Descent in Buildings (AM-190)
Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving nece...
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| Format: | eBook Book |
| Language: | English |
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Princeton University Press
2015
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| Edition: | 1 |
| Series: | Annals of Mathematics Studies |
| Subjects: | |
| ISBN: | 9781400874019, 1400874017, 9780691166902, 0691166900, 0691166919, 9780691166919 |
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| Abstract | Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings. |
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| AbstractList | Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings. Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings. No detailed description available for "Descent in Buildings (AM-190)". |
| Author | Weiss, Richard M Petersson, Holger P Mühlherr, Bernhard |
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| Copyright | 2016 Princeton University Press |
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| DOI | 10.1515/9781400874019 |
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| Editor | Weiss, Richard M Petersson, Holger P |
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| Keywords | quaternion division algebra discrete valuation anti-isomorphism abelian group building isomorphism separable quadratic extension trace map biquaternion division algebra pseudo-split building wild quadratic space split quadratic space ramified quadrangle relative Coxeter diagram unramified separable quadratic extension arctic region weak isomorphism scalar multiplication compatible representation vector space absolute Coxeter system Coxeter group Structure Theorem pseudo-quadratic space Moufang polygon exceptional quadrangle automorphism Moufang quadrangle affine building Coxeter diagram gem quadratic module semi-ramified quadrangle Clifford invariant tamely ramified division algebra involutory set polar space descent group Moufang building descent relative Coxeter group fixed point theory isotropic quadratic space generalized quadrangle proper involutory set absolute Coxeter diagram Moufang structure projection map Fundamental Theorem of Descent vertex non-abelian group round quadratic space quadratic space Coxeter system anisotropic quadratic space residual quadratic spaces relative rank unramified quadratic space unramified quadrangle chamber special vertex absolute rank Moufang set quadratic form standard involution Euclidean plane bilinear form algebraic group subbuilding of split type Moufang condition Tits index hyperbolic quadratic space anisotropic pseudo-quadratic space ramified separable quadratic extension fixed point building hyperbolic plane simplicial complex subbuilding root group sequence trace relative Coxeter system unramified quaternion division algebra thick building root spherical building Bruhat-Tits building parallel residues proper indifferent set ramified quaternion division algebra residue exceptional Moufang quadrangle hyperbolic quadratic module length function Pfister form canonical isomorphism thin T-building apartment finite dimension Hyperbolic geometry Composition algebra Bijection Subset Biquaternion Quaternion algebra Separable extension Subgroup Substructure Octonion Non-abelian Vector space Mathematical induction Simplicial complex Dynkin diagram Power set Addition Cardinality Algebraic group Permutation Homomorphism Affine transformation Hyperplane Existential quantification Dimension (vector space) Coset Convex hull Subring Special case Root system Summation Unit sphere Projective space Algebraic structure Equivalence class Algebraically closed field Metric space Quaternion Octonion algebra Residue field Set (mathematics) Embedding Diagram (category theory) Division algebra Linear subspace Linear map Linear combination Splitting field Empty set Theorem Euclidean space Quadratic form Galois group Automorphism Field extension Linear space (geometry) Discrete valuation Affine space Half-space (geometry) Additive group Purely inseparable extension Local field Finite set Module (mathematics) Scientific notation Surjective function Algebraic geometry |
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| Notes | Includes bibliographical references (p. [327]-331) and index |
| OCLC | 939554323 |
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| Snippet | Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their... No detailed description available for "Descent in Buildings (AM-190)". |
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| SubjectTerms | Addition Additive group Affine space Affine transformation Algebra & number theory Algebraic Algebraic geometry Algebraic group Algebraic structure Algebraically closed field Automorphism Bijection Biquaternion Buildings (Group theory) Cardinality Combinatorial geometry Combinatorics Composition algebra Convex hull Coset Coxeter group Diagram (category theory) Dimension (vector space) Discrete valuation Division algebra Dynkin diagram Embedding Empty set Equivalence class Euclidean space Existential quantification Field extension Finite set Galois group General Topics for Engineers Geometry Half-space (geometry) Homomorphism Hyperbolic geometry Hyperplane Linear combination Linear map Linear space (geometry) Linear subspace Local field Mathematical induction MATHEMATICS MATHEMATICS / General MATHEMATICS / Geometry / General Metric space Module (mathematics) Moufang polygon Moufang set Non-abelian Octonion Octonion algebra Permutation Power set Projective space Purely inseparable extension Quadratic form Quaternion Quaternion algebra Residue field Root system Scientific notation Separable extension Set (mathematics) Simplicial complex Special case Splitting field Subgroup Subring Subset Substructure Summation Surjective function Theorem Unit sphere Vector space |
| SubjectTermsDisplay | Algebraic Combinatorics Geometry Mathematics |
| TableOfContents | Descent in buildings (Annals of mathematics studies; 190) -- Contents -- Preface -- Part 1: Moufang Quadrangles -- Chapter One: Buildings -- Chapter Two: Quadratic Forms -- Chapter Three: Moufang Polygons -- Chapter Four: Moufang Quadrangles -- Chapter Five: Linked Tori, I -- Chapter Six: Linked Tori, II -- Chapter Seven: Quadratic Forms over a Local Field -- Chapter Eight: Quadratic Forms of Type E6, E7 and E8 -- Chapter Nine: Quadratic Forms of Type F4 -- Part 2: Residues in Bruhat-Tits Buildings -- Chapter Ten: Residues -- Chapter Eleven: Unramified Quadrangles of Type E6, E7 and E8 -- Chapter Twelve: Semi-ramified Quadrangles of Type E6, E7 and E8 -- Chapter Thirteen: Ramified Quadrangles of Type E6, E7 and E8 -- Chapter Fourteen: Quadrangles of Type E6, E7 and E8: Summary -- Chapter Fifteen: Totally Wild Quadratic Forms of Type E7 -- Chapter Sixteen: Existence -- Chapter Seventeen: Quadrangles of Type F4 -- Chapter Eighteen: The Other Bruhat-Tits Buildings -- Part 3: Descent -- Chapter Nineteen: Coxeter Groups -- Chapter Twenty: Tits Indices -- Chapter Twenty One: Parallel Residues -- Chapter Twenty Two: Fixed Point Buildings -- Chapter Twenty Three: Subbuildings -- Chapter Twenty Four: Moufang Structures -- Chapter Twenty Five: Fixed Apartments -- Chapter Twenty Six: The Standard Metric -- Chapter Twenty Seven: Affine Fixed Point Buildings -- Part 4: Galois Involutions -- Chapter Twenty Eight: Pseudo-Split Buildings -- Chapter Twenty Nine: Linear Automorphisms -- Chapter Thirty: Strictly Semi-linear Automorphisms -- Chapter Thirty One: Galois Involutions -- Chapter Thirty Two: Unramified Galois Involutions -- Part 5: Exceptional Tits Indices -- Chapter Thirty Three: Residually Pseudo-Split Buildings -- Chapter Thirty Four: Forms of Residually Pseudo-Split Buildings -- Chapter Thirty Five: Orthogonal Buildings -- Chapter Thirty Six: Indices for the Exceptional Bruhat-Tits Buildings -- Bibliography -- Index. Cover -- Title -- Copyright -- Dedication -- Contents -- Preface -- PART 1. MOUFANG QUADRANGLES -- Chapter 1. Buildings -- Chapter 2. Quadratic Forms -- Chapter 3. Moufang Polygons -- Chapter 4. Moufang Quadrangles -- Chapter 5. Linked Tori, I -- Chapter 6. Linked Tori, II -- Chapter 7. Quadratic Forms over a Local Field -- Chapter 8. Quadratic Forms of Type E6, E7 and E8 -- Chapter 9. Quadratic Forms of Type F4 -- PART 2. RESIDUES IN BRUHAT-TITS BUILDINGS -- Chapter 10. Residues -- Chapter 11. Unramified Quadrangles of Type E6, E7 and E8 -- Chapter 12. Semi-ramified Quadrangles of Type E6, E7 and E8 -- Chapter 13. Ramified Quadrangles of Type E6, E7 and E8 -- Chapter 14. Quadrangles of Type E6, E7 and E8: Summary -- Chapter 15. Totally Wild Quadratic Forms of Type E7 -- Chapter 16. Existence -- Chapter 17. Quadrangles of Type F4 -- Chapter 18. The Other Bruhat-Tits Buildings -- PART 3. DESCENT -- Chapter 19. Coxeter Groups -- Chapter 20. Tits Indices -- Chapter 21. Parallel Residues -- Chapter 22. Fixed Point Buildings -- Chapter 23. Subbuildings -- Chapter 24. Moufang Structures -- Chapter 25. Fixed Apartments -- Chapter 26. The Standard Metric -- Chapter 27. Affine Fixed Point Buildings -- PART 4. GALOIS INVOLUTIONS -- Chapter 28. Pseudo-Split Buildings -- Chapter 29. Linear Automorphisms -- Chapter 30. Strictly Semi-linear Automorphisms -- Chapter 31. Galois Involutions -- Chapter 32. Unramified Galois Involutions -- PART 5. EXCEPTIONAL TITS INDICES -- Chapter 33. Residually Pseudo-Split Buildings -- Chapter 34. Forms of Residually Pseudo-Split Buildings -- Chapter 35. Orthogonal Buildings -- Chapter 36. Indices for the Exceptional Bruhat-Tits Buildings -- Bibliography -- Index Chapter 17. Quadrangles of Type F4 PART 4. Galois Involutions -- Chapter 31. Galois Involutions PART 1. Moufang Quadrangles -- Chapter 6. Linked Tori, II Chapter 14. Quadrangles of Type E6, E7 and E8: Summary Chapter 4. Moufang Quadrangles Chapter 10. Residues Chapter 30. Strictly Semi-linear Automorphisms Chapter 36. Indices for the Exceptional Bruhat-Tits Buildings PART 3. Descent -- Index Chapter 34. Forms of Residually Pseudo-Split Buildings Chapter 11. Unramified Quadrangles of Type E6, E7 and E8 Chapter 25. Fixed Apartments Chapter 24. Moufang Structures Chapter 35. Orthogonal Buildings Chapter 26. The Standard Metric Chapter 22. Fixed Point Buildings Chapter 16. Existence Chapter 1. Buildings Chapter 9. Quadratic Forms of Type F4 Chapter 15. Totally Wild Quadratic Forms of Type E7 Preface Chapter 8. Quadratic Forms of Type E6, E7 and E8 Chapter 12. Semi-ramified Quadrangles of Type E6, E7 and E8 Chapter 21. Parallel Residues Chapter 32. Unramified Galois Involutions Chapter 18. The Other Bruhat-Tits Buildings Chapter 3. Moufang Polygons PART 2. Residues in Bruhat-Tits Buildings -- Chapter 13. Ramified Quadrangles of Type E6, E7 and E8 Chapter 20. Tits Indices - Chapter 33. Residually Pseudo-Split Buildings / Chapter 28. Pseudo-Split Buildings PART 5. Exceptional Tits Indices -- Contents Chapter 29. Linear Automorphisms Chapter 19. Coxeter Groups Frontmatter -- Chapter 27. Affine Fixed Point Buildings Chapter 5. Linked Tori, I Chapter 23. Subbuildings Chapter 2. Quadratic Forms Bibliography Chapter 7. Quadratic Forms over a Local Field |
| Title | Descent in Buildings (AM-190) |
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