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Full Description
From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new results are obtained, which went unnoticed even for fields, as in the case of involutions on 16-dimensional central simple algebras. The first chapter gives an introduction to the basic definitions and properties of hermitian forms which are used throughout the book.
Contents
I. Hermitian Forms over Rings.- §1. Rings with Involution.- §2. Sesquilinear and Hermitian Forms.- §3. Hermitian Modules.- §4. Symplectic Spaces.- §5. Unitary Rings and Modules.- §6. Hermitian Spaces over Division Rings.- §7. Change of Rings.- §8. Products of Hermitian Forms.- §9. Morita Theory for Hermitian Modules.- §10. Witt Groups.- §11. Cartesian Diagrams and Patching of Hermitian Forms.- II. Forms in Categories.- §1. Additive Categories.- §2. Categories with Duality.- §3. Transfer.- §4. Reduction.- §5. The Theorem of Krull-Schmidt for Additive Categories.- §6. The Krull-Schmidt Theorem for Hermitian Spaces.- §7. Some Applications.- III. Descent Theory and Cohomology.- §1. Descent of Elements.- §2. Descent of Modules and Algebras.- §3. Discriminant Modules.- §4. Quadratic Algebras.- §5. Azumaya Algebras.- §6. Graded Algebras and Modules.- §7. Universal Norms.- §8. Involutions on Azumaya Algebras.- §9. The Pfaffian.- IV. The Clifford Algebra.- §1. Construction of the Clifford Algebra.- §2. Structure of the Clifford Algebra, the Even Rank Case.- §3. Structure of the Clifford Algebra, the Odd Rank Case.- §4. The Discriminant and the Arf Invariant.- §5. The Special Orthogonal Group.- §6. The Spinors.- §7. Canonical Isomorphisms.- §8. Invariants of Quadratic Spaces.- §9. Quadratic Spaces with Trivial Arf Invariant.- V. Forms of Low Rank.- §1. Quadratic Modules of Rank 1.- §2. Quadratic Modules of Rank 2.- §3. Quadratic Modules of Rank 3.- §4. Quadratic Modules of Rank 4.- §5. Quadratic Spaces of Rank 5 and 6.- §6. Hermitian Modules of Low Rank.- §7. Composition of Quadratic Spaces.- VI. Splitting and Cancellation Theorems.- §1. Semilocal Rings, the Stable Range.- §2. The f-Rank.- §3. Serre's Splitting Theorem andCancellation.- §4. Unitary Groups.- §5. Cancellation for Unitary Spaces over Semilocal Rings.- §6. Cancellation and Stability for Unitary Spaces.- §7. A Splitting Theorem.- VII. Polynomial Rings.- §1. Principal Ideal Domains.- §2. Polynomial Rings.- §3. Bundles over $$\mathbb{P}^1_D$$.- §4. The Theorem of Karoubi.- §5. Quillen's Theorem.- §6. A Rigidity Theorem and the Horrocks Theorem.- §7. Isotropic Hermitian Spaces.- §8. Projective Modules over Polynomial Rings.- §9. Hermitian Spaces of Low Rank.- §10. Indecomposable Anisotropic Spaces.- §11. Hermitian Modules over Projective Spaces.- VIII. Witt Groups of Affine Rings.- §1. Witt Group of Schemes.- §2. Domains of Dimension ?3.- §3. Regular Local Rings Essentially of Finite Type.- §4. Real Smooth Surfaces.- §5. Real Curves.- §6. Examples.- §7. Symplectic Bundles over Affine Surfaces.
