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A primer of algebraic \({\mathcal D}\)-modules. (English) Zbl 0848.16019

London Mathematical Society Student Texts. 33. Cambridge: Cambridge Univ. Press. xii, 207 p. (1995).
This book is an introduction to the theory of modules over the Weyl algebra. In chapters I, II and III, the author introduces the Weyl algebra \(A_n(k)\) (where \(n\geq 1\) and \(k\) is a field of characteristic zero) in two different ways. First of all, as a subalgebra of \(\text{End}_k(k[X])\), where \(k[X]=k[x_1,\dots,x_n]\) is the polynomial ring with \(n\) indeterminates and coefficients in \(k\). Secondly, as a quotient of the free algebra in \(2n\) generators and coefficients in \(k\). Examples are given to show that these two definitions are not equivalent if the characteristic of the field is positive. By using the notion of degree of an operator in \(A_n(k)\) it is proved that the Weyl algebra is simple and that \(A_n(k)\) is the algebra of (linear) differential operators of \(k[X]\).
In chapter IV it is proved that if every endomorphism of \(A_n(k)\) is an automorphism then the Jacobian conjecture holds. Unfortunately nothing is known about the first condition.
Chapters V to X are devoted to the study of basic properties of (good) filtrations on (left) \(A_n(k)\)-modules (with respect to the Bernstein filtration of the Weyl algebra). Dimension and multiplicity of a finitely generated left \(A_n(k)\)-module are introduced by using the Hilbert polynomial of the associated graded module. Then Bernstein’s inequality (\(n\leq\dim(M)\leq 2n\)) is established for all f.g. left non-zero \(A_n(k)\)-modules \(M\). Such a module \(M\) is called holonomic if its dimension is \(n\) \((\dim(M) = n)\). The basic properties of holonomic modules are developed and the existence of the Bernstein polynomial is proved.
Chapter XI is devoted to the definition of the main geometric invariant of a f.g. left \(A_n(k)\)-module \(M\): its characteristic variety \(Ch(M)\) (with respect to the Bernstein filtration on the Weyl algebra). The (very) difficult proof of the involutivity of \(Ch(M)\) is not included. The author gives an example, due to Stafford, of an operator \(S \in A_n(k)\) such that the principal left ideal \(J = A_n(k)S\) is maximal (so, \(A_n(k)/J\) is an irreducible non holonomic module if \(n > 2\)). The work of Bernstein and Lunts on irreducible non holonomic \(A_n(\mathbb{C})\)-modules is summarized at the end of this chapter.
Operations (external products, inverse images and direct images) on \(A_n(k)\)-modules are studied in chapters XII to XVIII. In particular, the preservation of holonomicity by inverse and direct images is proved and a proof of Kashiwara’s theorem about the structure of \(A_n(\mathbb{C})\)-modules with support on a hyperplane (or a linear subspace) of \(\mathbb{C}^n\) is given. Chapter XIX is centered on the study of global asymptotic stability of a system of ordinary differential equations on the plane and its relation with the Jacobian conjecture. The last chapter (=XX) contains some results (due to Zeilberger) about automatic proof of identities using basic results on holonomic modules.
Each chapter ends by a list of exercises completing the results given in the text. The book includes a “coda” and two appendices. The first appendix describes how to define an action of \(A_n(k)\) on a \(k\)-vector space from the action of generators \(x_i\), \(\partial_i\) (i.e. compatibility with the relations of the generators). In appendix 2, a self-contained proof of the (formal) local inversion theorem is given, which is used in chapter IV.
The “coda” contains some comments and references about how to deal with modules over the ring of linear differential operators on a general algebraic variety. The references given for further lectures on analytic \(\mathcal D\)-modules are incomplete. For example, neither the book of Z. Mebkhout [Systèmes différentiels. Le formalisme de six opérations de Grothendieck pour les \({\mathcal D}_x\)-modules cohérents (Hermann, Paris, 1989; Zbl 0686.14020)] nor the recent one of J. E. Björk [Analytic \({\mathcal D}\)-modules and applications (Kluwer Academic Publishers, 1993; Zbl 0805.32001)] is cited. No comments and no references are given about effective or explicit computations in the Weyl algebra, with the little exception of a part of the last chapter and the brief comment about the explicit calculation of the Bernstein polynomial, in chapter X. Related to that, one can add to the references given by the author the work of J. Briançon, M. Granger, P. Maisonobe and M. Miniconi [Ann. Inst. Fourier 39, No. 3, 553-610 (1989; Zbl 0675.32008)].
In the reviewer’s opinion this book offers a very good introduction to the theory of \(\mathcal D\)-modules and it can be very useful for under- and post-graduate students with interest in the subject.

MSC:

16S32 Rings of differential operators (associative algebraic aspects)
16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces
17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
32C38 Sheaves of differential operators and their modules, \(D\)-modules
16P40 Noetherian rings and modules (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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