Algebraic theory of linear
differential operators and
introduction to geometry of jet spaces

Joseph KRASIL'SHCHIK
A program of the course at the 1st Italian Diffiety School
(Forino, July 17  - August 1, 1998)
and at the 1st Russian Diffiety School
(Pereslavl-Zalessky, January 26 - February 5, 1998)
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1.
Rings. Rings with unit. Associative rings. Commutative rings. Fields. Examples.
2.
Homomorphisms of rings. Kernels and images. Epi-, mono-, and isomorphisms. Algebras. Examples.
3.
Ideals. Quotient rings. Prime and maximal ideals.
4.
Modules. Generators. Submodules and quotient modules. Direct sums. Examples.
5.
Homomorphisms of modules. Kernels and images. Epi-, mono-, and isomorphisms. Exact sequences. Compositions of homomorphisms. Commutative diagrams.
6.
Homomorphisms of modules over commutative algebras. Bimodule structure.
7.
Free and projective modules.
8.
Tensor products. The tensor, symmetric and skew-symmetric algebras of a module.
9.
Categories and functors. Natural transformations of functors. Representable functors and representative objects.

$\displaystyle \star\ \star\ \star$

10.
Vector fields on manifolds as derivations of $ C^{\infty}(M)$. The Leibniz rule.
11.
Derivations of commutative algebras. Module and Lie algebra structures.
12.
Module-valued derivations. The modules $ \mathrm{D}(P)$. The commutator identity $ [a,[a',{\Delta}]]=0$. First order differential operators $ A\to {}P$ and their description.
13.
The algebraic definition of linear differential operators $ {\Delta}:P\to {}Q$. Equivalence of the algebraic and analytical definitions for the case $ A=C^{\infty}(\mathbb{R})$.
14.
The bimodules $ \mathrm{Dif{}f}_l^{(+)}(P,Q)$. The tautological operators of order $ l$ $ \i^{+,\bullet}: \mathrm{Dif{}f}_l^+(P,Q)\to {}\mathrm{Dif{}f}_l(P,Q)$ and $ \i^{\bullet,+}:\mathrm{Dif{}f}_l(P,Q)\to {} \mathrm{Dif{}f}_l^+(P,Q)$. The bimodule embeddings $ \mathrm{Dif{}f}_l^{(+)}(P,Q)\sbs \mathrm{Dif{}f}_{l+1}^{(+)}(P,Q)$, the module $ \mathrm{Dif{}f}_*^{(+)}(P,Q)$. The bimodules $ \mathrm{Dif{}f}_l^{(+)}(A,Q)= \mathrm{Dif{}f}_l^{(+)}(Q)$.
15.
Composition of linear differential operators. The structure of an associative algebra in $ \mathrm{Dif{}f}_*^{(+)}(P,P)$.
16.
The exact sequences

$\displaystyle 0\to {}Q\to {}\mathrm{Dif{}f}_1^{(+)}(Q)\to {}\mathrm{D}(Q)\to {}0$

and

$\displaystyle 0\to {}\mathrm{D}(Q)\to {}\mathrm{Dif{}f}_1(Q)\to {}Q\to {}0.$

The splitting $ \mathrm{Dif{}f}_1(Q)=Q\oplus\mathrm{D}(Q)$.
17.
The operators $ \rd_l:\mathrm{Dif{}f}_l^+(Q)\to {}Q$, their universal property. The natural isomorphism $ \mathrm{Dif{}f}_l^+(P,Q)=\mathrm{Hom}_A(P,\mathrm{Dif{}f}_l(Q))$. Representability of $ \mathrm{Dif{}f}_l^+(\bullet,Q)$.
18.
The universal composition $ c_{l,s}:\mathrm{Dif{}f}_l^+\mathrm{Dif{}f}_s^+(Q)\to {}\mathrm{Dif{}f}_{l+s}^+(Q)$ as a natural transformation of functors. Associativity of the universal composition.

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19.
The submodules $ \mu_l(P)\sbs A\ot_\Bbbk P$. The jet modules $ \mathcal{J}^l(P)=A\ot_\Bbbk P/ \mu_l(P)$. The operators $ j_l:P\to {}\mathcal{J}^l(P)$.
20.
The universal property of operators $ j_l$. The natural isomorphism of $ A$-modules $ \mathrm{Dif{}f}_l(P,Q)=\mathrm{Hom}_A(\mathcal{J}^l(P),Q)$. Representability of $ \mathrm{Dif{}f}_l(P,\bullet)$.
21.
Functorial nature of $ \mathcal{J}^l(\bullet)$. The exact sequences

$\displaystyle 0\to {}\mu_{l-1}(P)/\mu_l(P)\to {}\mathcal{J}^l(P)\to {\nu_{l,l-1}}\mathcal{J}^{l-1}(P) \to {}0$

22.
The module of infinite jets $ \mathcal{J}^\infty(P)$.
23.
The universal cocomposition $ c^{l,s}:\mathcal{J}^{l+s}(P)\to {}\mathcal{J}^l\mathcal{J}^s(P)$ as a natural transformation of functors. Coassociativity of the universal cocomposition.

$\displaystyle \star\ \star\ \star$

24.
The embedding $ i_1:A\to {}\mathcal{J}^1(A)$, $ a\mapsto aj_1(1)$, $ a\in A$. The module of 1-forms $ {\Lambda}^1(A)=\mathcal{J}^1(A)/\im i_1$. The de Rham operator $ d:A\to {}{\Lambda}^1(A)$, $ d\in\mathrm{D}({\Lambda}^1(A))$. The splitting $ \mathcal{J}^1(A)=A\oplus{\Lambda}^1(A)$.
25.
The universal property of $ d$. The natural isomorphism of $ A$-modules $ \mathrm{D}(P)=\mathrm{Hom}_A({\Lambda}^1(A))$. Representability of $ \mathrm{D}(\bullet)$.
26.
Description of $ {\Lambda}^1(A)$ by generators and relations.
27.
The modules $ {\Lambda}^l(A)={\Lambda}^1(A)\wg\dots\wg{\Lambda}^1(A)$, the module $ {\Lambda}^*(A)$. Wedge product $ \wg:{\Lambda}^l(A)\times{\Lambda}^s(A)\to {}{\Lambda}^{l+s}(A)$ in $ {\Lambda}^*(A)$.
28.
The de Rham differential $ d:{\Lambda}^l(A)\to {}{\Lambda}^{l+1}(A)$, its properties. The de Rham complex and the de Rham cohomology.
29.
The Spencer jet complex

\begin{multline*} 0\to A\overset{j_l}\to \mathcal{J}^l(A)\to {}\dotsb\\ \dots... ...i-1}({\Lambda}^{i+1}(A))\to {}\dotsb\to {}{\Lambda}^l(A)\to {}0 \end{multline*}

$\displaystyle \star\ \star\ \star$

30.
The modules $ \mathrm{D}_l(P)=\mathrm{Hom}_A({\Lambda}^l(A),P)$. The functors $ \mathrm{D}_l(\bullet)$.
31.
Elements of $ \mathrm{D}_l(P)$ as skew-symmetric $ P$-valued multiderivations.
32.
Wedge product $ \wg:\mathrm{D}_l(A)\times\mathrm{D}_s(P)\to {}\mathrm{D}_{l+s}(P)$, its properties. Structure of superalgebra in $ \mathrm{D}_*(A)=\bigoplus_{l\geq0}\mathrm{D}_l(A)$.
33.
Inner product (contraction) $ \mathrm{i}:\mathrm{D}(A)\times{\Lambda}^l(A)\to {}{\Lambda}^{l-1}(A)$, its properties.
34.
The formula

\begin{multline*} (d{\omega})(X_1,\dots,X_{l+1})=\sum_{{\alpha}=1}^{l+1}(-1)^{{... ...,\dots,\Hat{X}_{\alpha},\dots, \Hat{X}_{\beta},\dots,X_{l+1}). \end{multline*}

35.
Action of derivations on differential forms (Lie derivative), its properties.
36.
The formula

$\displaystyle (\mathrm{L}_X{\omega})(X_1,\dots,X_l)=X{\omega}(X_1,\dots,X_l)- \sum_{{\alpha}=1}^l{\omega}(X_1,\dots,[X,X_{\alpha}],\dots,X_l).$

$\displaystyle \star\ \star\ \star$

37.
Smooth algebras.

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38.
Motivations to consider fiber bundles: how to extend algebras of observables? First examples: Cartesian products of manifolds, the Moebius band, tangent and cotangent bundles.
39.
Locally trivial fiber bundles. Morphisms. Sections. Subbundles.
40.
Pull-backs. Restriction of a bundle to a submanifold. Whitney products.
41.
Vector bundles. Morphisms. Kernels and images. Modules of sections, the functor $ {\Gamma}(\bullet)$.
42.
Direct sums and tensor products of vector bundles.
43.
Evaluation of $ C^{\infty}(M)$-modules at points of $ M$. Isomorphism between the category of vector bundles over $ M$ and the category of finitely generated $ C^{\infty}(M)$-modules.

$\displaystyle \star\ \star\ \star$

44.
Example: evaluation of $ {\Lambda}^1(C^{\infty}(M))$ at points of $ M$. Ghosts. The geometrization functor $ \mathfrak{G}(\bullet)$ and geometrical modules. Geometrization of algebraic Calculus over $ C^{\infty}(M)$.
45.
Evaluation of $ \mathcal{J}^l(C^{\infty}(M))$ at point of $ M$. Identification with Taylor series of order $ l$.
46.
General geometric definition of the manifolds $ J^l(\pi)$. Points of $ J^l(\pi)$ as classes of tangent sections. Jets of sections. Local coordinates.
47.
Fiber bundles $ \pi_l:J^l(\pi)\to {}M$ and $ \pi_{l,l-1}:J^l(\pi)\to {} J^{l-1}(\pi)$. Vector bundle structure in $ \pi_l$. Fibers of the bundles $ \pi_{l,l-1}$.

$\displaystyle \star\ \star\ \star$

48.
Scalar nonlinear differential operators as functions on $ J^l(\pi)$. General nonlinear differential operators $ {\Gamma}(\pi)\to {}{\Gamma}(\xi)$.
49.
Nonlinear differential equations as submanifolds of $ J^l(\pi)$. Solutions.
50.
How to diminish the order of a differential equation using the universal cocomposition?
51.
The Cartan distribution on $ J^l(\pi)$, its maximal integral manifolds.
52.
Lie transformations and Lie fields. Symmetries.
53.
Passing to infinity.

References

  1. M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, Addison--Wesley Publ. Comp., Reading, Massachusetts, 1969.
  2. S. Lang, Algebra, Addison--Wesley Publ. Comp., Reading, Massachusetts, 1965.
  3. I. S. Krasil'shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, New York, 1986.
  4. I. S. Krasil'shchik, Calculus over commutative algebras: A concise user guide, Acta Appl. Math. 49 (1997), 235--248,
    URL: http://diffiety.ac.ru/preprint/96/01_96abs.htm.
  5. Jet Nestrujev, Smooth manifolds and observables, IUM, Moscow, 2000 (in Russian). English translation to appear.
    Questions and suggestions should go to J. S. Krasil'shchik, josephk @ diffiety.ac.ru.