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which is called the module of in¬nite jets for P . Denote by ν∞,k : J ∞ (P ) ’
J k (P ) the corresponding projections. Since νk+1,k —¦ jk = jk+1 for any k ≥ 0,
the system of operators jk induces the mapping j∞ : P ’ J ∞ (P ) satisfying
the condition ν∞,k —¦ j∞ = jk . Obviously, J ∞ (P ) is the representative object
for the functor Diff — (P, •) while the mapping j∞ possesses the universal
property similar to that of jk : for any ∆ ∈ Diff — (P, Q) there exists a unique
1
It makes no di¬erence whether we span µk by the left or the right multiplication due
to the identity la δ a (b — p) = ra δ a (b — p) + δ a δ a (b — p).
160 4. BRACKETS

homomorphism f ∆ : J ∞ (P ) ’ Q such that ∆ = f ∆ —¦ j∞ . Note that j∞ is
not a di¬erential operator in the sense of De¬nition 4.1.2
The functors J k (•) possess the properties dual to those of Diff + (•).
k
Namely, we can de¬ne the l-th Jet-prolongation of ∆ ∈ Diff k (P, Q) by
setting
def
∆(l) = jl —¦ ∆ : P ’ J l (Q)
and consider the commutative diagram
jk
’ J k (P )
P
jk (l)
jk+l jl
“ ’“
cl,k
J k+l (P ) ’ J l J k (P )
(l)
where cl,k = f jk is called the cogluing transformation. Similar to Diagram
(4.6), for any operators ∆ : P ’ Q, : Q ’ R of orders k and l respectively,
we have the commutative diagram
—¦∆
f
k+l
J ’R
(P )

cl,k f

J l (f ∆ ) l
lk
J J (P ) ’ J (Q)
and call cl,k the universal cocompositon operation. This operation is coasso-
ciative, i.e., the diagram
ck+l,s
k+l+s
’ J k+l J s (P )
J (P )

ck,l+s ck,l
“ “
J k (cl,s ) k l s
J k J l+s (P ) ’ J J J (P )
is commutative for all k, l, s ≥ 0.
1.3. Derivations. We shall now deal with special di¬erential operators
of order 1.
Definition 4.4. Let P be an A-module. A P -valued derivation is a
¬rst order operator ∆ : A ’ P satisfying ∆(1) = 0.
2
One might say that j∞ is a di¬erential operator of “in¬nite order”, but this concept
needs to be more clari¬ed. Some remarks concerning a concept of in¬nite order di¬erential
operators were made in Chapter 1, see also [51] for more details.
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 161

The set of such derivations will be denoted by D(P ). From the above
de¬nition and from De¬nition 4.1 it follows that ∆ ∈ D(P ) if and only if
a, b ∈ A.
∆(ab) = a∆(b) + b∆(a), (4.7)
It should be noted that the set D(P ) is a submodule in Diff 1 (P ) but not in
Diff + (P ).
1

Remark 4.2. In the case A = C ∞ (M ), M being a smooth manifold,
and P = A the module D(A) coincides with the module D(M ) of vector
¬elds on the manifold M .
For any A-homomorphism f : P ’ Q and a derivation ∆ ∈ D(P ), the
def
composition D(f ) = f —¦ ∆ lies in D(Q) and thus P ’ D(P ) is a functor
from the category of A-modules into itself. This functor can be generalized
as follows.
Let P be an A-module and N ‚ P be a subset in P . Let us de¬ne
def
D(N ) = {∆ ∈ D(P ) | ∆(A) ‚ N }.
def
Let us also set (Diff + )i = Diff + —¦ · · ·—¦Diff + , where the composition is taken
1 1 1
i times. We now de¬ne a series of functors Di , i ≥ 0, together with natural
embeddings Di (P ) ’ (Diff + )i (P ) by setting D0 (P ) = P , D1 (P ) = D(P )
1
and, assuming that all Dj (P ), j < i, were de¬ned,
Di (P ) = D(Di’1 (P ) ‚ (Diff + )i’1 (P )).
1
Since
D(Di’1 (P ) ‚ (Diff + )i’1 (P )) ‚ D((Diff + )i’1 (P )) ‚ (Diff + )i (P ), (4.8)
1 1 1

the modules Di (P ) are well de¬ned.
Let us show now that the correspondences P ’ Di (P ) are functors for
all i ≥ 0. In fact, the case i = 0 is obvious while i = 1 was considered
above. We use induction on i and assume that i > 1 and that for j < i all
j
Dj are functors. We shall also assume that the embeddings ±P : Dj (P ) ’
(Diff + )i (P ) are natural, i.e., the diagrams
1
j
±P
’ (Diff + )j (P )
Dj (P ) 1


(Diff + )j (f )
Dj (f ) (4.9)
1

“ j
±Q
’ (Diff + )j (Q)
Dj (Q) 1
are commutative for any homomorphism f : P ’ Q (in the cases j = 0, 1,
def
this is obvious). Then, if ∆ ∈ Di (P ) and a ∈ A, we set (Di (f ))(∆) =
Di’1 (∆(a)). Then from commutativity of diagram (4.9) it follows that Di (f )
takes Di (P ) to Di (Q) while (4.8) implies that ±P : Di (P ) ’ (Diff + )i (P ) is
i
1
a natural embedding.
162 4. BRACKETS

Note now that, by de¬nition, elements of Di (P ) may be understood as
K-linear mappings A ’ Di’1 (P ) possessing “special properties”. Given an
element a ∈ A and an operator ∆ ∈ Di (P ), we have ∆(a) ∈ Di’1 (P ), i.e.,
∆ : A ’ Di’1 (P ), etc. Thus ∆ is a polylinear mapping
∆ : A —K · · · —K A ’ P. (4.10)
i times
Let us describe the module Di (P ) in these terms.
Proposition 4.5. A polylinear mapping of the form (4.10) is an ele-
ment of Di (P ) if and only if
∆(a1 , . . . , a±’1 , ab, a±+1 , . . . , ai )
= a∆(. . . , a±’1 , b, a±+1 , . . . ) + b∆(. . . , a±’1 , a, a±+1 , . . . ) (4.11)
and
∆(. . . , a± , . . . , aβ , . . . ) = (’1)±β ∆(. . . , aβ , . . . , a± , . . . ) (4.12)
for all a, b, a1 , . . . , ai ∈ A, 1 ¤ ± < β ¤ i. In other words, Di (P ) consists of
skew-symmetric polyderivations (of degree i) of the algebra A with the values
in P .
Proof. Note ¬rst that to prove the result it su¬ces to consider the
case i = 2. In fact, the general case is proved by induction on i whose step
literally repeats the proof for i = 2.
Let now ∆ ∈ D2 (P ). Then, since ∆ is a derivation with the values in
Diff + (P ), one has
1
∆(ab) = a+ ∆(b) + b+ ∆(a), a, b ∈ A.
Consequently,
∆(ab, c) = ∆(b, ac) + ∆(a, bc) (4.13)
for any c ∈ A. But ∆(ab) ∈ D(P ) and thus ∆(ab, 1) = 0. Therefore, (4.13)
implies ∆(a, b) + ∆(b, a) = 0 which proves (4.12). On the other hand, from
the result proved we obtain that ∆(ab, c) = ’∆(c, ab) while, by de¬nition,
one has ∆(c) ∈ D(P ) for any c ∈ A. Hence,
∆(ab, c) = ’∆(c, ab) = ’a∆(c, b) ’ b∆(c, a) = a∆(b, c) + b∆(a, c)
which ¬nishes the proof.
To ¬nish this subsection, we establish an additional algebraic structure
in the modules Di (P ). Namely, we de¬ne by induction the wedge product
§ : Di (A) —K Dj (P ) ’ Di+j (P ) by setting
def
a § p = ap, a ∈ D0 (A) = A, p ∈ D0 (P ) = P, (4.14)
and
def
(∆ § )(a) = ∆ § (a) + (’1)j ∆(a) § (4.15)
for any ∆ ∈ Di (A), ∈ Dj (P ), i + j > 0.
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 163

Proposition 4.6. The wedge product of polyderivations is a well-
de¬ned operation.
Proof. It needs to prove that ∆ § de¬ned by (4.14) and (4.15) lies
in Di+j (P ). To do this, we shall use Proposition 4.5 and induction on i + j.
The case i + j < 2 is trivial.
Let now i + j ≥ 2 and assume that the result was proved for all k < i + j.
Then from (4.15) it follows that (∆ § )(a) ∈ Di+j’1 (P ). Let us prove that
∆ § satis¬es identities (4.11) and (4.12) of Proposition 4.5. In fact, we
have

(∆ § )(a, b) = (∆ § (a))(b) + (’1)j (∆(a) § )(b)
= ∆ § (a, b) + (’1)j’1 ∆(b) § (a) + (’1)j (∆(a) § (b)
+ (’1)j ∆(a, b) § ) = ’ ∆ § (b, a) + (’1)j’1 ∆(a) § (b)
+ (’1)j ∆(b) § (a) + ∆(b, a) § = ’(∆ § )(b, a),
where a and b are arbitrary elements of A.
On the other hand,

(∆ § )(ab) = ∆ § (ab) + (’1)j ∆(ab) §
= ∆ § a (b) + b (a) + (’1)j a∆(b) + b∆(a) §
= a ∆ § (b) + (’1)j ∆(b) § + b ∆ § (a) + (’1)j ∆(a) §
= a ∆ § )(b) + b(∆ § (a).
We used here the fact that ∆ § (a ) = a(∆ § ) which is proved by trivial
induction.
Proposition 4.7. For any derivations ∆, ∆1 , ∆2 ∈ D— (A) and , 1,
2 ∈ D— (P ), one has
(i) (∆1 + ∆2 ) § = ∆1 § + ∆2 § ,
(ii) ∆ § ( 1 + 2 ) = ∆ § 1 + ∆ § 2 ,
(iii) ∆1 § (∆2 § ) = (∆1 § ∆2 ) § ,
(iv) ∆1 § ∆2 = (’1)i1 i2 ∆2 § ∆1 ,
where ∆1 ∈ Di1 (A), ∆2 ∈ Di2 (A).
Proof. All statements are proved in a similar way. As an example, let
us prove equality (iv). We use induction on i1 + i2 . The case i1 + i2 = 0 is
obvious (see (4.14)). Let now i1 + i2 > 0 and assume that (iv) is valid for
all k < i1 + i2 . Then

(∆1 § ∆2 )(a) = ∆1 § ∆2 (a) + (’1)i2 ∆1 (a) § ∆2
= (’1)i1 (i2 ’1) ∆2 (a) § ∆1 + (’1)i2 (’1)(i1 ’1)i2 ∆2 § ∆1 (a)
= (’1)i1 i2 (∆2 § ∆1 (a) + (’1)i1 ∆2 (a) § ∆1 ) = (’1)i1 i2 (∆2 § ∆1 )(a)
for any a ∈ A.
164 4. BRACKETS

Corollary 4.8. The correspondence P ’ D— (P ) is a functor from the
category of A-modules to the category of graded modules over the graded
commutative algebra D— (A).
1.4. Forms. Consider the module J 1 (A) and the submodule in it gen-
erated by j1 (1), i.e., by the class of the element 1 — 1 ∈ A —K A. Denote by
ν : J 1 (A) ’ J 1 (A)/(A · j1 (1)) the natural projection of modules.
def
Definition 4.5. The quotient module Λ1 (A) = J 1 (A)/(A · j1 (1)) is
called the module of di¬erential 1-forms of the algebra A. The composition
def
d = d1 = ν —¦ j1 : A ’ Λ1 (A) is called the (¬rst) de Rham di¬erential of A.
Proposition 4.9. For any derivation ∆ : A ’ P , a uniquely de¬ned
A-homomorphism •∆ : Λ1 (A) ’ P exists such that the diagram
d
’ Λ1 (A)
A














P
Λ1 (A)
is commutative. In particular, is the representative object for the
functor D(•).
Proof. The mapping d, being the composition of j1 with a homomor-
phism, is a ¬rst order di¬erential operator and it is a tautology that f d (see
Proposition 4.4) coincides with the projection ν : J 1 (A) ’ Λ1 (A). On the
other hand, consider the diagram
f∆
1
J (A) ’P








ν









Λ1 (A)

j1





d

A
Since ∆ is a ¬rst order di¬erential operator, there exists a homomorphism
f ∆ : J 1 (A) ’ P satisfying the equality ∆ = f ∆ —¦ j1 . But ∆ is a derivation,
i.e., ∆(1) = 0, which means that ker(f ∆ ) contains A · j1 (1). Hence, there
exists a unique mapping •∆ such that the above diagram is commutative.

Remark 4.3. From the de¬nition it follows that Λ1 (A), as an A-module,
is generated by the elements da, a ∈ A, with the relations
d(±a + βb) = ±da + βdb, d(ab) = adb + bda,
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 165

±, β ∈ K, a, b ∈ A, while the de Rham di¬erential takes a to the coset
a mod (A · j1 (1)).
Let us set now
Λi (A) = Λ1 (A) § · · · § Λ1 (A) . (4.16)
i times

The elements of Λi (A) are called di¬erential i-forms of the algebra A. We
def
also formally set Λ0 (A) = A.
Proposition 4.10. The modules Λi (A), i ≥ 0, are representative ob-
jects for the functors Di (•).
Proof. The case i = 0 is trivial while the case i = 1 was proved already
(see Proposition 4.9). Let now i > 1 and a ∈ A. De¬ne the mappings
»a : homA (Λi (A), P ) ’ homA (Λi’1 (A), P ), ia : Di (P ) ’ Di’1 (P )
by setting
def def
(»a •)(ω) = •(da § ω), ia ∆ = ∆(a),
where ω ∈ Λi’1 (A), • ∈ homA (Λi (A), P ), and ∆ ∈ Di (P ).
Using induction on i, let us construct isomorphisms
ψi : homA (Λi (A), P ) ’ Di (P )
in such a way that the diagrams
ψi
homA (Λi (A), P ) ’ Di (P )

»a ia (4.17)
“ “
ψi’1
i’1
’ Di’1 (P )
homA (Λ (A), P )
are commutative for all a ∈ A.
The case i = 1 reduces to Proposition 4.9. Let now i > 1 and assume
that for i ’ 1 the statement is valid. Then from (4.17) we should have
• ∈ homA (Λi (A), P ),
(ψi (•))(a) = ψi’1 (»a (•)),
which completely determines ψi . From the de¬nition of the mapping »a it
follows that
»a —¦ »b = ’»b —¦ »a , a, b ∈ A,
»ab = a»b + b»a ,
i.e., im ψi ∈ Di (P ) (see Proposition 4.5).
Let us now show that ψi constructed in such a way is an isomorphism.
Take ∆ ∈ Di (P ), a1 , . . . , ai and set
def
¯ ’1
ψi (da1 § . . . dai ) = ψi’1 (X(a1 )) (da2 § · · · § dai ).
166 4. BRACKETS

’1
It may be done since ψi’1 exists by the induction assumption. Directly from
¯ ¯
de¬nitions one obtains that ψi —¦ ψi = id, ψi —¦ ψi = id. It is also obvious that
the isomorphisms ψi are natural, i.e., the diagrams
ψi
homA (Λi (A), P ) ’ Di (P )

homA (Λi (A), f ) Di (f )
“ “
ψi
homA (Λi (A), Q) ’ Di (Q)
are commutative for all homomorphisms f ∈ homA (P, Q).
From the result proved we obtain the pairing
·, · : Di (P ) —A Λi (A) ’ P (4.18)
de¬ned by
def
’1
ω ∈ Λi (A), ∆ ∈ Di (P ).
∆, ω = ψi (∆) (ω),
A direct consequence of the proof of Proposition 4.10 is the following
Corollary 4.11. The identity
∆, da § ω = ∆(a), ω (4.19)
holds for any ω ∈ Λi (A), ∆ ∈ Di+1 (A), a ∈ A.
Let us de¬ne the mappings d = di : Λi’1 (A) ’ Λi (A) by taking the ¬rst
de Rham di¬erential for d1 and setting
def
di (a0 da1 § · · · § dai ) = da0 § da1 § · · · § dai
for i > 1. From (4.16) and Remark 4.3 it follows that the mappings d i are
well de¬ned.
Proposition 4.12. The mappings di possess the following properties:
(i) di is a ¬rst order di¬erential operator acting from Λi’1 (A) to Λi (A);
(ii) d(ω § θ) = d(ω) § θ + (’1)i ω § d(θ) for any ω ∈ Λi (A), θ ∈ Λj (A);
(iii) di —¦ di’1 = 0.
The proof is trivial.
In particular, (iii) means that the sequence of mappings
d
d
0 ’ A ’1 Λ1 (A) ’ · · · ’ Λi’1 (A) ’i Λi (A) ’ · · ·
’ ’ (4.20)
is a complex.
Definition 4.6. The mapping di is called the (i-th) de Rham di¬er-
ential. The sequence (4.20) is called the de Rham complex of the algebra
A.
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 167

Remark 4.4. Before proceeding with further exposition, let us make
some important comments on the relation between algebraic and geometrical
settings. As we saw above, the algebraic de¬nition of a linear di¬erential
operator is in full accordance with the analytical one. The same is true if we
compare algebraic “vector ¬elds” (i.e., elements of the module D(A)) with
vector ¬elds on a smooth manifold M : derivations of the algebra C ∞ (M )
are identical to vector ¬elds on M .
This situation changes, when we pass to representative objects. A simple
example illustrates this e¬ect. Let M = R and A = C ∞ (M ). Consider the
di¬erential one-form ω = dex ’ ex dx ∈ Λ1 (A). This form is nontrivial as an
element of the module Λ1 (A). On the other hand, for any A-module P let
us de¬ne the value of an element p ∈ P at point x ∈ M as follows. Denote
by µx the ideal
def
µx = {f ∈ C ∞ (M ) | f (x) = 0} ‚ C ∞ (M )
def
and set px = p mod µx . In particular, if P = A, thus de¬ned value coincides
with the value of a function f at a point. One can easily see that ωx = 0
for any x ∈ M . Thus, ω is a kind of a “ghost”, not observable at any point
of the manifold. The reader will easily construct similar examples for the
modules J k (A). In other words, we can state that
Λi (M ) = Λi (C ∞ (M )), “(πk ) = J k (“(π))
for an arbitrary smooth manifold M and a vector bundle π : E ’ M .
Let us say that C ∞ (M )-module P is geometrical, if
µx · P = 0.

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