Quadratic algebras, Clifford algebras, and arithmetic Witt groups
``The goal of this volume is to introduce the reader in an elemental and accessible way to the large and dynamic area of algebras and forms over commutative rings. Quadratic algebras and their analysis give the volume its direction. Indeed, its defining moment is the fact that quadratic algebras lie at the heart of the theory of quadratic forms and Clifford algebras over commutative rings.'' The book is written in the style of lecture notes. Up to Chapter 12 it is nearly self-contained with full proofs. It can serve very well as an introductory text to more standard but also more abstract books like \textit{M.-A. Knus} [Quadratic and Hermitian forms over rings, Grundlehren Math. Wiss. 294 (1991; Zbl 0756.11008)]. One of the positive, if not the best, features of Hahn's book is the large number of exercises of varying degree of difficulty, and the hints for solving them. In the following I mention the headlines and main topics treated in the fifteen chapters of the book, occasionally with some additionally explanation. Ch. 1 studies free quadratic algebras \(R[X]/(X^ 2 - aX - b)\), their classification, involutions and gradings \(\pmod 2\). Ch. 2 is on separable \(R\)-algebras \(A\). \(A\) is called separable if the natural \(A^ e\)-module sequence \[ 0 \to \text{ker }\phi \to A^ e @>\phi>> A \to 0 \] splits. Here \(A^ e = A \otimes A^{\text{opp}}\) is the enveloping algebra of \(A\), \(\phi\) is given by \(\phi(a \otimes b^ 0) = ab\). This concept generalizes the usual separability of a field extension. The above free quadratic algebra is separable iff \(a^ 2 + 4b \in R^*\). Ch. 3 investigates the group structure of the set of isomorphism classes \([a,b]\) of separable free quadratic \(R\)-algebras, their discriminants and a related graded version. Ch. 4 introduces some standard facts on bilinear and quadratic forms over \(R\) via localization. An important topic is the group \(\text{Dis}(R)\) of ``discriminant modules'' \((P,f)\). Here \(P\) is a finitely generated projective \(R\)-module of rank 1, \(f: P \otimes P \to R\) is an isomorphism. Ch. 5 is a review on the Clifford algebras (and exterior algebras) of a quadratic module \(M\) over \(R\), mostly without proofs. Ch. 6 studies algebras with ``standard involution'' \(\sigma\) such that the reduced norm satisfies \(\text{nr}(a) = aa^ \sigma \in R\) for every \(a \in A\). Under fairly weak additional assumptions \(A\) is then either a quadratic algebra or almost a quaternion algebra. Ch. 7 opens the central part of the book with an investigation of the ``Arf algebra'' \(A(M) = \text{Cen}_{C(M)} C_ 0(M)\) of a quadratic module \(M\). In many cases \(A(M)\) is a free graded quadratic algebra \(R[X] / (X^ 2 - aX - b)\). A ``special element'' \(z \in C(M)\) is by definition an element such that \(\{1,z\}\) forms a basis of \(A(M)\) and \(z^ \alpha = a-z\) (where \(\alpha\) is the standard involution on \(C(M)\)). Ch. 8 handles some consequences of the existence of special elements, in particular the connection between the full Clifford algebra \(C(M)\) and its even part \(C_ 0(M)\). It also introduces ``Clifford modules'' and representations of Clifford algebras. Ch. 9 proves the main results on Clifford and Arf algebras: Let \(M\) be a finitely generated projective nonsingular and faithful quadratic module over an arbitrary commutative ring \(R\). Then \(C(M)\), \(C_ 0(M)\) and \(A(M)\) are separable \(R\)-algebras, and we have: \[ \text{Cen }C(M) = R \Leftrightarrow M \text{ has even rank } \Leftrightarrow \text{Cen }C_ 0(M) = A(M) \] \[ \text{Cen }C_ 0(M) = R \Leftrightarrow M\text{ has odd rank }\Leftrightarrow \text{Cen }C(M) = A(M). \] These results are due to De Meyer-Ingraham and Knus-Ojanguren. Some auxiliary results on Azumaya (= separable and central) algebras are only cited. Ch. 10 completes the above results by showing that special elements of \(C(M)\) exist whenever \(M\) is free. Ch. 11 treats the more classical matrix theory of Clifford algebras if \(R = F\) is a field. For instance \(C(M) \cong M_ k(D)\), where \(k \in \mathbb{N}\) and \(D\) is a central division algebra over \(F\), if \(\dim M\) is even. This chapter also contains the periodicity properties of Clifford algebras over \(\mathbb{R}\) and \(\mathbb{C}\) and collects some additional properties of \(C(M)\) if \(F\) is a local or global field. Ch. 12 returns to a commutative ring, constructs and investigates the group structure on projective separable quadratic algebras thereby generalizing the results for the free case obtained in Ch. 3. The Picard group \(\text{Pic}(R)\) and the group \(\text{Dis}(R)\) (see Ch. 4) play a considerable role here. For an integral domain \(R\) we have \(\text{Pic}(R) \cong \text{Cl}(R)\), the ideal class group of \(R\). The last three chapters are intended to give some applications of the earlier material. They are much more advanced and contain almost no proofs, the last chapter is very sketchy indeed. Ch. 13 collects material on Brauer, Brauer-Wall and Witt groups over rings. Ch. 14 studies the Witt group \(W_ q(R)\) of regular quadratic spaces over a so-called ``arithmetic Dedekind ring'' \(R\). The quotient field of \(R\) is a global field \(F\) which implies that \(\text{Cl}(R)\) is finite. If in addition every element of \(R\) is integral with respect to almost all valuations of \(F\) then \(R\) is called ``Hasse domain''. There are many commutative diagrams and exact sequences connecting \(W_ q(R)\) to other more accessible groups. In some typical cases the rank and the order of the torsion part of \(W_ q(R)\) can be computed. Ch. 15 concerns the applications of Clifford modules in topology. It mentions Radon-Hurwitz numbers, vector fields on spheres, the theorem of Adams, topological \(K\)-theory, Lie algebras and spin groups for the real Clifford algebra \(\text{Cl}_ n\). Dirac operators, a theorem of Stolz saying that every simply connected spin manifold of dimension \(\geq 5\) admits a metric with positive scalar curvature if and only if its ``Clifford index'' \(\alpha(X)\) vanishes, and results of Thorbergsson on isoparametric hypersurfaces in spheres, all this on less than 20 pages! As you can see from the above description the book is not very homogeneous. Some new results which cannot be found in other books are scattered over Chapters 6, 8, 10, 12 and 14. Nevertheless, the book is a welcome addition to the (scarce) literature in this area.
scientific article; zbMATH DE number 493075