Differential algebra and algebraic groups. (English) Zbl 0264.12102

Kolchin, E. R.

Pure and Applied Mathematics, 54. New York-London: Academic Press. xvii, 446 p. $ 24.50 (1973).

This book, published after years of careful preparation, is a tour de force of the highest proportions. The author, as is well known, is the leading authority in the field of differential algebra. There are few people working in this area who have not benefitted enormously through personal contact with him and none who have not been influenced by his publications. His goal here is to present a unified exposition of the subject, in an algebraic setting, presuming no more than a standard first year graduate course in algebra.
The chapter on algebraic preliminaries develops various notions for later use. Many of these, if not new, are set in a new form. A field extension \(L\) over \(K\) is quasi-separable if for every family of elements \((x_i)_{i\in I}\) in \(L\) that is separably in dependent over \(K\) (i.e. if \(f\in K[(X_i)_{i\in I}]\)annihilates \((x_i\) then so does every partial derivative \(\partial f/\partial X_i)\) there is a subset \(J\) of \(I\) such that \((x_i)_{i\in J}\) is a transcendence basis of \(K(x_i)_{i\in I}\) over \(K\) and \(I-J\) is finite. If \(R\) is a domain containing a subring \(R_0\), an ideal \(\mathfrak A\) of \(R\) is separable (resp. quasi-separable) if \(Q(f(R))\) is separable (resp. quasi-separable) over \(Q(f(R_0))\) where \(f: R\to R(\mathfrak A)\) is the canonical map and “\(Q\)” denotes quotient field. This notion is also extended to rings with zero divisors.
Another innovation is the concept of a “conservative system” to be used in the proof of the basis theorem. If \(R\) is a ring a set \(C\) of submodules of an \(R\)-module \(M\) is a conservative system if \(C\) is closed under arbitrary intersections and unions of chains (under inclusion). \(C\) is perfect if \(M=R\), every ideal in \(C\) is perfect (radical) and \(\mathfrak A:s\in C\) for all \(\mathfrak A\in C\), \(s\in R\). If every nonempty set of elements of \(C\) has a maximal member then \(C\) is Noetherian. In the case that \(C\) is Noetherian and perfect every ideal in \(C\) can be written uniquely as a finite intersection of prime ideals in \(C\) none of which contains any other.
There follows an interesting treatment of birational equivalence of ideals, specializations and dimension theory in polynomial algebras. The chapter also contains a lemma of Arnold Shapiro to be used later in the proof of the domination lemma. The latter is a major result one of whose applications deals with the distribution of singular zeros of a partial differential polynomial.
Chapter 1 introduces the basic concepts of differential algebra, that is, differential rings, ideals, fields, modules and algebras, as well as autoreduced sets and characteristic sets.
Chapter 2 is the theory of differential fields developed by Ritt for fields of functions, by Raudenbush and the author for abstract differential fields of characteristic zero, and by Seidenberg and the author for fields of positive characteristic. One finds for example the notions of differential inseparability base, and differential inseparability polynomial. This last is a numerical polynomial \(w\) attached to a finite extension \(\mathfrak F<\eta_1,\ldots, \eta_n>\) analogous to the Hilbert polynomial in algebraic geometry. The degree of \(w\) is less than or equal to the cardinality of \(\Delta\), the set of derivation operators, and for large \(s\) is \(w(s)\) the inseparability degree of \(\mathfrak F((\theta\eta_j)_{\text{ord }\theta\le s, 1\le j\le n})\) over \(\mathfrak F\). If the \(\eta_i\) are the residues of differential indeterminates \(y_i\) modulo a separable prime differential ideal \(P\) in \(\mathfrak F\{y_1,\ldots,y_n\}\), \(w=w_P\) is the differential dimension polynomial of \(P\) and \(P<Q\) implies \(w_P > w_Q\). Moreover for \(s > 0\), \(s\in\mathbb N\), \[ w_P = \dim (P \cap \mathfrak F [(\theta y_j)_{\text{ord }\theta\le s, 1\le j\le n}]). \] In Chapter 3 the author defines a differential conservative system \(C\) of a differential ring \(R\) to be a conservative system of differential ideals in \(R\). If \(\Sigma\subset R\), \(\{\Sigma\}\) denotes the smallest perfect differential ideal containing \(\Sigma\). Then he proves the basis theorem: Let \(R = R_0\{x_1,\ldots,x_n\}\) be an extension of differential rings and let \(C\) be a perfect differential conservative system of \(R\) with \(C\vert R_0\) Noetherian. If every prime \(C\)-ideal is quasi-separable then \(C\) is Noetherian. It follows that if \(R_0\) is a differential field and the \(x_i\) are differentially algebraically independent over \(R_0\) then a necessary and sufficient condition for \(R_0\) to be differentially quasi-perfect is that the set of perfect differential ideals in \(R\) be a Noetherian conservative system. If \(\mathfrak F\) is any differential field and \(\mathfrak A\) is an \(\mathfrak F\)-separable differential ideal in \(\mathfrak F\{y_1,\ldots,y_n\}\) (or \(\mathfrak A\) is prime and quasi-perfect over \(\mathfrak F)\) then \(\mathfrak A = \{\Phi\}\) for some finite set \(\Phi \subset\mathfrak A\). A differential field extension \(\mathfrak G\supset\mathfrak F\) is semiuniversal if every finitely generated separable extension of \(\mathfrak F\) can be embedded in \(\mathfrak G\). An extension \(\mathfrak U\) of \(\mathfrak F\) is universal if \(\mathfrak U\) is semiuniversal over every finitely generated extension of \(\mathfrak F\) in \(\mathfrak U\). Every differential field \(\mathfrak F\) has a universal extension.
The remainder of this chapter deals with differential specializations. The author generalizes results of Seidenberg and Rosenfeld on the possibility of extending such specializations and gives examples showing that the analogue for differential specializations of two well-known properties of algebraic specializations is false. A family \(\eta= (\eta_i)_{i\in I}\) of elements in an extension of a differential field \(\mathfrak F\) is constrained over \(\mathfrak F\) if \(\mathfrak F<\eta>\) is separable over \(\mathfrak F\) and there is an element \(B\) in the differential polynomial algebra \(\mathfrak F\{(y_i)_{i\in I}\}\) with \(B(\eta)\ne 0\) such that \(B(\eta') =0\) for every nongeneric differential specialization \(\eta'\) of \(\eta\) over \(\mathfrak F\). For-example if \(\eta\) is separably algebraic over \(\mathfrak F\), we can take \(B = 1\) or if \(\eta=e^x\) and \(\mathfrak F = \mathbb C(x)\), \(B=y\) will suffice.
Chapter 4, “Algebraic differential equations”, is divided into two parts. Part A deals with a universal differential field \(\mathfrak U\) (that is, \(\mathfrak U\) is universal over its prime field) of arbitrary characteristic. The differential Zariski topology on \(\mathfrak U^n\) is introduced: a set is closed if it is the set of zeros of a subset of \(\mathfrak F\{y_1,\ldots, y_n\}\). This makes \(\mathfrak U^n\) into a Noetherian space. If \(\mathfrak U\) is semiuniversal over \(\mathfrak F\) then every \(\mathfrak F\)-separable prime differential ideal \(P\) in \(\mathfrak F\{y_1,\ldots, y_n\} = R\) has a generic zero in \(\mathfrak U^n\). Moreover if \(\Sigma\subset R\) and \(B\) belongs to the \(\mathfrak F\)-separable differential ideal generated by \(\Sigma\), \(\{\Sigma\})/\mathfrak F\), then \(B\) vanishes at every \(\mathfrak F\)-separable zero of \(\Sigma\). Conversely if \(B\) vanishes at every \(\mathfrak F\)-constrained zero of \(\Sigma\) then \(B\in\{\Sigma\})/\mathfrak F\). Considering \(\mathfrak U^n\) as a vector space over \(K\), the differential field of constants of \(\mathfrak U\), any \(\ell\)-dimensional subspace is closed and corresponds to a differential ideal \(P\) in \(\mathfrak U\{y_1,\ldots, y_n\}\) generated by homogeneous linear elements \(L\). If \(\mathfrak U\{y_1,\ldots, y_n\}_1\) is the set of all such elements in \(\mathfrak U\{y_1,\ldots, y_n\}\) then \(P\cap \mathfrak U\{y_1,\ldots, y_n\}_1\). has codimension \(\ell\) in \(\mathfrak U\{y_1,\ldots, y_n\}_1\). Let \(A\in\mathfrak U\{y_1,\ldots, y_n\}\) and suppose \(A(\eta)=0\). If there is a ranking of \((y_1,\ldots, y_n)\)relative to which there is a derivative \(v\) of some \(y_j\) of highest rank such that \(\partial A/dv(\eta) \ne 0\) then \((\eta)\) is a nonsingular zero of \(A\). If \(A\) is irreducible over \(\mathfrak U\) the set of zeros of \(A\) has one irreducible component that contains all the nonsingular zeros of \(A\), namely the set of zeros of \(P_{\mathfrak U}(A)\), the general component of \(A\) in \(\mathfrak U\{y_1,\ldots,y_n\}\). The other irreducible components consist of singular zeros of \(A\). General components can be characterized in terms of their differential dimension polynomials.
In part B the characteristic is assumed to be zero in order to prove some of Ritt’s more profound results. There is a generalization of his so-called lowest degree theorem due to Hillman: Let \(\mathfrak F\) denote a differential subfield of \(\mathfrak U\), and let \(c\) be a transcendental constant in some extension of \(\mathfrak U<y_1,\ldots,y_n>\). For each power series \(P\) in \(\mathfrak F\{y_1,\ldots,y_n\}((c))\) write \(P= J_Pc^{v(P)} + \dots\) to indicate that \(J_P\) is the leading coefficient and \(v(P)\) is the order of \(P\). Let \(P,Q \in \mathfrak F\{y_1,\ldots,y_n\}((c)) - \{0\}\) such that \(P\in\{Q\}\). Then \(J_P\in\{J_Q\}\) in \(\mathfrak F\{y_1,\ldots,y_n\}\).
After proving Levi’s lemma the author establishes a generalization (the domination lemma) of a special case of Levi’s result. A short section on preparations is followed by Ritt’s famous component theorem: Let \(\mathfrak F\) be a differential subfield of \(\mathfrak U\) and let \(F\) be a nonzero differential polynomial in \(\mathfrak F\{y_1,\ldots,y_n\}\). If \(P\) is a component of \(\{F\}\) there is an irreducible \(B\) in \(\mathfrak F\{y_1,\ldots,y_n\}\) such that \(P= \mathcal P_{\mathfrak F}(B)\).
The low power theorem of Ritt gives a necessary and sufficient condition that the general component of an irreducible differential polynomial be a component of a given nonzero differential polynomial. Kolchin’s version is stronger than Ritt’s in one direction since he uses the domination lemma instead of Levi’s lemma.
Ritt also posed the following problem: if \(A\) is an irreducible element in \(\mathfrak F\{y_1,\ldots,y_n\}\) vanishing at \((0,\ldots,0)\) what are necessary and sufficient conditions for \((0,\ldots,0)\) to be a zero of \(\mathcal P_{\mathfrak F}(A)\)? If the order of \(A\) is 1 or 2 the solution is known but not in general (even if \(n = m = 1)\). In this book the author presents the strongest results to date. There is a section on systems of bounded order which gives a precise formulation and generalization of some work of Jacobi.
The chapter on algebraic groups presents an axiomatic treatment of the subject including a development of homogeneous spaces. An advantage of this approach is to make the Galois group of a differential field extension into an algebraic group and not merely isomorphic to one. There is an elegant treatment of Galois cohomology and Lie algebras with illustrative examples. The use of examples throughout the book is very effective. A section on logarithmic derivatives has found application in Kovačić’s work on the inverse problem in differential Galois theory and Cassidy’s on differential algebraic groups.
Chapter 6 is on the author’s Galois theory of differential fields. The idea is to make certain automorphisms of a differential field into an algebraic group. Accordingly the first part examines specializations of isomorphisms. Throughout this chapter the characteristic of \(\mathfrak U\) is assumed to be zero. An isomorphism \(\sigma\) of a differential field \(\mathfrak G\) is strong if \(\sigma\) fixes \(K\), the field of constants of \(\mathfrak G\) and \(\mathfrak G\cdot K= \sigma\mathfrak G\cdot K\). Then any specialization of \(\sigma\) is strong and the set of these \(\sigma\) can be identified with the group of automorphism of \(\mathfrak G\cdot K\) over \(K\). A finitely generated extension \(\mathfrak G\) of \(\mathfrak F\) is strongly normal if every isomorphism of \(\mathfrak G\) over \(\mathfrak F\) is strong and these form the Galois group, \(G(\mathfrak G/\mathfrak F)\) of \(\mathfrak G\) over \(\mathfrak F\).
An intermediate differential field \(\mathfrak F_1\) is strongly normal over \(\mathfrak F\) if and only if given \(\alpha\in\mathfrak F_1 - \mathfrak F\) there is a \(\sigma\) in \(G(\mathfrak F_1/\mathfrak F)\) such that \(\sigma\alpha\ne \alpha\). This is equivalent to \(G(\mathfrak G/\mathfrak F_1)\) being a normal subgroup of \(G(\mathfrak G/\mathfrak F)\). Linear groups correspond to extensions of the form \(\mathfrak F<\eta_1,\ldots,\eta_n> = \mathfrak G\) and \(\mathfrak F\) have the same constants and the \(n\) elements \(\eta_j = (\eta_{j1}, \ldots,\eta_{jr})\in\mathfrak B^r\) form a fundamental system of zeros of a linear differential ideal \(\mathfrak A\) of \(\mathfrak F\{y_1,\ldots,y_n\}\) of linear dimension \(n\). If the field of constants of \(\mathfrak F\) is not algebraically closed \(\mathfrak A\) need not have such a fundamental system of zeros. This theory has its roots in the work of Picard and Vessiot and was brought to fruition by the author.
The nonlinear case is described in detail in terms of his work on principal homogeneous spaces.
Despite the length of this review, many important results have been omitted and the reader is urged to consult the book for further details.

Reviewer: Peter Blum

MSC:

12H05	Differential algebra
14L10	Group varieties
12-02	Research exposition (monographs, survey articles) pertaining to field theory