##
**Kähler differentials.**
*(English)*
Zbl 0587.13014

Advanced Lectures in Mathematics. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. VII, 402 p. DM 76.00 (1986).

The book is based on lectures of the author for advanced students, explaining the role of Kähler differentials in commutative algebra and algebraic geometry. It starts from the very beginning, introducing the notion of derivations and differential algebras, and develops extensively the whole formalism of universal extensions, universal derivations, and universal differential algebras (de Rham algebras). Constructions of universal extensions are given in the most important cases: colimits, limits, polynomial algebras, residue class algebras, algebras of fractions. After these solid foundations the main part of the book (§ 5-17) is dedicated to applications in commutative algebra: In § 5 differential modules of (mostly finitely generated) field extensions are studied. Apart from the usual facts about separability, transcendence-, and p-bases the interesting notions of an ”admissible derivation” and an ”admissible subfield” are introduced (definition 5.19): Let L/K be a finitely generated field extension, \(\delta:\quad K\to K\delta K\) a derivation of K into a K-module. \(\delta\) is called ”admissible for L/K” if \(\dim_ KK\delta K<\infty\) and \(\dim_ LT(L/\delta)=p-\deg (L/K)- Tr\deg (L/K),\) where T(L/\(\delta)\) is the kernel of the canonical homomorphism \(L\otimes_ KK\delta K\to \Omega^ 1_{L/\delta}\), and \(\Omega^ 1_{L/\delta}\) the differential module of the universal L- extension of \(\delta\). (By Cartier’s inequality the \(\leq\) sign always holds). A subfield \(K_ 0\) of K is called ”admissible for L/K”, if \(K^ p\subseteq K_ 0\), \([K:K_ 0]<\infty\), and the universal derivation of \(K/K_ 0\) is admissible for L/K. These notions allow the formulations of many results without separability assumptions. An especially useful generalization of his notion to algebras of essentially finite type is given in § 6. Here the author first derives the well known exact sequences for the differential module of a local ring which allow to compute the minimal number \(\mu\) of generators of \(\Omega^ 1\) by means of the embedding dimension and the p-degree of the residue field extension in the absolute and relative case. He discusses the application to the notion of ramification and proves a structure theorem for unramified extensions. If S/R is an algebra essentially of finite type a derivation \(\delta:\quad R\to R\delta R\) is called ”admissible for \({\mathfrak P}\in Spec(S)''\) if \(R_{{\mathfrak p}}\delta R_{{\mathfrak p}}\) (\({\mathfrak p}:={\mathfrak P}\cap R)\) is a free \(R_{{\mathfrak p}}\)-module of finite rank and \(\mu_{{\mathfrak P}}(\Omega^ 1_{S/R})=e\dim (S_{{\mathfrak P}}/{\mathfrak p}\cdot S_{{\mathfrak p}})+Tr\deg (k({\mathfrak P})/k({\mathfrak p}))+rank(R\delta R).\) A differential algebra (\(\Omega\),d) of R is called ”admissible for \({\mathfrak P}''\) if \(\Omega_{{\mathfrak p}}\) is an exterior differential algebra and d an admissible derivation for \({\mathfrak P}\). Such admissible derivations exist in many cases. Finally, if A is an affine K-algebra, a subfield \(K_ 0\subset K\) is called ”admissible for A”, if \(K^ p\subseteq K_ 0\), \([K:K_ 0]<\infty\) and the universal differential algebra \(\Omega_{K/K_ 0}\) is admissible for all \({\mathfrak P}\in Spec(A)\). Admissible subfields are an improvement of what the author called ”differential constant field” in former papers. They play a decisive role for instance in all differential regularity criteria (Jacobi criteria) in prime characteristic p. (If \(p=0\), one can take \(K_ 0=K.)\) Such criteria for regularity and geometric regularity are the main subjects of § 7. In § 8 the author applies these criteria to the case of an algebra S/R that is smooth. Smoothness criteria in terms of the differential module are derived as well as results on the smooth locus of S/R in various situations. In § 9 local and global complete intersections are characterized by the differential module of certain admissible derivations. § 10 introduces the Kähler differents of S/R with respect to an admissible derivations of R and shows how the criteria of the preceding sections for regularity, smoothness, ramification, and complete intersection can be expressed in terms of these differents. Relations to the differents of Dedekind and Noether are also considered. In preparation of the treatment of completions and analytic algebras the notion of a universally finite differential algebra is introduced and discussed in § 11. In § 12 the behaviour of differential algebras under I-adic completions is studied. Then the existence of the universally finite differential algebras \({\tilde \Omega}\) for certain complete algebras is established. In § 13-14 many results of § 5-7 are generalized to the case of analytic or semianalytic k-algebras R over a complete noetherian local ring k, the maximal ideal of which is generated by the characteristic p of the residue field of k. Here R/k is called an analytic (semianalytic) k-algebra, if there is a power series ring P and a k-homomorphism \(P\to R\) such that R/P is finite (essentially of finite type). Every reduced semianalytic k-algebra S contains a unique maximal analytic subalgebra A. The universal S-extension of the universally finite differential module of A/k turns out to be the appropriate differential module for studying semianalytic algebras. If S is a field then \(Q(A)=k((X_ 1,...,X_ d))\) is a power series field and S/k is called an ”analytic (semianalytic) field extension”. For analytic field extensions the number d plays the same role as the transcendence degree does in the case of finitely generated field extensions. Therefore d is called the ”analytic transcendence degree” aTrdeg(S/k). In the case of semianalytic field extensions \(aTr\deg (S/k)+Tr\deg (S/Q(A))\) seems to be the appropriate substitute for the transcendence degree. The notion of admissible subfields is also generalized to the case of semianalytic field extensions. They play an essential role in the generalizations of the criteria for regularity and geometric regularity to the (semi-)analytic situation. In § 15 the case of a Frobenius sandwich \(S^ p\subset R\subset S\) (S containing a field of characteristic p\(>0)\) is considered with respect to the question when there are locally p-bases of S/R. There is a close connection between p-bases and differentially simple rings which leads to a criterion by Yuan for the existence of p-bases (theorem 15.5). From this a proof after Matsumura is given for a theorem by Kimura-Niitsuma stating that for regular S the existence of p-bases locally is equivalent to \(\Omega^ 1_{S/R}\) being projective and also to R being regular. There are applications to inseparable Galois theory.

The two final sections deal with the definition of traces (§ 16) and residues (§ 17) of differential forms: If S/R is a finite locally free algebra, \(\Omega\) a differential algebra of R and \(\Omega_ S\) the universal S-extension of \(\Omega\), is there a ”trace mapping” from \(\Omega_ S\) to \(\Omega\) ? The author first formulates 7 ”trace axioms” which should be satisfied by such mappings. He then constructs a trace for the case that R is noetherian and S/R is locally a complete intersection. This trace is uniquely determined by the first four trace axioms (linearity and compatibility with the canonical trace of S/R, base change, and direct products). As he points out and shows in an example in the exercises it is not possible to construct traces without restrictive conditions on S/R. Other cases in which traces have been constructed by different methods are mentioned at the end of the section. In § 17 the trace is used to define residues for differential forms in an algebraic function field L/K of one variable at each valuation ring R of L over K. First a residue map\(Res_ R:\quad \Omega^ 1_{L/K}\to K\) is defined, and the residue theorem is proved. But this is not sufficient in the inseparable case to prove the duality theorem. Here one must take an admissible subfield \(K_ 0\) instead of K as differential constant field and consider forms \(\omega \in \Omega^{r+1}_{L/K_ 0}\) with \(r:=\dim_ K\Omega^ 1_{K/K_ 0}.\) The residues for such forms are defined using the previously defined residues for \(\Omega^ 1_{L/K}\). Finally Serre’s duality theorems is proved for these differential forms. Connections with the Poincaré-residue and the Cartier-operator are given in the exercises.

Last but not least there are 7 appendices which take about one fourth of the book. They contain interesting material from commutative algebra for the convenience of the reader. Especially in appendices C and F on complete intersections and traces many results are presented that are difficult to find elsewhere. The appendices are independent of the rest of the book and constitute a useful supplement to the books of Matsumura and Bourbaki on commutative algebra that are used as a general reference.

The book is an excellent introduction into the theory of Kähler differentials and their application in commutative algebra. It leads from the fundamentals to current open problems. Much of the author’s own research work has entered into it. Of course some field could not be covered in an introductory book, like regular differentials residues in higher dimensions and duality theory. Many exercices that give additional material are another valuable asset. I recommend this book to all mathematicians working in commutative algebra as well as to advanced students.

The two final sections deal with the definition of traces (§ 16) and residues (§ 17) of differential forms: If S/R is a finite locally free algebra, \(\Omega\) a differential algebra of R and \(\Omega_ S\) the universal S-extension of \(\Omega\), is there a ”trace mapping” from \(\Omega_ S\) to \(\Omega\) ? The author first formulates 7 ”trace axioms” which should be satisfied by such mappings. He then constructs a trace for the case that R is noetherian and S/R is locally a complete intersection. This trace is uniquely determined by the first four trace axioms (linearity and compatibility with the canonical trace of S/R, base change, and direct products). As he points out and shows in an example in the exercises it is not possible to construct traces without restrictive conditions on S/R. Other cases in which traces have been constructed by different methods are mentioned at the end of the section. In § 17 the trace is used to define residues for differential forms in an algebraic function field L/K of one variable at each valuation ring R of L over K. First a residue map\(Res_ R:\quad \Omega^ 1_{L/K}\to K\) is defined, and the residue theorem is proved. But this is not sufficient in the inseparable case to prove the duality theorem. Here one must take an admissible subfield \(K_ 0\) instead of K as differential constant field and consider forms \(\omega \in \Omega^{r+1}_{L/K_ 0}\) with \(r:=\dim_ K\Omega^ 1_{K/K_ 0}.\) The residues for such forms are defined using the previously defined residues for \(\Omega^ 1_{L/K}\). Finally Serre’s duality theorems is proved for these differential forms. Connections with the Poincaré-residue and the Cartier-operator are given in the exercises.

Last but not least there are 7 appendices which take about one fourth of the book. They contain interesting material from commutative algebra for the convenience of the reader. Especially in appendices C and F on complete intersections and traces many results are presented that are difficult to find elsewhere. The appendices are independent of the rest of the book and constitute a useful supplement to the books of Matsumura and Bourbaki on commutative algebra that are used as a general reference.

The book is an excellent introduction into the theory of Kähler differentials and their application in commutative algebra. It leads from the fundamentals to current open problems. Much of the author’s own research work has entered into it. Of course some field could not be covered in an introductory book, like regular differentials residues in higher dimensions and duality theory. Many exercices that give additional material are another valuable asset. I recommend this book to all mathematicians working in commutative algebra as well as to advanced students.

Reviewer: R.Berger

### MSC:

13N05 | Modules of differentials |

12H05 | Differential algebra |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13B10 | Morphisms of commutative rings |

13H05 | Regular local rings |