Quadratic and Hermitian Forms over Rings

ISBN-13: 9783642754036 / Angielski / Miękka / 2012 / 524 str.

Max-Albert Knus

Opis

This book presents 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 properties of the theory. It is not an encyclopedic survey. It stresses the algebraic aspects of the theory and avoids - within reason - overlapping with other books on quadratic forms (like those of Lam, Milnor-Husemoller and Scharlau). One important tool is descent theory with the corresponding cohomological machinery. It is used to define the classical invariants of quadratic forms, but also for the study of Azmaya algebras, which are fundamental in the theory of Clifford algebras. Clifford algebras are applied, in particular, to treat in detail quadratic forms of low rank and their spinor groups. Another important tool is algebraic K-theory, which plays the role that linear algebra plays in the case of forms over fields. The book contains complete proofs of the stability, cancellation and splitting theorems in the linear and in the unitary case. These results are applied to polynomial rings to give quadratic analogues of the theorem of Quillen and Suslin on projective modules. Another, more geometric, application is to Witt groups of regular rings and Witt groups of real curves and surfaces."

Spis treści

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 $$\mathbb^1_D$$.- §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.