
of now corresponds to an -bilinear map which is symmetric and satisfies
(18.4.3.1); consequently, one has
H^2_A(B, L)^s = H^2(Hom_B(P'_•, L)).
(18.4.6) In the particular case where one considers a commutative field , an extension of , one has
whatever the -vector spaces , , and consequently (M, VI, 3.3 a)), one has a canonical
isomorphism
(18.4.6.1) H^i_k(K, L) ⥲ Hom_K(H_i(P_•), L)
and likewise
(18.4.6.2) H^2_k(K, L)^s ⥲ Hom_K(H_2(P'_•), L).
18.5. Case of topological rings
(18.5.1) Let , be two topological rings whose topology is linear, a continuous homomorphism, a topological -bimodule, and suppose that there exists an open two-sided ideal of such that , so that can be
considered as a -bimodule for every open ideal . Let then
be an open two-sided ideal of , an open two-sided ideal of
such that , so that can be considered as a discrete -ring. The preceding remark proves that the group is defined;
furthermore, if , are two open two-sided ideals
of and respectively such that , one has by (18.3.5.1) a canonical
homomorphism
(18.5.1.1) Exan_{A/𝔍}(B / 𝔎, L) → Exan_{A/𝔍'}(B / 𝔎', L).
The set of pairs of open ideals such that is evidently right-filtered for the relation " and
", and the maps (18.5.1.1) define an inductive system of additive groups with this
set as indexing set. One sets, by abuse of notation (for it is no longer a question of a group in natural bijective
correspondence with a set of extensions)
(18.5.1.2) Exantop_A(B, L) = lim Exan_{A/𝔍}(B / 𝔎, L).
─────→
To say that the second member of (18.5.1.2) is zero therefore means that, for every pair of open ideals , such that and
every -extension of by , there exist two open ideals , such that and
that the inverse image by the homomorphism of is trivial.
We leave to the reader the analogous definition of the inductive limits , starting from and , for the case where is commutative and a topological -algebra (resp. commutative -algebra).
(18.5.2) If one has a commutative diagram of continuous homomorphisms of rings
B' ──→ B
↑ ↑
│ │
A' ──→ A
one deduces from it canonically two homomorphisms of additive groups
Exantop_A(B, L) → Exantop_{A'}(B, L) → Exantop_{A'}(B', L)
by passage to the inductive limit starting from (18.3.5.1).
By virtue of the exactness of the functor lim in the category of commutative groups, the kernel of the homomorphism
Exantop_A(B, L) → Exantop_{A'}(B, L)
is the inductive limit of the kernels of the homomorphisms
Exan_{A/𝔍}(B / 𝔎, L) → Exan_{A'/𝔍'}(B / 𝔎, L)
where one has taken for the inverse image of ; one denotes this kernel by . One defines similarly and . Finally, if one has a homomorphism
continuous of -bimodules , one deduces from it canonically a homomorphism of additive groups
Exantop_A(B, L) → Exantop_A(B, L')
by passage to the inductive limit starting from (18.3.6.1).
(18.5.3) Given a topological ring and two topological -bimodules , , one denotes by the additive group of continuous -homomorphisms of into .
Lemma (18.5.3.1).
Let be a topological ring, , two topological -bimodules; one supposes that the topologies are linear, that is discrete and annihilated by an open two-sided ideal of . Then one has a canonical isomorphism
(18.5.3.2) lim Hom_{C/𝔎}(E / V, L) ⥲ Hom.cont_C(E, L)
─────→
where in the first member the inductive limit is taken following the right-filtered ordered set of pairs such that is an open two-sided ideal of , an open sub--bimodule of , such that , , .
As and are discrete, one has canonical homomorphisms
w_{𝔎, V} : Hom_{C/𝔎}(E / V, L) → Hom.cont_C(E, L) forming an inductive system, whence a homomorphism (18.5.3.2) by
passage to the inductive limit. As the homomorphism is surjective for , it follows at
once from the definition that the homomorphism (with , , ) is injective, and the same evidently holds for the homomorphism
Hom_{C/𝔎}(E / V, L) → Hom_{C/𝔎'}(E / V, L)
for ; one concludes that the homomorphism (18.5.3.2) is injective. On the other
hand, if is a continuous -homomorphism of into , its kernel is an open sub-bimodule V_0 of , and if
is an open two-sided ideal of such that
and , , it is clear that is the
canonical image of a -homomorphism of into , hence (18.5.3.2) is surjective.
This being so, the proposition (18.3.8) generalizes as follows to topological rings:
Proposition (18.5.4).
Let be a linearly topologized topological ring, a two-sided ideal of , the quotient topological ring; (where is endowed with the topology induced by that of and with the quotient topology of that of ) is then canonically endowed with a structure of topological -bimodule. For every discrete -bimodule annihilated by an open ideal of , there exists then a canonical isomorphism
(18.5.4.1) Hom.cont_C(𝔍 / 𝔍^2, L) ⥲ Exantop_B(C, L).
Indeed, for every open ideal of such that annihilates
, one has, by (18.3.9), a canonical isomorphism
Hom_{B/(𝔍 + 𝔎)}((𝔍 + 𝔎) / (𝔍^2 + 𝔎), L) ⥲ Exan_B(B / (𝔍 + 𝔎), L)
and it suffices to pass to the inductive limit, taking account of (18.5.3.1).