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AuthorKnus, Max-Albert. author
TitleQuadratic and Hermitian Forms over Rings [electronic resource] / by Max-Albert Knus
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1991
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Descript XI, 524 p. online resource


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


I. Hermitian Forms over Rings -- ยง1. Rings with Involution -- ยง2. Sesquilinear and Hermitian Forms -- ยง3. Hermitian Modules -- ยง4. Symplectic Spaces -- ยง5. Unitary Rings and Modules -- ยง6. Hermitian Spaces over Division Rings -- ยง7. Change of Rings -- ยง8. Products of Hermitian Forms -- ยง9. Morita Theory for Hermitian Modules -- ยง10. Witt Groups -- ยง11. Cartesian Diagrams and Patching of Hermitian Forms -- II. Forms in Categories -- ยง1. Additive Categories -- ยง2. Categories with Duality -- ยง3. Transfer -- ยง4. Reduction -- ยง5. The Theorem of Krull-Schmidt for Additive Categories -- ยง6. The Krull-Schmidt Theorem for Hermitian Spaces -- ยง7. Some Applications -- III. Descent Theory and Cohomology -- ยง1. Descent of Elements -- ยง2. Descent of Modules and Algebras -- ยง3. Discriminant Modules -- ยง4. Quadratic Algebras -- ยง5. Azumaya Algebras -- ยง6. Graded Algebras and Modules -- ยง7. Universal Norms -- ยง8. Involutions on Azumaya Algebras -- ยง9. The Pfaffian -- IV. The Clifford Algebra -- ยง1. Construction of the Clifford Algebra -- ยง2. Structure of the Clifford Algebra, the Even Rank Case -- ยง3. Structure of the Clifford Algebra, the Odd Rank Case -- ยง4. The Discriminant and the Arf Invariant -- ยง5. The Special Orthogonal Group -- ยง6. The Spinors -- ยง7. Canonical Isomorphisms -- ยง8. Invariants of Quadratic Spaces -- ยง9. Quadratic Spaces with Trivial Arf Invariant -- V. Forms of Low Rank -- ยง1. Quadratic Modules of Rank 1 -- ยง2. Quadratic Modules of Rank 2 -- ยง3. Quadratic Modules of Rank 3 -- ยง4. Quadratic Modules of Rank 4 -- ยง5. Quadratic Spaces of Rank 5 and 6 -- ยง6. Hermitian Modules of Low Rank -- ยง7. Composition of Quadratic Spaces -- VI. Splitting and Cancellation Theorems -- ยง1. Semilocal Rings, the Stable Range -- ยง2. The f-Rank -- ยง3. Serreโ{128}{153}s Splitting Theorem and Cancellation -- ยง4. Unitary Groups -- ยง5. Cancellation for Unitary Spaces over Semilocal Rings -- ยง6. Cancellation and Stability for Unitary Spaces -- ยง7. A Splitting Theorem -- VII. Polynomial Rings -- ยง1. Principal Ideal Domains -- ยง2. Polynomial Rings -- ยง3. Bundles over $$\mathbb{P}̂1_D$$ -- ยง4. The Theorem of Karoubi -- ยง5. Quillenโ{128}{153}s Theorem -- ยง6. A Rigidity Theorem and the Horrocks Theorem -- ยง7. Isotropic Hermitian Spaces -- ยง8. Projective Modules over Polynomial Rings -- ยง9. Hermitian Spaces of Low Rank -- ยง10. Indecomposable Anisotropic Spaces -- ยง11. Hermitian Modules over Projective Spaces -- VIII. Witt Groups of Affine Rings -- ยง1. Witt Group of Schemes -- ยง2. Domains of Dimension ?3 -- ยง3. Regular Local Rings Essentially of Finite Type -- ยง4. Real Smooth Surfaces -- ยง5. Real Curves -- ยง6. Examples -- ยง7. Symplectic Bundles over Affine Surfaces

Mathematics Algebraic geometry Number theory Mathematics Number Theory Algebraic Geometry


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