Quadratic and Hermitian Forms Over Rings: (294 Grundlehren der Mathematischen Wissenschaften)

This book presents the theory of quadratic and hermitian forms over rings in a very general setting. It avoids any restriction on characteristics and takes full advantage of the functorial properties of the theory. Emphasis is placed on algebraic aspects of the theory. The text also defines descent theory, with its corresponding cohomological machinery. Clifford algebras are additionally applied, specifically to treat in detail the quadratic forms of low rank and their spinor groups. The text also covers the stability, cancellation and splitting theorems in unitary K-theory. These theorems are applied to polynomial rings, to give quadratic analogues of the theorem of Quillen and Suslin on projective modules, and to Witt groups of regular rings.

Table of 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 and Cancellation.- 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.