Quadratic and Hermitian Forms over Rings

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.