Lectures

on Linear Differential Operators

over Commutative Algebras The 1st Diffiety School

Forino, Italy

July, 1998

Joseph Krasil’shchik The Independent University of Moscow

in collaboration with Barbara Prinari Università di Lecce

This work was partially supported by the INTAS Grant N 96-0793. The author is indebted to L’istituto Italiano per gli Studi Filosofici for the financial support of the 1st Italian Diffiety School, where these lectures were given, and to the Community of Forino for the hospitality and

stimulating conditions. It is also my pleasure to express gratitude to the Department of Applied Mathematics of the University of Seville, where this

text was written in the final form.

Contents

Foreword v Lecture 1 1 Lecture 2 9 Lecture 3 17 Lecture 4 23 Lecture 5 33 Exercises 39 Bibliography 47 Index 49

iv CONTENTS

Foreword

These Lecture Notes contain introductory material to algebraic theory of linear differential operators over commutative algebras. The basics of this theory were exposed in a short paper by A.M. Vinogradov [11]. Later, a much more extensive exposition was published in the first chapter of [4]. A very concise version can also be found in [2].

The initial course, held in Forino (Summer 1998) at the First Italian Diffiety School, consisted of ten lectures and a number of practical lessons, but in preparation of the printed version it became clear that a more logical structure should contain five parts (which are also called lectures below):

• general introduction to commutative algebra (rings, algebras, and modules) together with main concepts of the category theory — first two lectures,

• main functors of differential calculus over commutative algebras — lecture 3,

• their representative objects (jets and differential forms) — lecture 4,

• relations to geometry serving as a bridge to the next course in geom-

etry of differential equations — the last lecture. We conclude these notes with a series of exercises which are an essential part of the main text. A lot of them were analyzed and solved during practical lessons held by M.M. Vinogradov and V.A. and V.N. Yumaguzhins.

As additional reading, we strongly recommend the books by I. MacDonald and M. Atiyah [7] and S. Lang [6] (commutative algebra), S. MacLane [8, 9] (category theory and homology), M. Atiyah [1] and D. Husemoller [3] (vector bundles), and Jet Nestruev [10] (relations between commutative algebra and geometry of smooth manifolds). An extended discussion of algebraic theory of differential calculus together with its relation to differential equations can be found in [5].

⋆ ⋆ ⋆

This text is based on the notes made by Barbara Prinari during the lectures. She not only fixed the material with great accuracy, but also typeset the initial TEXfile. Without her help these lectures would hardly be published.

vi FOREWORD

Lecture 1

We start with the maim concept lying in the base of all our future constructions. This is a notion of a ring.

Let A be an Abelian group. The operation in the group will be denoted by the symbol “+”:

+: A × A → A, (a, b) ↦ a + b.

Suppose there is another operation in A denoted by “dot”:

·: A × A → A, (a, b ↦ a · b ≡ ab,

such that one has the distributivity law

a(b + c) = ab + ac  (1)
(b + c)a = ba + ca.  (2)

for all a, b, c ∈ A. Then one says that A is a ring. If in addition we have associativity

a(bc) = (ab)c,

then A is called an associative ring. If there is commutativity

ab = ba,

it is called commutative. If there exists an element 1 ∈ A such that

1 · a = a, a · 1 = a

for all a ∈ A, the ring is called unitary and the element 1 is called the unity of the ring. It is easy to check that if it exists it is unique.

Example 1. The groups ℤ, ℝ, ℂ are rings with respect to “ordinary” operations of summation and multiplication of numbers. They are all associative, unitary and commutative rings.

Remark 1. The rings ℝ and ℂ possess an additional property: any element a ≠ 0 is invertible in them, i.e., there exists an element b ≡ a^{−1} such that a · a^{−1} = a^{−1} · a = 1. Such rings are called fields. Obviously, ℤ is not a field.

Example 2. Let V be a vector space over ℝ. The set of all linear operators acting on V,

L: V → V,

forms a ring with composition playing the role of the product and it is denoted by Mat(V) (since, in a chosen basis, linear operators correspond to matrices). Then we know that

L_{1} ∘ (L_{2} ∘ L_{3}) = (L_{1} ∘ L_{2}) ∘ L_{3},

i.e., Mat(V) is an associative ring, but since in general,

L_{1} ∘ L_{2} ≠ L_{2} ∘ L_{1}

it is not commutative.

Example 3. Let E be a set and consider the set of all subsets of E denoted by 2^{E}. It can be shown that 2^{E} forms a commutative ring with respect to the operations

a + b =^{def} a ∩ b, a + b =^{def} ā ∩ b) ∪ (a ∩ b̄), a, b ∈ 2^{E},

where ā =^{def} E \ a (see Exercise 2).

Example 4. Let C(ℝ) be the set of all continuous functions on the real line. With respect to the operations

(f + g)(x) =^{def} f(x) + g(x)
(f · g)(x) =^{def} f(x)g(x)

C(ℝ) is a commutative and unitary ring (the same is valid for the set of smooth functions C^{∞}(ℝ)).

Example 5. Let G be a finite group and A a commutative ring with unit. Consider the set A[G] consisting of all formal sums

A[G] = { ∑_{g_{i}∈G} a_{i}g_{i} | a_{i} ∈ A}.

Then, having two such expressions, we can define their sum and product

∑_{g_{i}∈G} a_{i}g_{i} + ∑_{g_{i}∈G} b_{i}g_{i} =^{def} ∑_{g_{i}∈G}(a_{i} + b_{i})g_{i}
∑_{g_{i}∈G} a_{i}g_{i} · ∑_{g_{j}∈G} b_{i}g_{j} =^{def} ∑(a_{i}b_{j})g_{i}g_{j}.

Then this object becomes a ring which is associative but in general not commutative (if G is not commutative). A[G] is called the group ring of G (with coefficients in A).

Example 6. Let V be a vector space and assume that there exists a multiplication in V denoted by [ , ] such that

1. [a, b] = −[b, a],

2. [a, b + c] = [a, b] + [a, c],
3. [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0.

Then V is called a Lie algebra (it is neither commutative, nor associative). If A is a ring, a Lie algebra structure can naturally be induced by defining

[a, b] =^{def} ab − ba.

for all a, b ∈ A. It is easy to check that it satisfies properties 1-3. If A is a commutative ring, then [ , ] will be trivial, i.e., [a, b] = 0 for any a, b ∈ A.

Example 7. Let V be a vector space over ℝ (or any other field). Then we can define the tensor product V ⊗ V. This is a vector space. In a similar way we can consider the spaces V ⊗ V ⊗ V = (V ⊗ V) ⊗ V and so on:

\underbrace{V ⊗ V ⊗ ⋯ ⊗ V}_{n times} = V^{⊗n}

which is called the n-th tensor power of V and, if we put V^{⊗0} =^{def} ℝ, we can define

T (V) = ∑_{n≥0} V^{⊗n}.

Elements of V^{⊗n} are linear combinations of homogeneous elements v = v_{1} ⊗ ⋯ ⊗ v_{n}, v_{i} ∈ V. If w = w_{1} ⊗ ⋯ ⊗ w_{m} ∈ V^{⊗m} is another homogeneous element, we define their product v ⊗ w as a simple concatenation:

v ⊗ w =^{def} v_{1} ⊗ ⋯ ⊗ v_{n} ⊗ w_{1} ⊗ ⋯ ⊗ v_{m} ∈ V^{⊗(n+m)}.

This operation is extended to arbitrary elements of T (V) in an obvious way. Thus T (V) becomes an associative algebra with respect to the tensor product. It is called the tensor algebra of the space V. In a sense, all rings can be obtained from tensor algebras by a regular procedure.

Let now A and B be two rings and consider a map

f: A → B  (3)

such that for all a, a′ ∈ A one has

f(a + a′) = f(a) + f(a′)  (4)
f(aa′) = f(a)f(a′)  (5)

(if A and B are unitary we should add also

f(1_{A}) = 1_{B},  (6)

where 1_{A} are 1_{B} units in A and B respectively.). Then f is called a ring homomorphism.

The subset of B

im f =^{def} {b ∈ B | b = f(a), a ∈ A}

is a ring and is called the image of f and the subset of A

ker f =^{def} {a ∈ A | f(a) = 0}

is a ring too, and is called the kernel of f. If ker f = 0, the homomorphism f is called a monomorphism. In the case im f = B, it is called an epimorphism. If a homomorphism is both epi- and monomorphism, it is called an isomorphism.

Elements of ker f satisfy the following property: for all a ∈ A and a′ ∈ ker f one has aa′ ∈ ker f and a′a ∈ ker f.

Now let A be a ring and P ⊂ A be a subgroup of A with respect to addition. If

A · P ⊂ P  (7)

(i.e., ap ∈ P for all a ∈ A and p ∈ P), P is called a left ideal of A. If

P · A ⊂ P,  (8)

then P is called a right ideal of A. If both (7) and (8) are valid, then P is called a two-sided ideal. If A is a commutative ring, all ideals are two-sided ones.

Consider a pair, a ring A and a two-sided ideal P. Then the quotient group A/P is defined

A/P = {[a] = a mod P | a ∈ A}.

Two elements a, a′ are in the same coset in A/P if and only if a − a′ ∈ P. Then we can define sum and product of cosets in the natural way

[a] + [b] =^{def} [a + b], [a] · [b] =^{def} [a · b].

If P is a two-sided ideal, these operation are well defined. With respect to these operations the quotient set is a ring. It is the quotient ring of A with respect to the two-sided ideal P.

There is a natural epimorphism π: A → A/P taking any element a ∈ A to the corresponding coset [a] ∈ A/P. Thus we have the sequence of homomorphisms

0 → P ⟶^{i} A ⟶^{π} A/P → 0,

where i is the natural embedding and the kernel of π coincides with the image of i. This is the so-called short exact sequence of rings.

Example 8. Consider the real line ℝ and the ring of smooth functions C^{∞}(ℝ). Fix a point x ∈ ℝ and consider the set

μ_{x} = {f ∈ C^{∞}(ℝ) | f(x) = 0}  (9)

It is easy to check that μ_{x} is an ideal. The quotient ring C^{∞}(ℝ)/μ_{x} is isomorphic to ℝ (see Exercise 7). If we choose two points x, y ∈ ℝ the set μ_{x,y} of functions vanishing at both points is an ideal too, and so on.

Example 9. Consider an integer n ∈ ℤ and the set

nℤ = {na | a ∈ ℤ}  (10)

It is an ideal in the ring of integer numbers so we can consider the quotient ring ℤ_{n} =^{def} ℤ/nℤ. This ring consists of n elements

ℤ_{n} = {[0], [1], …, [n − 1]}

which are residues modulo n. In fact, we can prove (see Exercise 9) that any ideal of ℤ is of the form (10), that is the following fact is valid:

Proposition 1. A subset P ⊂ ℤ is an ideal of the ring ℤ if and only if there exists n ∈ ℤ such that P = nℤ.

Note that ℤ_{n} is a field if and only if n is a prime number.

Example 10. Consider the tensor algebra T (V) (see Example 7) and the two-sided ideal P_{−} of this algebra generated by the elements of the form a⊗b−b⊗a, a, b ∈ V = T^{1}(V). This means that P_{−} consists of the elements of the form

∑_{i} v_{i} ⊗ (a_{i} ⊗ b_{i} − b_{i} ⊗ a_{i}) ⊗ w_{i}, v_{i}, w_{i} ∈ T (V).

Then the quotient ring is commutative and is called the symmetric algebra of V and is denoted by S(V). If we chose a basis x_{1}, …, x_{n} in V, then any element of S(V) can be uniquely represented in the form of finite sum

∑_{0≤i_{1},...,i_{n}} a_{i_{1},...,i_{n}}x_{1}^{i_{1}} … x_{n}^{i_{n}},

where a_{i_{1},...,i_{n}} are elements of the basic field, while S(V) is identified with the ring of polynomials in the variables x_{1}, …, x_{n}.

Example 11. Consider another ideal P_{+} in T (V) generated by the elements a ⊗ b + b ⊗ a. The corresponding quotient algebra is denoted by ⋀(V) and is called the external algebra of V. The multiplication in this algebra is denoted by ∧: ⋀(V) ⊗ ⋀(V) → ⋀(V). Under the natural projection π: T (V) → ⋀(V), the images of elements from V^{⊗n} constitute elements of degree n. If v, w ∈ ⋀(V) are elements of degrees n and m respectively, the one can show that v ∧ w = (−1)^{nm}wv.

If x_{1}, …, x_{n} is a basis in V, then any element of ⋀(V) is uniquely represented in the form

∑_{1≤i_{1}<...i_{k}≤n} a_{i_{1},...,i_{n}}x_{i_{1}} ∧ ⋯ ∧ x_{i_{k}}.

Example 12. Now let A be the ring of all real polynomials in one variable A = ℝ[x]. Any ideal P in A is of the form A · f, i.e., is generated by some polynomial. The quotient ring is a field in this case if and only if the polynomial f is indecomposable. This field is ℝ, if f = x − λ, λ ∈ ℝ, and ℂ, if f has complex roots.

The last example, together with Example 9, leads to the following

Definition 1. Let A be a ring and P ⊂ A be its ideal. P is called maximal, if for any other ideal P′ the embedding P′ ⊃ P implies P′ = P.

Theorem 2. Let A be a commutative and unitary ring. Let P ⊂ A be an ideal. Then the quotient ring A/P is a field if and only if P is a maximal ideal. (For the proof see Exercise 15).

Let A be a commutative ring with unity (from now on we shall always consider such rings only) and let a, b ∈ A. We say that a ≠ 0 is a zero divisor, if

ab = 0  (11)

for some b ≠ 0.

Definition 2. An ideal P ⊂ A is called a prime ideal, if ab ∈ P implies that either a ∈ P or b ∈ P.

Proposition 3. An ideal P ⊂ A is prime if and only if the quotient ring A/P possesses no zero divisors (see Exercise 16).

Proposition 4. Any maximal ideal P ⊂ A is a prime ideal (see Exercise 17).

Proposition 5. Any ideal of A is contained in a maximal ideal (for the proof Exercise 18).

Example 13. Let A be the ring of polynomials over ℝ in three variables, A = ℝ[x, y, z]. Consider the ideals

P_{1} = {x − λ}, P_{2} = {x − λ, y − μ}, P_{3} = {x − λ, y − μ, z − ν}.

Then both P_{1} and P_{2} are prime ideals, P_{3} is a maximal and consequently a prime ideal too.

Let us now consider two rings A and B and a ring homomorphism φ: A → B. Then we can define an action of A over B in this way: for any a ∈ A and b ∈ B we set

a ∘ b =^{def} φ(a)b.  (12)

Usually we shall omit the “∘” symbol and denote the action of a over b simply as ab.

It is easy to check, using (1), (2) and (4), (5), that this action possesses the following properties

a ∘ (b + b′) = a ∘ b + a ∘ b′  (13)
(a ∘ b)b′ = a ∘ (bb′)  (14)
(aa′) ∘ b = a ∘ b + a′ ∘ b  (15)
(a + a′) ∘ b = a ∘ b + a′ ∘ b  (16)

and, if A is a unitary ring, one has also

1_{A} ∘ b = b.  (17)

In this case, B is called an algebra over A or an A-algebra.

Example 14. The ring of smooth functions C^{∞}(ℝ^{n}) is an algebra over the ring ℝ.

Example 15. The polynomial ring 𝕜[x_{1}, …, x_{n}] is an algebra over the field 𝕜.

Take now instead of B just an Abelian group and let us denoted by P. Let us define an action of A over P in a reasonable way. “Reasonable” in this context means that for all a ∈ A and p ∈ P one has a correspondence

(a, p) ↦ ap ∈ P

and this correspondence possesses the following properties: for all a, a′ ∈ A and p, q ∈ P one has

a(p + q) = ap + aq  (18)
(a + a′)p = ap + a′p  (19)
(a + a′)p = ap + a′p.  (20)

Then we say that P is an A-module. We shall always assume the modules under consideration to be unitary, which means that

1_{A}p = p.  (21)

Example 16. Any vector space over a field 𝕜 is a module over this field.

Example 17. Consider the algebra C^{∞}(ℝ) and the set of vector valued functions A = {(f_{1}, …, f_{s}) | f_{i} ∈ C^{∞}(ℝ)}. We can multiply such functions by any function g ∈ C^{∞}(ℝ) in a natural way

g(f_{1}, …, f_{s}) =^{def} (gf_{1}, …, gf_{s})

and add them to each other as vectors. So A is a C^{∞}(ℝ)-module, that is a module over the ring of smooth functions.

Example 18. Let V be a real vector space (i.e., a space over ℝ) and φ: V → V be a linear operator. Consider the ring of all real polynomials in one variable, ℝ[x]. If p ∈ ℝ[x],

p = a_{0} + a_{1}x + ⋯ + a_{n}x^{n}, a_{0}, …, a_{n} ∈ ℝ,

then we can define the action of this polynomial over V by

p(v) =^{def} (a_{0} + a_{1}φ + ⋯ + a_{n}φ^{n}) (v)

that is

p(v) = a_{0}v + a_{1}φ(v) + ⋯ + a_{n}φ^{n}(v).

It is easy to check that V is a module over the ring ℝ[x] with respect to this action.

Example 19. Let now G be a finite group and V be a vector space over a field 𝕜. We say that G is represented in V, if a group homomorphism of G to the group of endomorphisms of V is given, that is for any g ∈ G one has φ(g): V → V satisfying

φ(gg′) = φ(g) ∘ φ(g′).

To say that G is represented in V is the same as to say that we have a 𝕜[G]-module structure on V (see Example 5).

Let A be a ring and P and Q be two A-modules. A mapping f: P → Q is called an A-homomorphism if for any elements p, p′ ∈ P and a ∈ A one has

f(p + p′) = f(p) + f(p′)  (22)
f(ap) = af(p).  (23)

If A is a field and P, Q are vector spaces over A, an A-homomorphism is just a linear operator. The set of all A-homomorphisms is denoted by Hom_{A}(P, Q). If f ∈ Hom_{A}(P, Q) and a ∈ A, we can define an action of a over f as

(af)(p) =^{def} af(p)  (24)

and the sum of two homomorphisms by

(f + f′(p) =^{def} f(p) + f′(p).

This definition introduces an A-module structure in Hom_{A}(P, Q). The ring A itself is an A-module, so we can consider the module Hom_{A}(A, P) It is easy to check that

Hom_{A}(A, P) ≃ P.  (25)

Let us also introduce the module P^{∗} =^{def} Hom_{A}(P, A) which is called the dual or adjoint module of P.

At the end of the previous lecture we introduced the notion of a homomorphism between A-modules and showed that the set Hom_{A}(P, Q) is an A-module as well.

Let us now consider three A-modules P, Q, and R and two A-module homomorphisms f, g

P ⟶^{f} Q ⟶^{g} R.

Then we can consider the composition h = g ∘ f: P → R. It is easy to check that h is an A-homomorphism from P to R. A way to express that h is the composition of f and g is to say that the diagram

P  \overline{f →} Q

[figure: g]

R \begin{aligned}h \\ & ↓ \end{aligned}

is commutative. We shall often use the diagram language to make exposition more clear and intuitive.

Let us consider a more complicated situation: namely let A be a ring and B be an A-algebra. Let φ: A → B be the corresponding homomorphism and P, Q be B-modules. Then they are A-modules as well: the A-module structure is defined by

ap =^{def} φ(a)p, a ∈ A, p ∈ P.

Example 20. Obviously, the field of complex numbers ℂ is an ℝ-alge-bra. Therefore any complex vector space can be considered as a real vector space using the previous definition with φ: ℝ ↪ ℂ being the standard embedding.

So, P and Q are both A- and B-modules and, consequently, Hom_{A}(P, Q) is an A-module while Hom_{B}(P, Q) is both a B-module and an A-module as well according to the previous definition. But we can define also a B-module structure in Hom_{A}(P, Q): given f ∈ Hom_{A}(P, Q) and b ∈ B, we can set

(bf)(p) =^{def} bf(p),  p ∈ P  (26)

and we can also introduce a different action, which we shall denote by ^{+}:

(b^{+}f)(p) =^{def} f(bp).  (27)

It is easy to check that both bf and b^{+}f are A-homomorphism from P to Q and they are different since f is an A-homomorphism and not a Bhomomorphism.

Example 21. Let B = ℝ[x]. Then B an ℝ-algebra and is a module over itself. As for any ring, Hom_{B}(B, B) = B. On the other hand, elements of Hom_{ℝ}(B, B) are uniquely determined by images of 1, x, x^{2}, … and are arbitrary infinite-dimensional ℝ-matrices.

Thus, two module structures exist in Hom_{A}(P, Q) defined by the actions (26) and (27), and these structures commute in the following sense:

b_{1}(b_{2}^{+}f) = b_{2}^{+}(b_{1}f).

for any b_{1}, b_{2} ∈ B and f ∈ Hom_{A}(P, Q). In this case we say that a bimodule structure is given.

Now let A be a ring and P, Q two A-modules. We say that a mapping f ∈ Hom_{A}(P, Q) is an epimorphism, if it is surjection, that is for any q ∈ Q there exists p ∈ P such that f(p) = q. In this case we also say that the sequence P ⟶^{f} Q → 0 is exact. We say that f is an embedding (or monomorphism), if f(p) = 0 implies p = 0 and represent this fact by the exact sequence 0 → P ⟶^{f} Q. Epimorphic embeddings are called isomorphisms.

If P = Q, elements of module Hom_{A}(P, P) are called endomorphisms and the notation

End_{A} P =^{def} Hom_{A}(P, P)

is used. Note that End_{A} P is an associative A-algebra with respect to composition.

Now let P be an A-module and P′ ⊂ P a subset of P. We say that P′ is a submodule in P if

p_{1} + p_{2} ∈ P′, ap ∈ P′

for all p, p_{1}, p_{2} ∈ P′ and a ∈ A.

Example 22. Let P = A. Then submodules of A are just its ideals.

Let P′ be a submodule of P. Then p ∼ q ⇔ p − q ∈ P′ is an equivalence relation and we can consider the quotient P/P′. Evidently, the operations

(p + P′) + (q + P′) =^{def} (p + q) + P′
a(p + P′) =^{def} ap + P′

determine a well-defined module structure in P/P′ and one has a natural epimorphism P → P/P′, p ↦ p + P′.

Let P, Q be A-modules and f: P → Q be an A-module homomorphism. Consider the sets

ker f = {p ∈ P | f(p) = 0}
im f = {q ∈ Q | q = f(p) for some p ∈ P } .

Both are submodules (of P and Q respectively) called kernel and image of f. We can consider the quotient P/ ker f and it is easy to check that P/ ker f is isomorphic to im f.

Now given two modules P and Q we can consider their Cartesian product

P × Q = {(p, q) | p ∈ P, q ∈ Q}

and introduce an A-module structure to it by

(p, q) + (p′, q′) =^{def} (p + p′, q + q′)
a(p, q) =^{def} (ap, aq).

The module obtained is denoted by P ⊕ Q and is called the direct sum of P and Q. The simplest A-module is A itself, so we can construct the module

A^{n} = \underbrace{A ⊕ A ⊕ ⋯ ⊕ A}_{n times} .

which is called a free module with n generators (or of rank n). In A^{n} we can consider the free generators

e_{1} = (1_{A}, 0, …, 0)
e_{2} = (0, 1_{A}, …, 0)
. . . . . . . . . . . . . . . . .
e_{n} = (0, 0, …, 1_{A})

and any element a ∈ A^{n} can be uniquely written in the form

a = a_{1}e_{1} + ⋯ + a_{n}e_{n}.

The following result explains why the modules and generators above are called free.

Proposition 6. Let P be an arbitrary A-module and A^{n} be the free module of rank n with free generators e_{1}, …, e_{n}. Then for any elements p_{1}, …, p_{n} ∈ P there exists a unique homomorphism f: A^{n} → P such that f(e_{i}) = p_{i}, i = 1, …, n.

For an arbitrary A-module P, we say that elements p_{1}, …, p_{n} are generators of p, if any element p ∈ P can be represented as a linear combination p = ∑_{i=1}^{n} a_{i}p_{i}, a_{i} ∈ A (of course, this representation may not be unique). From Proposition 6 we obtain the following

Corollary 7. Any A-module P with n generators can be represented as a quotient of the free module A^{n} of rank n.

Thus, given a module P, we have P ≃ A^{n}/R for some submodule R ⊂ A^{n}. If r_{1}, …, r_{k} are generators of R, we say that P is described by relations r_{1} = 0, …, r_{k} = 0.

Example 23. The module ℤ_{m} is the quotient of ℤ described by the relation m = 0.

Now let A be a commutative ring and P, Q, R be A-modules. Consider the set P × Q and a mapping f: P × Q → R. We say that f is bilinear if

1. f(ap, q) = af(p, q),

2. f(p, aq) = af(p, q),
3. f(p + p′, q) = f(p, q) + f(p′, q),
4. f(p, q + q′) = f(p, q) + f(p, q′).

for any a ∈ A, p, p′ ∈ P, q, q′ ∈ Q. In other words, the mappings f_{p}: q ↦ f(p, q) and f_{q}: p ↦ f(p, q) are A-homomorphisms from Q to R and from P to R respectively.

Let now T be an A-module and t: P ×Q → T be a bilinear map. We say that T is the tensor product of P and Q if for any bilinear map f: P ×Q → R to an arbitrary A-module R, there exists a unique homomorphism f′ such that the diagram

\begin{matrix}P × Q & \overline{f →} & R \\  & [figure] & f′ \\ T & \begin{aligned}t \\ & ↓ \end{aligned} \end{matrix}  (28)

is commutative.

The tensor product T of P and Q is denoted by P ⊗_{A} Q. If it exists, it is unique up to isomorphisms.

Consider the free module generated by the formal expressions p ⊗ q with p ∈ P and q ∈ Q and denote it by L(P, Q). Let R ⊂ L(P, Q) be the submodule generated by the elements

(p + p′) ⊗ q − p ⊗ q − p′ ⊗ q
p ⊗ (q + q′) − p ⊗ q − p ⊗ q′
ap ⊗ q − p ⊗ aq.

One can show that the quotient module L(P, Q)/R satisfies property (28) and thus we have L(P, Q)/R ≃ P ⊗_{A} Q. This proves existence of tensor product. Its main properties are formulated in the following

Proposition 8. For any A-modules the following isomorphisms take place:

1. P ⊗_{A} (Q ⊗_{A} R) ≃ (P ⊗_{A} Q) ⊗_{A} R,

2. A ⊗_{A} Q ≃ Q ⊗_{A} A ≃ Q,
3. P ⊗_{A} (Q ⊕ R) ≃ P ⊗_{A} Q ⊕ P ⊗_{A} R,

4. (P ⊕ Q) ⊗_{A} R ≃ P ⊗_{A} R ⊕ Q ⊗_{A} R,

5. Hom_{A}(P ⊗_{A} Q, R) ≃ Hom_{A}(P, Hom_{A}(Q, R)).

Now we shall introduce an important class of modules.

Definition 3. An A-module P is called a projective if for any A-mod-ules M and M′, any homomorphism f ∈ Hom_{A}(P, M′), and any epimorphism h: M → M′ there exists a homomorphism f′ ∈ Hom_{A}(P, M) such that the diagram

P

[figure: f′]

M  \overline{h →} M′ \begin{aligned}& f \\ ↓ \end{aligned}  \overline{→} 0

is commutative.

Theorem 9. An A-module P is projective if and only if there exists a free module F and a submodule P′ ⊂ F such that F = P ⊕ P′

In particular, all free modules are projective, but not vice versa.

Now let us consider a commutative ring A and the set of all A-modules, which we denote by Mod(A). We can define some nice operations in this set; for example,

P, Q ∈ Mod(A) ⟹ Hom_{A}(P, Q) ∈ Mod(A).

What are the properties of this correspondence? First fix P and consider the correspondence Hom_{A}(P, ·): Q ↦ Hom_{A}(P, Q). Then you have a correspondence from Mod(A) to itself

Q ∈ Mod(A) ⟹ Hom_{A}(P, Q) ∈ Mod(A).

Now, consider another module Q′ and a homomorphism g ∈ Hom_{A}(Q, Q′). Then to any f ∈ Hom_{A}(P, Q) we can put into correspondence the composition g ∘ f ∈ Hom_{A}(P, Q′). Denoting g ∘ f by Hom_{A}(P, g)(f), we obtain the mapping

Hom_{A}(P, g): Hom_{A}(P, Q) → Hom_{A}(P, Q′).

So, for a fixed P, Hom_{A}(P, ·) is a correspondence taking A-modules to Amodules and A-homomorphisms to A-homomorphisms. Moreover, if we have a sequence

Q ⟶^{g} Q′ ⟶^{g′} Q′′,

then it can be easily seen that

Hom_{A}(P, g′ ∘ g) = Hom_{A}(P, g′) ∘ Hom_{A}(P, g)  (29)

and

Hom_{A}(P, id_{Q}) = id_{Hom_{A}(P,Q)}.  (30)

Let us consider another example of a similar nature. Fix a module P construct the correspondence

T_{P} : Mod(A) → Mod(A)

by setting T_{P} (Q) = P ⊗_{A} Q. If we have a homomorphism g ∈ Hom_{A}(Q, Q′), we can define T_{P} (g): P ⊗_{A} Q → P ⊗_{A} Q′ by

T_{P} (g)(p ⊗ q) =^{def} p ⊗ g(q).

Similar to the previous case, we have T_{P} (g′ ∘ g) = T_{P} (g′) ∘ T_{P} (g), if the composition g′ ∘ g is defined, and T_{P} (id_{Q}) = id_{P ⊗Q}.

Let us now fix a module Q and consider the correspondence

Hom_{A}(·, Q): Mod(A) → Mod(A)

taking a module P to the module Hom_{A}(P, Q). Then for any homomorphism f ∈ Hom_{A}(P′, P) we have the homomorphism Hom_{A}(f, Q): Hom_{A}(P, Q) → Hom_{A}(P′, Q) defined by Hom_{A}(f, Q)(g) =^{def} f ∘ g, where g: P → Q. In this case we have

Hom_{A}(f ∘ f′, Q) = Hom_{A}(f′, Q) ∘ Hom_{A}(f, Q),  (31)

whenever the composition f ∘ f′ is defined. Note that the order in the right-hand side is reversed with respect to (29).

Let us generalize these constructions. Consider “something” (a sort of Universum) denoted by C, which consists of the collection O of some objects such that, given a pair of objects O_{1}, O_{2} ∈ O, we can put it into correspondence with a set Mor(O_{1}, O_{2}), whose elements are called morphisms from O_{1} to O_{2}. Suppose that the correspondence between the pair O_{1}, O_{2} and this set Mor(O_{1}, O_{2}) satisfies the following properties:

1. For any object O there exists a morphism id_{O} ∈ Mor(O, O);

2. Given a triple O_{1}, O_{2}, O_{3}, one has the composition mapping

∘: Mor(O_{1}, O_{2}) × Mor(O_{2}, O_{3}) → Mor(O_{1}, O_{3})

such that: (a) For any φ ∈ Mor(O_{1}, O_{2}) one has id_{O_{2}} ∘φ = φ and φ∘id_{O_{1}} = φ;

(b) If φ ∈ Mor(O_{1}, O_{2}), ψ ∈ Mor(O_{2}, O_{3}), and ζ ∈ Mor(O_{3}, O_{4}),

the associative rule holds: ζ ∘ (ψ ∘ φ) = (ζ ∘ ψ) ∘ φ.

Then C is called a category. Let us consider some examples.

Example 24. Consider the category whose objects are sets. If we have two sets S_{1} and S_{2}, and we define

Mor(S_{1}, S_{2}) = Maps(S_{1}, S_{2}),

i.e., objects are mappings of sets. We have a category of sets.

Example 25. Consider all finite groups as objects. If we have two finite groups G_{1} and G_{2} and define the set of morphisms as the set of all group homomorphisms between G_{1} and G_{2} we get a category. The same can be done with Abelian groups or with all groups.

Example 26. Consider all modules over a commutative ring A and, as above, denote them by Mod(A). In this case

Mor(P_{1}, P_{2}) = Hom_{A}(P_{1}, P_{2}).

Example 27. In the case of vector spaces over a field 𝕜, morphisms will be linear maps between these spaces.

How to establish relations between different categories? Let C_{1} and C_{2} be two categories. Consider a correspondence

F: C_{1} ⟹ C_{2}  (32)

taking any object of C_{1} to an object of C_{2},

O ↦ F (O)

and if we have a pair of objects O and O′ in C_{1}, then F takes

Mor(O, O′) ⟹ Mor(F (O), F (O′)).

Assume that three objects of C_{1} are given, O_{1}, O_{2}, O_{3}, together with two morphisms φ ∈ Mor(O_{1}, O_{2}) and ψ ∈ Mor(O_{2}, O_{3}). Then F is called a covariant functor between C_{1} and C_{2}, if for all morphisms φ, ψ

F (φ ∘ ψ) = F (φ) ∘ F (ψ).

whenever the composition φ ∘ ψ is defined. If F takes

Mor(O, O′) ⟹ Mor(F (O′), F (O)).

and

F (φ ∘ ψ) = F (ψ) ∘ F (φ),

then it is called is called a contravariant functor.

Example 28. The correspondence T_{P} : Q ↦ P ⊗_{A}Q is a covariant functor from the category Mod(A) to itself.

Example 29. For a fixed P, the correspondence Q ↦ Hom_{A}(P, Q) is also a covariant functor from the category Mod(A) to itself.

Example 30. If we fix Q, the correspondence P ↦ Hom_{A}(P, Q) is a contravariant functor.

Example 31. Let G be a finite group. Consider the normal subgroup generated by g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} with g_{1}, g_{2} ∈ G, and denoted by [G, G]:

[G, G] = {g_{1}g_{2}g_{1}^{−1}g_{2}^{−1}} ⊂ G.

Then G/[G, G] is an Abelian group while the correspondence

G ↦ G/[G, G]

is a functor from the category of finite groups to the category of Abelian groups.

Example 32. Smooth manifolds make a category with smooth maps playing the role of morphisms. Then the correspondence M ↦ C^{∞}(M) is a contravariant functor from this category to the category of ℝ-algebras.

Let now C_{1} and C_{2} be two categories and F, F: C_{1} ⟹ C_{2} be two (say, covariant) functors. Assume that for any object O of C_{1} a morphism T_{O}: F (O) → G(O) is given. Then the correspondence O ↦ T_{O} is called a natural transformation of functors F and G if the diagram

\begin{matrix}F (O) & \overline{\underrightarrow{F (φ)}} & F (O′) \\ G(O) & \begin{aligned}T_{O} \\ & ↓ \end{aligned} & \overline{\underrightarrow{G(φ)}} & G(O′) & \begin{aligned}& T_{O′} \\ ↓ \end{aligned} \end{matrix}

is commutative for all objects O, O′ and all morphisms φ ∈ Mor(O, O′). For example, the isomorphism Hom_{A}(P ⊗_{A} Q, R) ≃ Hom_{A}(P, Hom(Q, R)) gives a natural transformation of the functors R ↦ Hom_{A}(P ⊗_{A} Q, R) and R ↦ Hom_{A}(P, Hom(Q, R)). We shall see other examples of natural transformations in next lectures.

Using a geometrical analogy now, we shall start to construct differential calculus in the category Mod(A).

Let U be a domain in ℝ^{n}. Let us fix a point x ∈ U; a tangent vector ξ_{x} in x is a map

ξ_{x}: C^{∞}(U) → ℝ

which is linear and such that for all f, g ∈ C^{∞}(U) one has

ξ_{x}(fg) = ξ_{x}(f)g(x) + f(x)ξ_{x}(g).  (33)

Suppose we have a tangent vector at any point of the domain. Then we can consider the mapping

ξ: C^{∞}(U) → C^{∞}(U)

such that ξ(f)(x) =^{def} ξ_{x}(f) smoothly depending on x. Then (33) acquires the from

ξ(fg) = ξ(f)g + fξ(g).  (34)

It is easy to check that for all f, g ∈ C^{∞}(U) and α ∈ ℝ one has also 1. ξ(fg) = ξ(f)g + fξ(g),

2. ξ(αf) = αξ(f).

Thus, smooth family of tangent vectors is a ℝ-linear map from the ring of smooth functions to itself that satisfies the Leibniz rule (34).

Now let us try to generalize this concept. Let K be a commutative unitary ring and A be a K-algebra. A map X: A → A is called a derivation, if it is K-linear and satisfies the Leibniz rule, that is

1. X(κa) = κX(a);

2. X(a + b) = X(a) + X(b);
3. X(ab) = X(a)b + bX(a).

for all a, b ∈ A and W κ ∈ K. The set of all derivations will be denoted by D(A). The classical geometric case corresponds to K = ℝ and A = C^{∞}(ℝ^{n}).

For any X, Y ∈ D(A) and a, b ∈ A, we can set

(X + Y)(a) =^{def} X(a) + Y (a)
(aX)(b) =^{def} aX(b),

and it is easy to check that X +Y and aX are derivations as well. So the set of derivations D(A) in an A-module. In fact it is more than just a module.

For any X, Y ∈ D(A) we can consider the compositions X ∘ Y and Y ∘ X and define

[X, Y] =^{def} X ∘ Y − Y ∘ X.  (35)

It is easy to check that it is a derivation; it is called the Lie bracket of X and Y. One can show that for all X, Y, Z ∈ D(A) and κ ∈ K the following identities hold

1. [X, Y] + [Y, X] = 0;

2. [X + Y, Z] = [X, Z] + [Y, Z];
3. [κX, Y] = κ[X, Y];
4. [X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0.

Thus, D(A) is a Lie algebra over K.

Consider two domains W ⊂ ℝ^{n} and V ⊂ ℝ^{m} and let F: W → V be a smooth mapping. Then this mapping generates a ring homomorphism F^{∗}: C^{∞}(V) → C^{∞}(W) which is defined by

(F^{∗}(f))(b) = f(F (b)).  (36)

So every function f on V is pulled back to a smooth function on W in this way. Note that F^{∗} is also a homomorphism of ℝ-algebras.

We can generalize this concept. Let us denote A = C^{∞}(V) and B = C^{∞}(W). Having a mapping F: W → V, we can construct F^{∗} and so, having a derivation ξ in A we can define a mapping \overline{ξ}: A → B such that for all f ∈ A one has

\overline{ξ}(f) =^{def} F^{∗}(ξ(f)).

One can check that \overline{ξ} satisfies the Leibniz rule, that is

\overline{ξ}(fg) = \overline{ξ}(f)F^{∗}(g) + F^{∗}(f)\overline{ξ}(g)

for any f, g ∈ A (action by F^{∗} on B determines an A-algebra structure in B).

Let A be a K-algebra and P be an A-module. We can define

D(P) = {f ∈ Hom_{K}(A, P) | f(ab) = af(b) + bf(a) ∀a, b ∈ A}.  (37)

If we have such an f ∈ D(P) and a ∈ A, then

(af)(b) =^{def} a(f(b))

and it is easy to check that af ∈ D(P), so D(P) is an A-module. However it is not a Lie algebra, since there is no natural way to define a commutator.

On the other hand, if we have two A-modules P, Q, an A-module homomorphism φ: P → Q and a derivation with values in P, we can consider the composition D(φ) =^{def} φ ∘ f: A → Q. It is K-linear and it satisfies the same identities, so φ ∘ f ∈ D(Q). Then to any module P we put into correspondence the module D(P). Moreover, having two homomorphisms P ⟶^{φ} Q ⟶^{ψ} R, it results that

D(ψ ∘ φ) = D(ψ) ∘ D(φ),

so D(·) a covariant functor. We say that elements of D(P) are P -valued derivation of A.

Fix a commutative ring A which is an algebra over a ring K. Let P be an A-module and X be a derivation, that is X ∈ Hom_{K}(A, P) and

X(ab) = aX(b) + bX(a).  (38)

for any a, b ∈ A. Any element X of D(P) can be multiplied by an element of A in two ways, from the left, that is

(aX)(b) =^{def} aX(b)  (39)

or from the right

(a^{+}X)(b) =^{def} X(ab).  (40)

Now, (a^{+}X)(b) = X(ab) = aX(b) + X(a)b = (aX)(b) + X(a)b, so we can write

(a^{+}X − aX)(b) = X(a)b

and this suggests a way to define a commutator

[a, X](b) =^{def} X(a)b.  (41)

Now let a be fixed; then if we have another element \overline{a}, we can easily check that for all b one has [\overline{a}, [a, X]](b) = 0, that is

[\overline{a}, [a, X]] = 0.  (42)

Let now X: A → P be such that [\overline{a}, [a, X]] = 0 for all a, \overline{a} ∈ A. Then the difference X −X(1) satisfies (38) provided X itself satisfies (42). Then, if we put X − X(1) = \overline{X}, then X = \overline{X} + f, where \overline{X} is a derivation and f = X(1) is a constant. We conclude that any element satisfying (42) is the sum of a derivation and a constant. Standard first order differential operators (on ℝ) are of this form,

Δ = g \frac{∂}{∂x} + f.

Let A be a K-algebra and P, Q be A-modules. Then we can consider the bimodule Hom_{K}(P, Q). Let Δ ∈ Hom_{K}(P, Q) and consider

[a, Δ] = a^{+}Δ − Δa = Δa − aΔ.

This difference belongs to Hom_{K}(P, Q) so we can take \overline{a} ∈ A and consider [\overline{a}, [a, Δ]]. We say that Δ is a first order differential operator acting from P to Q if for all a, \overline{a} ∈ A one has

[\overline{a}, [a, Δ]] = 0.

Consider two A-modules P, Q and a homomorphism Δ ∈ Hom_{K}(P, Q). We say that Δ is a differential operator of order ≤ k if for all a_{0}, a_{1}, …, a_{k} ∈ A it results

[a_{0}, [a_{1}, … [a_{k}, Δ] …]] = 0.

Let us denote the set of all such maps as D̃iff_{k}(P, Q), k ∈ ℕ. What are the structures living in this set? If we have two differential operators Δ, ∇ ∈ D̃iff_{k}(P, Q), then it is easy to check that Δ+∇ ∈ D̃iff_{k}(P, Q), so D̃iff_{k}(P, Q) is an Abelian group. Moreover, given a ∈ A, it results that

[a_{k}, aΔ] = a[a_{k}, Δ]

and so D̃iff_{k}(P, Q) is also an A-module, which will be denoted by Diff_{k}(P, Q). If we consider the right multiplication, we get the second module structure denoted by Diff_{k}^{+}(P, Q), and finally Diff_{k}^{(+)}(P, Q) will denote the bimodule structure.

Consider three modules and two linear maps P ⟶^{Δ} Q ⟶^{∇} R. If you take a ∈ A and consider [a, ∇ ∘ Δ], you will have

[a, ∇ ∘ Δ] = [a, ∇] ∘ Δ + ∇ ∘ [a, Δ].

Let Δ be a K-linear map. You can prove by induction or directly from the definition that

\underbrace{[a, [a, … [a, Δ] …]]}_{k+1 times} = ∑_{i=0}^{k+1}(−1)^{k+1−i}\binom{k + 1}{i}a^{i}∇a^{k+1−i}.

Consider f ∈ C^{∞}(ℝ) and x_{0} ∈ ℝ. Then that f can be represented in the form

f(x) = f(x_{0}) + (x − x_{0})f′(x_{0}) + ⋯ + \frac{(x − x_{0})^{k}}{k!} f^{(k)}(x_{0}) + (x − x_{0})^{k+1}g(x)

with g ∈ C^{∞}(ℝ). Now let Δ: C^{∞}(ℝ) → C^{∞}(ℝ) satisfy the algebraic definition of a differential operator of order ≤ k and apply Δ to the function f. Because of linearity, we have

Δ(f) = f(x_{0})Δ(1) + f′(x_{0})Δ(x − x_{0}) + ⋯

⋯ + \frac{f^{(k)}(x_{0})}{k!}Δ [(x − x_{0})^{k}] + Δ [(x − x_{0})^{k+1}g(x)] .

If we compute this expression at x = x_{0} it suffices to notice that the last term in the right-hand side vanishes and so the value of Δ(f) at the point x_{0} is determined by the set of coefficients Δ_{0} = Δ(1), Δ_{1} = Δ(x − x_{0}), …, Δ_{k} = Δ(x − x_{0})^{k} and

Δ = Δ_{0} + Δ_{1} \frac{d}{dx} + ⋯ + \frac{Δ_{k}}{k!} \frac{d^{k}}{dx^{k}} .

A similar proof is valid for arbitrary differential operators Δ: C^{∞}(ℝ^{n}) → C^{∞}(ℝ^{n}). So from the algebraic definition in the case of smooth functions we obtain the usual expression and, vice versa, if we have

Δ = ∑ Δ_{σ} \frac{∂^{σ}}{∂x^{σ}} ,

where Δ_{σ} ∈ C^{∞}(ℝ), this Δ satisfies our algebraic definition of a differential operator. Thus we can think of differential operators as of purely algebraic objects.

Consider the bimodule Diff_{k}^{(+)}(P, Q). If we have a differential operator of order k, i.e.,

[a_{0}, [a_{1}, … [a_{k}, Δ] …]] = 0,

then we can add an arbitrary a′ and obtain

[a′, [a_{0}, [a_{1}, … [a_{k}, Δ] …]]] = 0,

so Δ is a differential operator of order k + 1 as well and consequently

Diff_{k}^{(+)}(P, Q) ↪ Diff_{k+1}^{(+)}(P, Q).

Operators of zero order are those which satisfy [a_{0}, Δ] = 0 for all a_{0} ∈ A, that is a_{0}Δ = Δa_{0} and so elements of Diff_{0}(P, Q) are just A-homomorphisms and the two module structures coincide in this case.

Having the sequence of embeddings

Hom_{A}(P, Q) = Diff_{0}(P, Q) ↪ ⋯ ↪ Diff_{k}^{(+)}(P, Q) ↪ Diff_{k+1}^{(+)}(P, Q) ↪ ⋯,

we can define a new object

Diff_{∗}^{(+)}(P, Q) =^{def} ⋃_{k≥0} Diff_{k}^{(+)}(P, Q).

It is a bimodule consisting of all differential operators of any finite order.

Let us fix a module P. Then, having a homomorphism f: Q → Q′ and an operator Δ ∈ Diff_{k}^{(+)}(P, Q), we can consider the composition P ⟶^{Δ} Q ⟶^{f} Q′ and associate to any differential operator Δ ∈ Diff_{k}^{(+)}(P, Q) the operator f ∘ Δ ∈ Diff_{k}^{(+)}(P, Q′). This map, which depends of P and on f, will be denoted by

Diff_{k}^{(+)}(P, f): Δ ∈ Diff_{k}^{(+)}(P, Q) ↦ f ∘ Δ ∈ Diff_{k}^{(+)}(P, Q′),

and it defines a functor from the category of A-modules to the category of A-bimodules.

Fix Q now. Then, given a homomorphism f: P → P′, to any operator Δ ∈ Diff_{k}^{(+)}(P′, Q) we can associate the operator P ⟶^{f} P′ ⟶^{Δ} Q. If P ⟶^{f} P′ ⟶^{g} P′′ are two homomorphisms, then

Diff_{k}^{(+)}(f ∘ g, Q) = Diff_{k}^{(+)}(g, Q) ∘ Diff_{k}^{(+)}(f, Q)

and so it is a contravariant functor.

Consider the modules Diff_{k}(P, Q) and Diff_{k}^{+}(P, Q). They are isomorphic as Abelian groups, but they are different as far as module structures are concerned. Then we can consider the identical mappings

Diff_{k}(P, Q) ⟶^{id^{+}} Diff_{k}^{+}(P, Q)

and

Diff_{k}(P, Q) ⟵^{id_{+}} Diff_{k}^{+}(P, Q).

The mappings id_{+} and id^{+} are differential operators of order ≤ k, but there compositions id_{+} ∘ id^{+} and id^{+} ∘ id_{+} are operators of zero order (identical homomorphisms).

What happens in general if we consider the compositions of differential operators P ⟶^{Δ} Q ⟶^{∇} R with Δ ∈ Diff_{k}^{(+)}(P, Q) and ∇ ∈ Diff_{l}^{(+)}(Q, R)? Generally speaking, ∇ ∘ Δ ∈ Hom_{K}(P, R). Let us prove that it is a differential operator. To this end, take a ∈ A and consider the identity

[a, ∇ ∘ Δ] = [a, ∇] ∘ Δ + ∇ ∘ [a, Δ].

Hence, for m = k + l one has

[a_{0}, [a_{1}, … [a_{m}, ∇ ∘ Δ] …]]

= ∑[a_{i_{0}}, [a_{i_{1}}, … [a_{i_{s}}, ∇] …]] ∘ [a_{j_{0}}, [a_{j_{1}}, … [a_{j_{t}}, Δ] …]].

The whole number of a’s is k +l+1; then either in the first term the number of commutators is ≥ l or in the second one it is ≥ k. In any case, one of the two components vanishes and hence ∇ ∘ Δ ∈ Diff_{k+l}^{(+)}(P, R), just as in the standard case.

Consider a particular case P = Q. Taking the composition of two differential operators from P to itself, we obtain a new differential operator from P to itself. Composition is obviously associative so Diff_{∗}^{(+)}(P, P) is an associative algebra with respect to composition (but not commutative!).

To deal with differential operators from A (as A-module over itself) to an A-module Q, we shall use the notation

Diff_{k}^{(+)}(Q) =^{def} Diff_{k}^{(+)}(A, Q).

Derivations with values in Q are first order differential operators so we have D(Q) ↪ Diff_{1}^{(+)}(Q). Note that the embedding D(Q) ↪ Diff_{1}^{+}(Q) ia a first order monomorphic differential operator, while D(Q) ↪ Diff_{1}(Q) is an Amodule monomorphism.

Let us consider the sequence 0 → D(Q) → Diff_{1}(Q). If we take the quotient of Diff_{1}(Q) by the image, we obtain Q (we are killing the differential part), so we have the exact sequence of homomorphisms

0 → D(Q) → Diff_{1}(Q) → Q → 0.  (43)

On the other hand, Diff_{0}^{(+)}(Q) ↪ Diff_{1}^{(+)}(Q) and, since we established that Diff_{0}^{(+)}(Q) ≡ Diff_{0}^{(+)}(A, Q) = Hom_{A}(A, Q) = Q, we have the exact sequence of differential operators

0 → Q → Diff_{1}^{+}(Q) → D(Q) → 0.  (44)

From these two sequences, (43) and (44), we get two important theories: Spencer cohomology and algebraic model of Hamiltonian mechanics.

In the previous lecture we established that

Q ↦ Diff_{l}^{(+)}(P, Q),  (45)
P ↦ Diff_{l}^{(+)}(P, Q)  (46)

are covariant and contravariant functors from Mod(A) to the category of A-bimodules.

Let us fix the second module structure and consider the mapping

�_{l}: Diff_{l}^{+} Q → Q

defined by

�_{l}(Δ) =^{def} Δ(1), Δ ∈ Diff_{l}^{+} Q.  (47)

Then �_{l} is a differential operator of order l (note that in the first module structure it is a homomorphism). In fact, for any a ∈ A one has �_{l}(a^{+}Δ) = Δ(a).

Consider an arbitrary differential operator of order Δ ∈ Diff_{l}^{+}(P, Q) and let us establish a correspondence between this module and the module Hom_{A}(P, Diff_{l}^{+} Q). Let us construct a mapping φ: A → Q by setting

(φ(Δ)(p))(a) = Δ(ap), a ∈ A, p ∈ P.

One can easily check that it is a homomorphism and, moreover, it is an isomorphism. In fact if f ∈ Hom_{A}(P, Diff_{l}^{+} Q) then the inverse mapping is f ↦ �_{l} ∘f. This means that for any differential operator Δ of order l acting from P to Q, there exists a homomorphism φ(Δ) such that the diagram

P  \overline{Δ  →} Q

[figure: φ(Δ); �_{l}]

Diff_{l}^{+} Q

is commutative.

In fact, we established that the functors

P ↦ Diff_{l}^{+}(P, Q) and P ↦ Hom_{A}(P, Diff_{l}^{+} Q)

act in the same way, they are identical to each other. In category theory this means that the functor Diff_{l}^{+}(·, Q) is representable. The module Diff_{l}^{+} Q is called the representative object for this functor.

Consider now the operator �_{l}: Diff_{l}^{+}(Q) → Q. Then we can also consider the composition

Diff_{s}^{+}(Diff_{l}^{+}(Q)) ⟶^{�_{s}} Diff_{l}^{+}(Q) ⟶^{�_{l}} Q,

which is a differential operator of order l + s. Hence, by the universal property of the operator Diff_{l+s}^{+} Q ⟶^{�_{l+s}} Q there exists a unique homomorphism

φ(�_{l} ∘ �_{s}) =^{def} c_{l,s} such that the following diagram

Diff_{s}^{+}(Diff_{l}^{+} Q) \overline{�_{s}→} Diff_{l}^{+} Q

Diff_{s+l}^{+} Q \begin{aligned}c_{l,s} \\ & ↓ \end{aligned}  \overline{�_{s+l} →} Q \begin{aligned}& �_{l} \\ ↓ \end{aligned}

is commutative. The mapping c_{l,s} is called the universal composition. Universal composition is associative in the sense that the diagram

Diff_{l}^{+}(Diff_{s}^{+}(Diff_{m}^{+} Q)) \overline{\underrightarrow{c_{l,s}}} Diff_{l+s}^{+}(Diff_{m}^{+} Q)

Diff_{l}^{+}(Diff_{s+m}^{+} Q) \begin{aligned}Diff_{l}^{+}(c_{s,m}) \\ & ↓ \end{aligned}  \overline{c_{l,s+m} →} Diff_{s+l+m}^{+} Q \begin{aligned}& c_{l+s,m} \\ ↓ \end{aligned}

is commutative.

Note also that the universal composition is related to any module Q and, strictly speaking, we must write c_{l,s} = c_{l,s}(Q). Let φ: Q → Q′ be a homomorphism. Then the diagram

Diff_{l}^{+}(Diff_{s}^{+} Q) \overline{\underrightarrow{c_{l,s}(Q)}} Diff_{l+s}^{+} Q

Diff_{l}^{+}(Diff_{s}^{+} φ) \begin{aligned}Diff_{l}^{+}(c_{s,m}) \\ & ↓ \end{aligned}  \overline{\underrightarrow{c_{l,s}(Q′)}} Diff_{l+s}^{+} Q′ \begin{aligned}& Diff_{l+s}^{+}(φ) \\ ↓ \end{aligned}

is commutative. It means that c_{l,s}: Diff_{l}^{+} Diff_{s}^{+} ⇒ Diff_{l+s}^{+} is a natural transformation of functors.

Now consider the functor

Q ↦ Diff_{l}(P, Q)

for a fixed P and let us try to find the representative object for it. We need to find something, ?(P), which possesses the property

Diff_{l}(P, Q) ≃ Hom_{A}(?(P), Q).

Consider the tensor product A ⊗_{K} P and let us introduce two A-module structures into this object. Namely, we set

b(a ⊗ p) = (ba) ⊗ p, b_{+}(a ⊗ p) = a ⊗ (bp).

In such a way we obtain bimodule since, given b, b′, we have

b(b′_{+}(a ⊗ p)) = b′_{+}(b(a ⊗ p)).

Let us measure the difference between these two multiplications. If θ ∈ A ⊗_{K} P, we set

[a, θ] =^{def} aθ − a_{+}θ.

Taking arbitrary elements a_{0}, a_{1}, … a_{l} ∈ A, we can consider the iterated commutator

[a_{0}, [a_{1}, … [a_{l}, θ] …]] ∈ A ⊗_{K} P

and generate, for a fixed l, the submodule

μ_{l}(P) =^{def} { [a_{0}, [a_{1}, … [a_{l}, θ]]] }

taking all a_{i} ∈ A and θ ∈ A ⊗_{K} P. The quotient module

J^{l}(P) = A ⊗_{K} P/μ_{l}(P).

is called the module of l-jets.

Let us take an element a ⊗ p ∈ A ⊗_{K} P and denote its coset in J^{l}(P) by [a ⊗ p]. Having an element p ∈ P, we can consider the element 1 ⊗ p and the coset [1 ⊗ p] thus defining the mapping

j_{l}: P → J^{l}(P), p ↦ j_{l}(p) =^{def} [1 ⊗ p].

The element j_{l}(p) is called the l-jet of p.

The set J^{l}(P) is an A-module with respect to multiplication b[a ⊗ p] =^{def} [ba⊗p]. Note that J^{0}(P) = P. It can be easily proved that j_{l} is a differential operator of order l.

Theorem 10. Let Δ: P → Q be a differential operator of order ≤ l. Then there exists a uniquely defined homomorphism ψ(Δ) such that the diagram

P  \overline{Δ  →} Q

[figure: j_{l}; ψ(Δ)]

J^{l}(P)

is commutative.

From this theorem we obtain the isomorphism

Diff_{l}(P, Q) ≃ Hom_{A}(J^{l}(P), Q)

which means that the module J^{l}(P) is the representative object for the functor Diff_{l}(P, ·).

Consider the operator j_{l}. Being of order ≤ l, it is also of order ≤ l + 1 and consequently we have the commutative diagram

J^{l}(P) ← ν_{l+1,l} J^{l+1}(P)

[figure: j_{l+1}]

P^{j_{l}↑}

The mapping ν_{l+1,l} is dual to the embedding Diff_{l}(P, Q) ↪ Diff_{l+1}(P, Q) and, in fact, is an epimorphism. So, we have the sequence of mappings

P ≡ J^{0}(P) ← J^{1}(P) ← ⋯ ← J^{l}(P) ← J^{l+1}(P) ← ⋯  (48)

The kernel of the projection ν_{l+1,l} coincides with the quotient μ_{l}(P)/μ_{l+1}(P) and thus we have the exact short sequence of modules

0 → μ_{l}(P)/μ_{l+1}(P) → J^{l+1}(P) ⟶^{ν_{l+1,l}} J^{l}(P) → 0.

Now consider (48) and a sequence of elements θ_{l}, l = 0, 1, ⋯, such that θ_{l} ∈ J^{l}(P) and ν_{l+1,l}(θ_{l+1}) = θ_{l}. These sequences may be added to each other and multiplied by elements a ∈ A component-wise. So they form an A-module which is denoted by J^{∞}(P) and called the module of infinite jets. If we know the sequence as a whole, we, in particular, know its l-th term and thus get the mapping

ν_{∞,l}: J^{∞}(P) → J^{l}(P).

From our definition it follows also that we can construct the diagram J^{∞}(P)

[figure: ν_{∞,l+1}]

J^{l}(P) \begin{aligned}ν_{∞,l} \\ & ↓ \end{aligned} ← ν_{l+1,l} J^{l+1}(P)

and it is commutative.

In particular, if p ∈ P, then the sequence {j_{l}(p)}_{l≥0}, is a element of J^{∞}(P). It is denoted by j_{∞}(p) and is called th infinite jet of the element p. Obviously, ν_{∞,l}(j_{∞}(p)) = j_{l}(p). Formally speaking, the mapping j_{∞}: P → J^{∞}(P) is not a differential operator, that is there always exist a sequence a_{0}, …, a_{l} of an arbitrary high length such that

[a_{0}, [a_{1}, … [a_{l}, j_{∞}] …]] ≠ 0,

but if we restrict ourselves to a finite part, it becomes a differential operator.

The correspondence P ↦ J^{l}(P) is a covariant functor. In fact, if we consider a homomorphism f: P → Q, then the composition f ∘ j_{l} is a differential operator of order l, so there exists a unique homomorphism from J^{l}(P) to J^{l}(Q), denoted by J^{l}(f), such that the diagram

\begin{matrix}P & \overline{j_{l}→} & J^{l}(P) \\ Q & \begin{aligned}f \\ & ↓ \end{aligned} & \overline{j_{l}→} & J^{l}(Q) & \begin{aligned}& J^{l}(f) \\ ↓ \end{aligned} \end{matrix}

is commutative. Due to uniqueness, we have

J^{l}(f ∘ g) = J^{l}(f) ∘ J^{l}(g)

whenever it makes sense.

Now consider a module P and for any l the operator j_{l}: P → J^{l}(P). Since J^{l}(P) is an A-module as well, we can take the composition

P ⟶^{j_{l}} J^{l}(P) ⟶^{j_{s}} J^{s}(J^{l}(P)),

which is a differential operator of order l + s. By the universal property, we can construct the commutative diagram

\begin{matrix}P & \overline{\underrightarrow{j_{l+s}}} & J^{l+s}(P) \\ J^{l}(P) & \begin{aligned}j_{l} \\ & ↓ \end{aligned} & \overline{\underrightarrow{j_{s}}} & J^{s}(J^{l}(P)) & \begin{aligned}& c^{l,s} \\ ↓ \end{aligned} \end{matrix}

and this mapping c^{l,s} is dual to the universal composition map c_{l,s}. The homomorphisms c^{l,s} and c_{l,s} are adjoint in categorical sense and c^{l,s} is called the universal co-composition. This operation possesses the co-associativity property expressed by commutativity of the diagram

\begin{matrix}J^{l+s+m}(P) & \overline{\underrightarrow{c^{s,l+m}}} & J^{s}(J^{l+m}(P)) \\ J^{s+l}(J^{m}(P)) & \begin{aligned}c^{l+s,m} \\ & ↓ \end{aligned} & \overline{\underrightarrow{c^{s+l,m}}} & J^{s}(J^{l}(J^{m}(P))) & \begin{aligned}& J^{s}(c^{l,m}) \\ ↓ \end{aligned} \end{matrix}

Consider now the case l = 1 and take the module J^{1}(A). Then we have the mapping

i_{1} : A → J^{1}(A)

defined by i_{1}(a) =^{def} [a ⊗ 1]. It is a homomorphism, so its image is a submodule. Let us set

Λ^{1}(A) =^{def} J^{1}(A)/ im i_{1}.  (49)

Then we obtain the exact sequence of modules

0 → A ⟶^{i_{1}} J^{1}(A) ⟶^{π} Λ^{1}(A) → 0,

where π is the natural projection. The module Λ^{1}(A) is called the module of 1-forms of algebra A. The composition

d ≡ π ∘ j_{1} : A → Λ^{1}(A)

is called the first de Rham differential and is a derivation of A with values in Λ^{1}(A). It is easy to check that Λ^{1}(A) is generated by the elements d_{a}, a ∈ A, with the relations

d_{ab} = ad_{b} + bd_{a}, d_{αa+βb} = αd_{a} + βd_{b},

where a, b ∈ A, α, β ∈ K.

We continue to deal with differential forms. An immediate consequence of Theorem 10 is

Theorem 11. For any derivation X: A → P there exists a unique homomorphism ψ(X): Λ^{1}(A) such that the diagram

A  \overline{d →} Λ^{1}(A)

[figure: ψ(X)]

P \begin{aligned}X \\ & ↓ \end{aligned}

is commutative.

Hence, we have the isomorphism

Hom_{A}(Λ^{1}(A), P) = D(P),

and the module of 1-forms is the representative object of the covariant functor D(·).

Now let us define the module Λ^{l}(A) of differential l-forms as the l-th exterior power of Λ^{1}(A) (see Example 11), that is

Λ^{l}(A) = ⋀^{l} Λ^{1}(A)

(when A is fixed, we shall skip it as argument and use a simpler notation Λ^{l} = Λ^{l}(A)). Thus, we have a series of modules

Λ^{0} = A, Λ^{1}, Λ^{2}, …, Λ^{l}, …

The module Λ^{l} is generated by the elements a_{0}, da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{l} and we set by definition

d(a_{0} da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{l}) = da_{0} ∧ da_{1} ∧ da_{2} ∧ ⋯ ∧ da_{l}.  (50)

We take (50) for a definition of d and thus obtain the sequence of first order differential operators

0 → A ⟶^{d} Λ^{1} ⟶^{d} … ⟶^{d} Λ^{l−1} ⟶^{d} Λ^{l} ⟶^{d} ⋯  (51)

We call the mapping d the de Rham differentials and the sequence (51) is called the de Rham complex of the algebra A.

Remark 2. This is a “bad” definition of the de Rham complex. A “good” one refers to rather complicated categorial construction, which are beyond the scope of these lectures.

The basic properties of the de Rham differential are as follows:

1. As it was already mentioned, it is a first order differential operator.

2. It differentiates the wedge product, i.e.,

d(ω ∧ θ) = d(ω) ∧ θ + (−1)^{l}ω ∧ d(θ)  (52)

where l is the degree of ω.

3. d ∘ d = 0

From the last property it follows that

im d ⊂ ker d

at all terms in the chain (48). Hence, we can introduce the K-modules

H^{l}(A) =^{def} ker d/ im d,

which are called the de Rham cohomology of the algebra A.

Remark 3. Note that if you have an arbitrary sequence of homomorphisms

P_{0} ⟶^{∂_{0}} P_{1} ⟶^{∂_{1}} ⋯ → P_{l−1} ⟶^{∂_{l}} P_{l} ⟶^{∂_{l+1}} ⋯

with the property ∂_{l+1} ∘ ∂_{l} = 0, you can realize the same construction. Such sequences are called complexes of modules and the modules of the form ker ∂_{l+1}/ im ∂_{l} are called homologies (or cohomologies) associated to these complexes.

Now consider the composition j_{s} ∘ d: Λ^{l−1} → J^{s}(Λ^{l}). It is a differential operator of order s + 1. Due to the universal property of operators j_{s}, we obtain the following commutative diagram

\begin{aligned}⋯ & \overline{→} & Λ^{l−1} & \overline{d →} & Λ^{l} & \overline{d →} & Λ^{l+1} & \overline{→} & ⋯ \\ ⋯ & \overline{→} & J^{s+1}(Λ^{l−1}) & \begin{aligned}j_{s+1} \\ & ↓ \end{aligned} & \overline{S→} & J^{s}(Λ^{l}) & \begin{aligned}j_{s} \\ & ↓ \end{aligned} & \overline{\underrightarrow{S}} & J^{s−1}(Λ^{l+1}) & \begin{aligned}j_{s−1} \\ & ↓ \end{aligned} & \overline{→} & ⋯ \end{aligned}

A simple exercise is to prove that the sequence

0 → A ⟶^{j_{k}} J^{k}(A) ⟶^{S} J^{k−1}(Λ^{1}) → … J^{1}(Λ^{k−1}) ⟶^{S} Λ^{k} → 0

is a complex. It is called the Spencer complex of A and its cohomologies are called the Spencer cohomologies.

We have established already that the module Λ^{1} is the representative object for the functor D(·), i.e., D(P) = Hom_{A}(Λ^{1}, P). Let us now try to understand what functors are represented by the modules Λ^{l}. Let us set Hom_{A}(Λ^{l}, P) =^{def} D_{l}(P) and start with the case l = 2. Thus,

D_{2}(P) = Hom_{A}(Λ^{2}, P).

Note that there exists a natural homomorphism from this module to the module Hom_{A}(Λ^{1}, Hom_{A}(Λ^{1}, P)); let us denote it by η. To construct this η, we set

[η(f)](ω)(θ) =^{def} f(ω ∧ θ),

where f ∈ Hom_{A}(Λ^{2}, P), ω, θ ∈ Λ^{1}. Hence, D_{2}(P) lies in D_{1}(D_{1}(P)). Consequently, we can take an element ∇ ∈ D_{2}(P) and consider ∇(a), obtaining a new derivation lying in D_{1}(P). Obviously,

∇(a)(b) = −∇(b)(a).

and thus we have

D_{2}(P) = {∇: A → D_{1}(P) | ∇(a, b) = −∇(b, a)}.

In a similar way, for an arbitrary l, we have

Hom_{A}(Λ^{l}, P) → Hom_{A} (Λ^{1}, \underbrace{Hom_{A}(Λ^{l−1}, P)}_{D_{l}(P)} ),

which means that we have a mapping from D_{l}(P) to D_{1}(D_{l−1}(P)). It means that any element ∇ ∈ D_{l}(P) can be evaluated at a_{1}, …, a_{l} ∈ A and by simple induction on l we obtain that ∇: A × ⋯ × A → P lies in D_{l}(P) if and only if

1. It is a derivation with respect to all arguments,

∇(a_{1}, …, a_{i−1}, ab, a_{i+1}, …, a_{l}) = a∇(a_{1}, …, a_{i−1}, b, a_{i+1}, …, a_{l}) + b∇(a_{1}, …, a_{i−1}, a, a_{i+1}, …, a_{l})

2. It is skew-symmetric,

∇(a_{1}, …, a_{l}) = (−1)^{|σ|}∇(a_{σ(1)}, …, a_{σ(l)}),

where |σ| denotes the parity. So we have two ways to describe the module D_{l}(P): the first one is to

identify D_{l}(P) with Hom_{A}(Λ^{l}, P) thus obtaining the coupling ⟨∇, ω⟩ ∈ P; the second one is to describe D_{l}(P) as skew-symmetric derivations of the form A × ⋯ × A → P. The relation between the two interpretation is

∇(a_{1}, …, a_{l}) = ⟨∇, da_{1} ∧ ⋯ ∧ da_{l}⟩.

Let us now define inductively the wedge product operation

∧: D_{l}(A) ⊗ D_{s}(P) → D_{l+s}(P).

When l = s = 0, we have D_{0}(A) = A, D_{0}(P) = P and we set a ∧ p = ap. Now, by induction on l + s we set for Δ ∈ D_{l}(A) and ∇ ∈ D_{s}(P)

(Δ ∧ ∇)(a) = Δ ∧ ∇(a) + (−1)^{|∇|}Δ(a) ∧ ∇.  (53)

We formulate the final result in the following

Proposition 12. Let A be a commutative K-algebra and P be an Amodule. Then:

1. The wedge product ∧: D_{l}(A) ⊗ D_{s}(P) → D_{l+s}(P) is well defined by (53).

2. The module D_{∗}(A) = ⨁_{l≥0} D_{l}(A) is a graded associative commutative algebra with respect to the wedge product, i.e.,

Δ ∧ ∇ = (−1)^{|Δ|·|∇|}∇ ∧ Δ.

3. The module D_{∗}(P) = ⨁_{l≥0} D_{l}(P) is a graded module over D_{∗}(A).

Consequently, the correspondence P ↦ D_{∗}(P) is a covariant functor from the category of all A-modules to the category of all graded D_{∗}(A)- modules.

We shall now define a coupling between the modules D_{1}(A) and Λ^{l} (the inner product) Namely, if ω ∈ Λ^{l} and X ∈ D_{1}(A), then for any ∇ ∈ D_{l−1}(P) we set

⟨∇, X ω⟩ = (−1)^{|∇|}⟨X ∧ ∇, ω⟩.

Thus, the inner product is the adjoint to the wedge product ∧ of polyderivations. We shall also use the notation i_{X}ω instead of X ω. The main properties of the inner product are formulated in the following

Proposition 13. For any X, Y ∈ D(A) and ω, θ ∈ Λ^{∗} one has:

1. The mapping i_{X}: Λ^{l} → Λ^{l−1} is A-linear.

2. The action of is anticommutative, i.e.,

i_{X} ∘ i_{Y} = −i_{Y} ∘ i_{X}

(in particular i_{X} ∘ i_{X} = 0).

3. For all a ∈ A one has i_{X}(da) = X(a).

4. Finally, it is a derivation of Λ^{∗} = ⨁_{l≥0} Λ^{l}, i.e.,

i_{X}(ω ∧ θ) = (i_{X}ω) ∧ θ + (−1)^{|ω|}ω ∧ i_{X}θ.

Now, we shall write the action of the de Rham differential in terms of inner product. Introduce the notation

ω(X_{1}, …, X_{l}) = i_{X_{l}}(… (i_{X_{1}}ω) …) ∈ A.

Let ω be a form of degree l; then

(dω)(X_{1}, …, X_{l+1}) = ∑_{α=1}^{l+1}(−1)^{α−1}X_{α}ω(X_{1}, …, X̂_{α}, …, X_{l+1})

+ ∑_{α<β}(−1)^{α+β}ω([X_{α}, X_{β}], X_{1}, …, X̂_{α}, … X̂_{β}, …, X_{l+1}),

and it is the usual form of the de Rham differential.

Finally, we define the Lie derivative L_{X} of a form ω ∈ Λ^{l} with respect to a derivation X ∈ D_{1}(A). Namely, we set

L_{X}(ω) =^{def} d(L_{X}ω) + L_{X}(dω) = [d, L_{X}]ω,

It is easy to check that L_{X}: Λ^{l} → Λ^{l} satisfies the following properties:

Proposition 14. For any X, Y ∈ D_{1}(A), ω, θ ∈ Λ^{∗}, and a ∈ A one has 1. L_{X}: Λ^{∗} → Λ^{∗} is a K-linear mapping.

2. L_{X} is a derivation of Λ^{∗}, i.e.,

L_{X}(ω ∧ θ) = (L_{X}ω) ∧ θ + ω ∧ (L_{X}θ).

3. L_{X}(dω) = d(L_{X}ω), i.e., [L_{X}, d] = 0.

4. L_{aX}(ω) = aL_{X}ω + da ∧ L_{X}ω.

5. L_{[X,Y]} = [L_{X}, L_{Y}].

6. [L_{X}, i_{Y}] = i_{[X,Y]}.

Our last step is to construct a bridge between algebraic theory of linear differential operators and geometry of nonlinear differential equations. We start with general considerations.

Consider an abstract mechanical system. To observe properties of the system, you need the notion of observables and these observables are real functions on the manifold M of positions. But the set of initial positions is not sufficient to describe system dynamics: we must also know, for example, momenta or velocities of the points, which adds to the picture cotangent or tangent bundle of M. If particles under consideration possess additional properties (e.g., spin),we should also add new variables corresponding to these observables.

Let us begin with simplest situation. Suppose we have a smooth manifold M (which can be understood as a set of states of a physical system). What are smooth functions on this manifold? Of course, there is an analytical definition, but how can it be understood in a geometric way? A function is a smooth map from M to ℝ. But one can consider the Cartesian product M × ℝ and the canonical projection π: M × ℝ → M. Then to say that f is a smooth function on M is the same as to say that f: M → M × ℝ is a smooth mapping satisfying

π ∘ f = id_{M}.  (54)

This definition can be generalized. Consider an additional manifold N and the Cartesian product M ×N. An N-valued function f is a smooth mapping from M to M ×N satisfying the same identity (54). For instance if N = ℝ^{s}, we obtain the concept of smooth vector-valued functions on our manifold M, i.e., functions represented by vectors (f_{1}, …, f_{s}), whose components are just smooth real-valued functions.

Consider now a more complicated situation. Let M be a smooth manifold. Then, by definition, it is locally diffeomorphic to ℝ^{n}. If a point moves in the neighborhood U ⊂ M and x = (x_{1}, …, x_{n}) are local coordinates of this point in U, the velocity is just a tangent vector v_{1}∂/∂x_{1}+⋯+v_{n}∂/∂x_{n}. So at one point all possible velocities are described by points of ℝ^{n} with coordinates (v_{1}, …, v_{n}). If we consider two intersecting local charts U and V and a point in their intersection, we can express the velocities in both local coordinates. Then we have the sets of all possible velocities ℝ^{n} × U and ℝ^{n} × V and these two objects, if U ∩ V is not void, are related to each other by the Jacobi matrix. In this way, spaces of “local velocities” in different neighborhoods are glued together to a new manifold T M called the tangent manifold. There is a natural projection π: T M → M and locally the projection looks as the projection ℝ^{n} × ℝ^{n} → ℝ^{n}. Any smooth mapping f: M → T M satisfying (54) may be understood as a field of velocities on M.

This construction is a particular case of the following one. Consider three smooth manifolds M, O, and F let π: O → M be a surjective smooth mapping. Suppose any point x ∈ M possesses a neighborhood U_{x} = U such that there exists a diffeomorphism φ = φ_{U}: π^{−1}(U_{x}) → U_{x} × F satisfying π |_{π^{−1}(U_{x})} = pr_{U_{x}} ∘ φ, where pr_{U_{x}}: U_{x} × F → U_{x} is the projection to the first component. In this case we say that this π is a locally trivial fiber bundle over M, M is called the base, F the fiber, and O is called the total space of the bundle π. The set F_{x} = π^{−1}(x) is called the fiber over the point x. From the definition it follows that for any x ∈ M there exists a diffeomorphism

φ_{x}: F_{x} → F.  (55)

Example 33. For any two manifolds, M and F, the projection M×F → M is a fiber bundle.

Example 34. The Möbius band is the total space of the bundle over the circle S^{1} with the fiber ℝ.

Example 35. The natural projection T M → M is a fiber bundle with the fiber ℝ^{n}, n = dim M.

Example 36. The natural projection T^{∗}M → M, where T^{∗}M is the cotangent manifold, is a fiber bundle with the fiber ℝ^{n}, n = dim M.

Now fix M and consider two fiber bundles π and π′. A morphism of π to π′ is a smooth mapping φ: O → O′ such that the diagram

\begin{aligned}O & & \overline{φ  →} & O′ \\ & [figure: π] & [figure: π′] \\ & & M \end{aligned}  (56)

is commutative. A morphism of bundles which is a diffeomorphism is called an isomorphism. A bundle isomorphic the Cartesian product M ×F is called trivial. If the mapping φis an embedding, we say that π is a subbundle in π′.

Fiber bundles over a manifold M together with their morphisms form a category. To any bundle π: O → M one can put into correspondence the set smooth mappings

Γ(π) =^{def} {f: M → O | π ∘ f = id_{M}}.

The elements of Γ(π) are called sections. The correspondence π ↦ Γ(π) is a functor from the category of fiber bundles to the category of sets. In fact, if φ is a morphism of the form (56), we can set Γ(φ): f ↦ φ ∘ f ∈ Γ(π′) for any f ∈ Γ(π).

We can generalize definition of a morphism in a following way. Let π: O → M and π′: O′ → M′ be two fiber bundles and ψ: M → M′ be a smooth mapping. Then φ: O → O′ is a morphism, if the diagram

\begin{matrix}O & \overline{φ →} & O′ \\ M & \begin{aligned}π \\ & ↓ \end{aligned} & \overline{ψ →} & M′ & \begin{aligned}& π′ \\ ↓ \end{aligned} \end{matrix}

is commutative.

Now let us consider a smooth mapping g: M′ → M and the subset in Cartesian product M′ × O, which consists of pairs defined by

g^{∗}(O) = {(x′, o) | g(x′) = π(o)}

It is a submanifold in M′ × O and can be mapped both to O and to M′:

\begin{matrix}(x′, o) ∈ g^{∗}(O) & \underrightarrow{π^{∗}(g)} & O ∋ o \\ x′ ∈ M′ & \begin{aligned}g^{∗}(π) \\ & ↓ \end{aligned} & \overline{g →} & M & \begin{aligned}& π \\ ↓ \end{aligned} \end{matrix}

By definition of the space g^{∗}O, it is a commutative diagram and the mapping g^{∗}(π): g^{∗}(O) → M′ is a fiber bundle. Is called the pullback (or induced bundle). In particular, if g is an embedding, the bundle g^{∗}(π) is called the restriction of π to the submanifold M′.

When g is a fiber bundle too, the diagonal arrow is a fiber bundle as well:

g^{∗}(O)  \overline{→} O

[figure: W]

M′ \begin{aligned}g^{∗}(π) \\ & ↓ \end{aligned}  \overline{g →} M \begin{aligned}& π \\ ↓ \end{aligned}

and is called the Whitney product of bundles g and π. Note that if F is the fiber of the bundle π and G is that of the bundle g, then the fiber of Whitney product will be F × G.

In what follows, we shall deal with a special type of fiber bundle.

Definition 4. A fiber bundle π: O → M is called a vector bundle, if 1. The fiber F is a vector space.

2. For any U, V ⊂ M such that φ_{U}: π^{−1}(U) → U ×F and φ_{V}: π^{−1}(V) → V × F the mappings

φ_{U} |_{π^{−1}(U∩V)} ∘ φ_{V}^{−1}|_{(U∩V)×F} : U ∩ V) × F → (U ∩ V) × F

are fiber-wise linear.

We shall consider the case when F is an ℝ-vector space. The dimension of F as a vector space is called the dimension of a fiber bundle.

Let π: O → M and π′: O′ → M be two vector bundles and φ: O → O′ be a morphism. We say that φ is a morphism of vector bundles if it is fiber-wise linear, i.e., φ(a + b) = φ(a) + φ(b) for any a, b ∈ F_{x} and x ∈ M. Obviously, vector bundles over M together with vector bundle morphisms form a category which we denote by Vect(M).

Consider now two sections f, g ∈ Γ(π) of a vector bundle π. Then, due to vector space structure in the fibers, we can set

(f + g)(x) =^{def} f(x) + g(x), x ∈ M.

Moreover, by the same reason, for any smooth function r ∈ C^{∞}(M) we set

(rf)(x) =^{def} r(x)f(x), x ∈ M.

Now, we see that the correspondence Γ: π ↦ Γ(π) is a functor from the category Vect(M) to the category Mod(C^{∞}(M)) of C^{∞}(M)-modules. An amazing fact is that for any finite-dimensional manifold M and finite-di-mensional vector bundle π the module Γ(π) is projective with finite number of generators. Moreover, any such a module over the algebra C^{∞}(M) can be realized as Γ(π) for some vector bundle π! A detailed discussion of these matters can be found in [10].

Let us fix now a smooth manifold M and establish relations between calculus in the category Mod(C^{∞}(M)) and analytical constructions over M needed for the geometrical theory of differential equations.

Consider two vector bundles, π and ξ, over M. Then a linear differential operator of order k acting from π to ξ is an element of the module Diff_{k}(Γ(π), Γ(ξ)). Note that this definition is in full agreement with the classical one. In a similar way, π-valued derivation on M are derivations C^{∞}(M) → Γ(π) and, in particular, C^{∞}(M)-valued derivations coincide with vector fields on M, or with section of the tangent bundle T M. We shall meet no problem in carrying over the theory of functors Diff_{k}^{(+)}, D_{i}, etc. to the geometric situation.

The problem will arise when passing to representative objects. Consider a projective module P with finite number of generators over C^{∞}(M). Then, as we already know, a locally trivial vector bundle π over M corresponds to P such that Γ(π) = P. Consider a point x ∈ M and the ideal μ_{x} =^{def} {f ∈ C^{∞}(M) | f(x) = 0}. The value of an element p ∈ P, as of a section of π, at x is its coset p(x) in the quotient C^{∞}(M)/μ_{x}, the latter being a finite-dimensional space over the field C^{∞}(M) = ℝ. Projective modules over C^{∞}(M) are characterized by the fact that their elements are completely determined by the values p(x) at all points x ∈ M. It follows from Theorem 9 and the fact that the bundles corresponding to free modules are trivial.

Example 37. Let M = ℝ and A = C^{∞}(M). The module Λ^{1}(A) contains, for example, the element d(e^{x}) − e^{x} dx. This element vanishes at all points of ℝ, but is nontrivial in Λ^{1}(A). It is easy to find similar examples for the modules of jets. Thus, in representative objects a sort of “ghost” elements arise, which are not observed geometrically.

This example shows that the category of projective C^{∞}(M)-modules is not closed with respect to construction of representative objects for the basic functors of differential calculus and a straightforward approach does not lead to geometrical constructions parallel to these objects.

To kill the “ghost” similar to the one arising in Example 37, we shall apply the following procedure. For a smooth manifold M and a C^{∞}(M)- module P, let us set 𝔐(P) =^{def} ⋂_{x∈M} μ_{x} · P. Let us say that a module is geometrical, if 𝔐(P) = 0. The we have the functor of geometrization 𝔊: P ↦ P/𝔐(P) acting from the category of all C^{∞}(M)-modules to the category of geometrical modules over the same algebra.

Proposition 15. The functors D_{i}(·) and Diff_{k}(P, ·) are representable in the category of geometrical modules over C^{∞}(M). Moreover, for any geometrical module Q one has

D_{i}(Q) = Hom_{C^{∞}(M)}(𝔊(Λ^{i}(C^{∞}(M)), Q)

and

Diff_{k}(P, Q) = Hom_{C^{∞}(M)}(𝔊(J^{k}(P)), Q).

Corollary 16. For any smooth manifold M one has

Λ^{i}(M) = 𝔊(Λ^{i}(C^{∞}(M)).

To formulate a result similar to Corollary 16, we need a new geometrical construction.

Let π: E → M be locally trivial vector bundle. Consider a point θ ∈ E, π(θ) = x ∈ M, and a section f ∈ Γ(π) whose graph passes through θ: f(x) = θ. Denote by [f]_{x}^{k} the set of sections whose graphs are tangent to the graph of f at θ with order k. This class is called the k-jet of f at x. Obviously, it is completely determined by the Taylor expansion of f at x of order k. Let us choose a coordinate neighborhood U ∋ x in such a way that the bundle π becomes trivial over U. Let x_{1}, …, x_{n} be local coordinates in U and e_{1}, …, e_{m} be a basis in the fiber of π. Then any section is locally represented as f = f^{1}e_{1} +⋯+f^{m}e_{m}, where f^{j} = f^{j}(x_{1}, …, x_{n}) are smooth functions on U. Hence, the class [f]_{x}^{k} is completely determined by the values of all partial derivatives ∂^{|σ|}f^{j}/∂x_{σ} at x, j = 1, …, m, 0 ≤ |σ| ≤ k. Denote the number of these partial derivatives by N = N(n, m, k).

Now, let us take an atlas {U_{α}} in M consisting of coordinate neighborhoods of the above described type. Consider the set

J^{k}(π) =^{def} {[f]_{x}^{k} | x ∈ M, f ∈ Γ(π)}.  (57)

It is covered by the subsets

Ũ_{α} =^{def} {[f]_{x}^{k} | x ∈ U_{α}, f ∈ Γ(π)}.  (58)

Define coordinate functions x_{i} and p_{σ}^{j} in U_{α} by

x_{i}([f]_{x}^{k}) = x_{i}, p_{σ}^{j}([f]_{x}^{k}) = \frac{∂^{|σ|}f^{j}}{∂x_{σ}} (x).  (59)

Then functions (59) establish a one-to-one correspondence between Ũ_{α} and the space U_{α} × ℝ^{N}.

Proposition 17. The system of sets Ũ_{α} together with coordinate functions (59) constitute an atlas in the set J^{k}(π). Thus J^{k}(π) becomes a smooth manifold. Moreover, the projection π_{k}: J^{k}(π) → M, π_{k}([f]_{x}^{k}) =^{def} x, is a smooth locally trivial vector bundle.

Definition 5. The manifold J^{k}(π) is called the manifold of k-jets emphmanifold of jets for the bundle π. The bundle π_{k} is called the bundle of k-jetsemphbundle of jets for the bundle π.

We can now formulate a statement similar to Corollary 16:

Proposition 18. Let M be a smooth manifold and π: E → M be a locally trivial vector bundle. Let P = Γ(π). Then

Γ(π_{k}) = 𝔊(J^{k}(P)).

Manifolds of jets is a natural environment for geometrical theory of differential equations. This theory will be discussed in the next lecture course.

Exercises

Exercise 1. Let A be a ring. Prove that 0 · a = a · 0 = 0 for all a ∈ A.

Exercise 2. Let A be a set and 2^{E} the set of all subsets of A. Two natural binary operations exist in 2^{E}: the union ∪ and the intersection ∩ operations.

1. Show that 2^{E} is not a ring with respect to these operations.

2. Prove that if we define

a · b = a ∩ b  (60)
a + b = (\overline{a} ∩ b) ∪ (a ∩ \overline{b})  (61)

where the “ bar” denotes complement in E, then 2^{E} is a commutative unitary ring.

Exercise 3. Consider the set ℤ_{2} = {0, 1}. Check that it is a commutative ring with the following rules of summation

0 + 0 = 1, 1 + 0 = 1, 1 + 1 = 0

and the usual ones for multiplication.

Let now E be a set. The characteristic functions of a subset a ∈ 2^{E} is defined as

χ_{a}(x) = \begin{cases}1 & x ∈ a \\ 0 & x ∉ a \end{cases}

Since there exists a one-to-one correspondence between 2^{E} and the set of characteristic functions X(E), we can introduce the structure of a ring in the latter by setting

χ_{a} + χ_{b} = χ_{a+b}, χ_{a}χ_{b} = χ_{a·b},

where a, b ∈ 2^{E} and sum and product of sets are defined by (60) and (61).

Exercise 4. Prove that the correspondence 0 ↦ χ_{∅}, 1 ↦ χ_{E} is a homomorphism of ℤ_{2} to X(E) and thus X(E) is a ℤ_{2}-algebra.

Exercise 5. Let A and B be two rings and f: A → B be a ring homomorphism. Prove that ker f is a two-sided ideal in A.

Exercise 6. Let A be a ring and I be a two-side ideal of A. Consider the quotient A/I and prove that the operations

[a] + [b] =^{def} [a + b], [a] · [b] =^{def} [ab]

determine a well-defined ring structure in A/I. What structure will be carried by A/I, if I is a left or right ideal?

Exercise 7. Let A = C^{∞}(ℝ) and x ∈ ℝ. Consider the set

μ_{x} = {f ∈ A | f(x_{)} = 0} .

Prove that μ_{x} is an ideal and that A/μ_{x} is isomorphic to ℝ.

Exercise 8. Let again A = C^{∞}(ℝ) and x, y ∈ ℝ be two different points. Consider the set

μ_{xy} = {f ∈ A | f(x) = f(y) = 0} .

Prove that μ_{xy} is an ideal and that A/μ_{xy}, as an ℝ-vector space is isomorphic to ℝ^{2} and describe the ring structure in terms of a basis in ℝ^{2}.

This problem has an interesting prolongation. Of course, if we set x = y, the result will the same as in Exercise 7. But we can act in a different way. Namely, let d = d(x, y) be the distance between the points x and y. Then, as d → 0, the ideal I_{xy} becomes

I = \{f ∈ C^{∞}(ℝ) | f(x) = \frac{df}{dx}(x) = 0\}

and you can prove that A/I is still isomorphic to ℝ^{2} as a vector space, but sum and product will be defined as follows

(a_{1}, b_{1}) + (a_{2}, b_{2}) = (a_{1} + a_{2}, b_{1} + b_{2})
(a_{1}, b_{1}) · (a_{2}, b_{2}) = (a_{1}a_{2}, a_{1}b_{2} + a_{2}b_{1}).

Now we shall need the following

Definition 6. Let A be a ring and I ⊂ A an ideal. The ideal I is called principal, if there exists an element a ∈ A such that I = {aA}.

If all ideals of A are principal ideals then A is called principal ideal domain. We shall denote by I_{n} the principal ideals of ℤ (that is ideals of the form

I_{n} = nℤ with n ∈ ℤ).

Exercise 9. Prove that ℤ is a principal ideal domain.

Exercise 10. Let p ∈ ℤ Prove that ℤ_{p} =^{def} ℤ/I_{p} is a field if and only if p is prime.

Let us, for example, consider the case p = 3. Then

ℤ/3ℤ = {[0], [1], [2]}

and we can write the tables of sum and multiplication:

+ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1  \begin{matrix}· & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 1 \end{matrix}

Exercise 11. Prove that nℤ ⊂ mℤ if and only if m divides n.

For an arbitrary ring A and its ideals I and I′ we can define the ideal I + I′ by setting

I + I′ =^{def} {a + b | a ∈ I, b ∈ I′}.

Exercise 12. Consider the ideals nℤ and mℤ with m, n ∈ ℤ. Then nℤ + mℤ = lℤ for some l ∈ ℤ. Prove that l is maximal common divisor (mcd) of m and n.

Exercise 13. Prove also that nℤ ∩ mℤ = sℤ with s being the minimal common multiple (mcm) of m and n.

Exercise 14. Let A be a field. Prove that I ⊂ A is an ideal if and only if I = {0} or I = A. Thus, fields, and they only, have no nontrivial ideals.

Exercise 15. Let A be a (commutative unitary) ring. Prove I ⊂ A is a maximal ideal if and only if A/I is a field.

Exercise 16. Prove that an ideal I ⊂ A is prime if and only if the quotient A/I possesses no zero divisors.

Exercise 17. Prove that any maximal ideal is prime.

Exercise 18. Prove that any ideal is contained in a maximal ideal. (Hint: Use the Zorn Lemma.)

Exercise 19. Let A be a ring and I ⊂ A be its ideal. Consider the natural projection π: A → A/I, π(a) = [a]. Prove that:

1. if J ⊂ A/I is an ideal then π^{−1}(J) ⊂ A is an ideal in A containing I;

2. if I′ ⊂ A is an ideal, then π(J) ⊂ A/I is an ideal too.

Exercise 20. Construct an example of projective module which is not free.

Exercise 21. Let x ∈ ℝ and μ_{x} ⊂ C^{∞}(ℝ) be the ideal of functions vanishing at x. Prove that μ_{x} is a projective C^{∞}(ℝ)-module. Is it true for the case ℝ^{n}?

Exercise 22. Let x ∈ S^{1} and μ_{x} ⊂ C^{∞}(S^{1}) be the ideal of functions vanishing at x. Prove that μ_{x} is a projective C^{∞}(S^{1})-module.

Exercise 23. Prove that P is projective if and only if the exists P′ such that P′ ⊕ P is a free module.

Exercise 24. Prove that P is a projective module if and only if for any A-modules Q and R an exact sequence

0 → Q ⟶^{i} R ⟶^{π} P → 0

splits, i.e., there exists a homomorphism j: P → R such that π ∘ j = id_{P}.

Exercise 25. Let

0 → R → S → T → 0

Be an exact sequence of modules. Prove that P is projective if and only if the sequence

0 → Hom_{A}(P, R) → Hom_{A}(P, S) → Hom_{A}(P, T)

is an exact as well.

Exercise 26. Consider C^{∞}(ℝ) and vector valued functions (i.e., rows consisting of smooth functions)

C^{∞}(ℝ) ⊕ ⋯ ⊕ C^{∞}(ℝ).

Compute derivations of C^{∞}(ℝ) with values in this module.

Exercise 27. Let A = C^{∞}(ℝ), P = A ⊕ A be the set of vector valued functions with two components. Describe first order differential operators acting from P to itself.

Exercise 28. Consider the cross M = {(x, y) | xy = 0} ⊂ ℝ^{2}} and let us define the algebra A = C^{∞}(M) by

C^{∞}(M) =^{def} {f: M → ℝ | f = f′|_{M}, f′ ∈ C^{∞}(ℝ^{2})}

1. Describe all tangent vectors at the point 0.

2. Describe all derivations A → A.

3. Describe Diff_{∗}(A). Show that this algebra is not generated by functions Diff_{0}(A) = A and derivations D(A) (contrary to the case of smooth manifolds).

Exercise 29. Let A = C(ℝ) the algebra of all continuous functions on ℝ. Prove that D(A) = 0.

Exercise 30. Let A = ℝ[x]/(x^{n}). Describe D(A), Diff_{∗}(A).

Exercise 31. Let A = ℤ_{m}[x]/(x^{n}). Describe D(A), Diff_{∗}(A).

For any commutative K-algebra A, let us introduce the quotient modules

Smbl_{k} A =^{def} Diff_{k}^{(+)} A/ Diff_{k−1}^{(+)} A

and denote by smbl_{k} Δ the coset of Δ ∈ Diff_{k}^{(+)} A in Smbl_{k} A. This coset is called the symbol of the operator Δ.

Exercise 32. Prove that the module structures inherited by Smbl_{k} A from Diff_{k}^{+} A and from Diff_{k} A coincide.

Let f = smbl_{k} Δ and g = smbl_{l} ∇, Δ ∈ Diff_{k} A, ∇ ∈ Diff_{l} A. Let set f · g =^{def} smbl_{k+l}(Δ ∘ ∇). Then Smbl_{∗} A = ⨁_{i≥0} Smbl_{i} A becomes an algebra, which is call the algebra of symbols for A. Let us also set {f, g} =^{def} smbl_{k+l−1}(Δ ∘ ∇ − ∇ ∘ Δ).

Exercise 33. Prove that Smbl_{∗} A is a commutative algebra with respect to the multiplication (f, g) ↦ f · g.

Exercise 34. Prove that Smbl_{∗} A is a Lie K-algebra with respect to the multiplication (f, g) ↦ {f, g} and

{f, g · h} = {f, g} · h + g · {f, h}

for any f, g, h ∈ Smbl_{∗} A.

Exercise 35. Write down the action of c_{l,s} in terms of elements and prove commutativity of all the diagrams above.

Exercise 36. Prove the associativity law for the universal composition c_{l,s}.

Exercise 37. Prove that the universal composition c_{l,s} is a natural transformation of functors.

Exercise 38. Let Δ ∈ Diff_{l}^{(+)}(A) with A = C^{∞}ℝ). Prove in coordinate form that ∇ = [a, Δ] lies in Diff_{l−1}^{+} A.

Exercise 39. Prove that id_{+} and id^{+} are differential operators.

Exercise 40. Consider the operator �_{l}: Diff_{l}^{+} A → A defined in (47). Check in coordinate form that it is a differential operator of order l (in algebraic form it has already been proved) for A = C^{∞}(ℝ^{n}).

Exercise 41. Prove the isomorphism between the modules Diff_{l}^{+}(P, Q) and Hom_{A}(P, Diff_{l}^{+} Q).

Exercise 42. Let P = Q = \{\binom{f_{1}}{f_{2}} | f_{2} ∈ C^{∞}(ℝ) ≡ A\}. Describe the homomorphism φ(Δ): P → Diff^{+} P for an arbitrary differential operator Δ: P → P.

Exercise 43. Prove that J^{k}(P) is the representative object for the functor Diff_{k}(P, ·) (Theorem 10).

Exercise 44. Prove the embeddings μ_{l+1} ⊂ μ_{l} for the submodules μ_{l} ⊂ A ⊗_{K} P defining J^{l}(P).

Exercise 45. Prove that μ_{l}/μ_{l+1} = ker ν_{l+1,l}, where the projections ν_{l+1,l}: J^{l+1}(P) → J^{l}(P) is defined by the universal property of J^{l}(P).

Exercise 46. Prove co-associativity of the co-composition c^{k,l}: J^{k+l} → J^{k}(J^{l}(P)).

Exercise 47. Prove that the co-composition c^{k,l} is a natural transformation of functors J^{k+l} and J^{k}(J^{l}(P)).

Exercise 48. Prove that Λ^{1} is the representative object for the functor D_{1}(·) (Theorem 11).

Let P be an A-module and N ⊂ P be a subset in P. Introduce the notation

D(N) =^{def} {Δ ∈ D(P) | im Δ ⊂ N}.

Of course, in general, D(N) does not possess an A-module structure.

Exercise 49. Prove that the modules D_{i}(P) may be defined in the following inductive way: D_{0}(P) = P, D_{1}(P) = D(P) and

D_{i+1}(P) = D(D_{i}(P) ⊂ (Diff_{1}^{+})^{i}(P))

Exercise 50. Prove the basic properties inner product (Proposition 13):

1. The mapping i_{X}: Λ^{l} → Λ^{l−1} is A-linear.

2. The action of is anticommutative, i.e.,

i_{X} ∘ i_{Y} = −i_{Y} ∘ i_{X}

(in particular i_{X} ∘ i_{X} = 0).

3. For all a ∈ A one has i_{X}(da) = X(a).

4. Finally, it is a derivation of Λ^{∗} = ⨁_{l≥0} Λ^{l}, i.e.,

i_{X}(ω ∧ θ) = (i_{X}ω) ∧ θ + (−1)^{|ω|}ω ∧ i_{X}θ.

Exercise 51. Prove the basic properties of the de Rham differential d:

1. It is a first order differential operator.

2. It differentiates the wedge product, i.e.,

d(ω ∧ θ) = d(ω) ∧ θ + (−1)^{l}ω ∧ d(θ)

3. d ∘ d = 0

Exercise 52. Prove the formula

dω(X_{1}, …, X_{n}) = ∑_{i} X_{i}(ω(X_{1}, …, X̂_{i}, …, X_{n}))

+ ∑_{i<j}(−1)^{i+j}ω([X_{i}, X_{j}], X_{1}, …, X̂_{i}, … X̂_{j}, …, X_{n}).

Exercise 53. Prove the basic properties of Lie derivatives (Proposition 14):

1. L_{X}: Λ^{∗} → Λ^{∗} is a K-linear mapping.

2. L_{X} is a derivation of Λ^{∗}, i.e.,

L_{X}(ω ∧ θ) = (L_{X}ω) ∧ θ + ω ∧ (L_{X}θ).

3. L_{X}(dω) = d(L_{X}ω), i.e., [L_{X}, d] = 0.

4. L_{aX}(ω) = aL_{X}ω + da ∧ L_{X}ω.

5. L_{[X,Y]} = [L_{X}, L_{Y}].

6. [L_{X}, i_{Y}] = i_{[X,Y]}.

Exercise 54. Prove the formula

(L_{X}ω)(X_{1}, …, X_{n}) = Xω(X_{1}, …, X_{n}) + ∑_{i=1}^{n} ω(X_{1}, …, [X, X_{i}], …, X_{n}).

A K-algebra A is called smooth if (a) K is an algebra over the field ℚ of rational numbers and (b) Λ^{1}(A) is a projective module with finite number of generators.

Exercise 55. Prove that when A is a smooth algebra, then one has the isomorphism

ker ν_{l,l−1} = S^{⊗k}Λ^{1}(A),

where S^{⊗k} denotes the k-th symmetric power of a module.

Exercise 56. Prove that when A is a smooth algebra and P is a projective module, then J^{k}(P) is projective as well.

Exercise 57. Prove that for smooth algebras the algebra Diff_{∗} A is multiplicatively generated by elements of Diff_{1} A (cf. Exercise 28).

Let A be a commutative ring. The set Spec A consisting of all its prime ideals is called the spectrum of A. Let a ∈ A. Let us define

U_{a} =^{def} {I ∈ Spec A | a ∉ I}

and take the system {U_{a}}_{a∈A} for a base of open sets in Spec A. Thus, Spec A becomes a topological space and the corresponding topology in Spec A is called the Zarissky topology. If f: A → B is an algebra homomorphism, then for any I ∈ Spec B the set f^{−1}I ⊂ A is a prime ideal in A. Thus we obtain the mapping f^{∗}: Spec B → Spec A which is continuous in Zarissky topology. It is easy to see that the correspondence A ↦ Spec A determines a contravariant functor from the category of commutative rings to that of topological spaces.

The set Spec_{m} A of maximal ideals contains in Spec A, but the correspondence A ↦ Spec_{m} A is not functorial. Nevertheless, we can consider the following construction. Fix a field 𝕜 and call a homomorphism φ: A → 𝕜 a 𝕜-point. Denote the set of 𝕜-points by Spec_{𝕜} A. Let a ∈ A and U =^{def} {φ ∈ Spec_{𝕜} A | φ(a) ≠ 0. Taking {U_{a}}_{a∈A} for a base of open sets, we make a topological space of Spec_{𝕜} A. The correspondence A ↦ Spec_{𝕜} A determines a functor and there exists natural continuous mappings Spec_{𝕜} A → Spec_{m} A → Spec A (the composition being a natural transformation of functors Spec_{𝕜} A and Spec A!).

Exercise 58. Consider the polynomial algebra A = ℝ[x]. Describe Spec A, Spec_{ℝ} A and Spec_{ℂ} A.

Exercise 59. Consider the ring of periodic functions

A = {f ∈ C^{∞}(ℝ) | f(x) = f(x + 1) ∀x ∈ ℝ}.

Describe Spec_{ℝ} A.

Exercise 60. Let

A = {f ∈ C^{∞}(ℝ^{2}) | f(x, y) = f(x, y + 1) ∀(x, y) ∈ ℝ^{2}}.

Prove that Spec_{ℝ} A ≃ S^{1} × ℝ.

Exercise 61. Let

A = {f ∈ C^{∞}(ℝ^{2}) | f(x + 1, y) = f(x, y) = f(x, y + 1) ∀(x, y) ∈ ℝ^{2}}.

Prove that Spec_{ℝ} A ≃ S^{1} × S^{1}.

46 EXERCISES

Exercise 62. Let

A = {f ∈ C^{∞}(ℝ^{2}) | f(x + 1, y) = f(x, y) = f(x, −y) ∀(x, y) ∈ ℝ^{2}}. Prove that Spec_{ℝ} A is the Klein bottle.

Exercise 63. Let

A = {f ∈ C^{∞}(ℝ^{2}) | f(x, y) = f(x + 1, −y) ∀(x, y) ∈ ℝ^{2}}.

Prove that Spec_{ℝ} A is the Möbius band.

Exercise 64. Let

A = {f ∈ C^{∞}(ℝ3 \ {0}) | f(x, y, z) = f(λx, λy, λz) ∀λ > 0}. Prove that Spec_{ℝ} A ≃ S^{2}.

Exercise 65. Let

A = {f ∈ C^{∞}(ℝ^{3} \ {0}) | f(x, y, z) = f(λx, λy, λz) ∀λ ≠ 0}. Prove that Spec_{ℝ} A ≃ ℝℙ^{2}.

Exercise 66. Let

A = C^{∞}(ℝ^{3})/(x^{2} + y^{2} + z^{2}).

Prove that Spec_{ℝ} A ≃ S^{2}.

Exercise 67. Consider the ring of characteristic functions (see Exercise 3) and describe its spectrum and Zarissky topology.

Exercise 68. Consider the ring of smooth bounded functions on ℝ^{n} and describe its spectrum and Zarissky topology.

Exercise 69. Prove that functions (59) on p. 38 determine a smooth manifold structure on J^{k}(π) (Proposition 17).

Exercise 70. Prove that π_{k}: J^{k}(π) → M is a smooth locally trivial vector bundle.

Exercise 71. Let M be a smooth manifold and τ^{∗}: T^{∗}M → M be its cotangent bundle. Prove that Γ(τ^{∗}) = 𝔊(Λ^{i}(C^{∞}(M)), where 𝔊 is the geometrization functor.

Exercise 72. Let π: E → M be a smooth finite-dimensional vector bundle and π_{k}: J^{k}(π) → M be the bundle of its k-jets. Prove that Γ(π_{k}) = 𝔊(J^{k}(P)), where P = Γ(π).

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[6] S. Lang, Algebra, Addison-Wesley, 1992.

[7] I.G. MacDonald and M.F. Atiyah, Introduction to Commutative Algebra, Perseus Pr., 1994.

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48 BIBLIOGRAPHY

Index

adjoint module, 8 algebra of symbols, 42 algebra over a ring, 6 associative ring, 1

base of a fiber bundle, 34 bilinear mapping, 12 bimodule, 10

category, 14 characteristic functions of a subset, 39 cohomology, 29 commutative diagram, 9 commutative ring, 1 complex, 29 contravariant functor, 15 covariant functor, 15

derivation, 19 derivation of an algebra, 17 de Rham cohomology, 29 de Rham complex, 29 de Rham differential, 28, 29 differential operator, 19 dimension of a fiber bundle, 36 direct sum of modules, 11 dual module, 8

endomorphism, 10 epimorphism, 4, 10 external algebra, 5

fiber of a fiber bundle, 34 field, 1 field of velocities, 34 free module, 11 functor, 15

generators of a module, 11 geometrical module, 37 geometrization functor, 37 group ring, 2

homology, 29 image of a homomorphism, 3, 11 induced bundle, 35 infinite jet of an element, 26 inner product, 31 isomorphism, 4, 10 isomorphism of fiber bundles, 34

jet of a section at a point, 37

jet of an element, 25

kernel of a homomorphism, 4, 11

left ideal of a ring, 4

Lie algebra, 3 Lie derivative, 32 linear differential operator in fiber bun-

dles, 36 locally trivial fiber bundle, 34

maximal ideal, 5 module of 1-forms, 28 module of infinite jets, 26 module of jets, 25 module over a ring, 7 monomorphism, 4, 10 morphism of fiber bundles, 34, 35 morphism of vector bundles, 36

natural transformation of functors, 16

prime ideal, 6 principal ideal, 40 projective module, 13 pullback, 35

quotient ring, 4

representable functor, 24 representative object, 24 right ideal of a ring, 4 ring, 1 ring homomorphism, 3 ring of polynomials, 5

section of a bundle, 35 short exact sequence, 4 smooth algebra, 44 spectrum of a ring, 45 Spencer cohomologies, 30 Spencer complex, 30 subbundle of a fiber bundle, 34 submodule, 10 symbol of a differential operator, 42 symmetric algebra, 5

tensor algebra of a vector space, 3 tensor product, 12 to-sided ideal of a ring, 4 total space of a fiber bundle, 34 trivial fiber bundle, 34

unitary module, 7 unitary ring, 1 universal co-composition, 27 universal composition, 24

wedge product in polyderivations, 31 Whitney product, 35

Zarissky topology, 45 zero divisor, 6