Localization for
Blumberg, Andrew J.
Localization for - 1st ed. - 1 online resource (112 pages) - Memoirs of the American Mathematical Society Series ; v.265 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- Chapter 1. Review of , , and -- 1.1. Review of spectral categories -- 1.2. Review of the construction of , , and -- 1.3. Review of the invariance properties of -- 1.4. The Dennis-Waldhausen Morita Argument -- Chapter 2. and of simplicially enriched Waldhausen categories -- 2.1. Simplicially enriched Waldhausen categories -- 2.2. Spectral categories associated to simplicially enriched Waldhausen categories -- 2.3. The \Sdot and Moore nerve constructions -- 2.4. The Moore \Spdot construction -- 2.5. , , and the cyclotomic trace -- Chapter 3. -theory theorems in and -- 3.1. The Additivity Theorem -- 3.2. The Cofiber Theorem -- 3.3. The Localization Theorem -- 3.4. The Sphere Theorem -- 3.5. Proof of the Sphere Theorem -- Chapter 4. Localization sequences for and -- 4.1. The localization sequence for of a discrete valuation ring -- 4.2. The localization sequence for ( ) and related ring spectra -- 4.3. Proof of the Dévissage Theorem -- Chapter 5. Generalization to Waldhausen categories with factorization -- 5.1. Weakly exact functors -- 5.2. Embedding in simplicially tensored Waldhausen categories -- 5.3. Spectral categories and Waldhausen categories -- Bibliography -- Index -- Back Cover.
The authors develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. They resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized dévissage theorem. As applications, the authors prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding THH(ku), and more generally establish a framework for advancing the Rognes program for studying Waldhausen's chromatic filtration on A(*).
9781470461409
K-theory.
Algebraic topology.
Cobordism theory.
Homology theory.
Electronic books.
QA612.33 .B586 2020
512/.66
Localization for - 1st ed. - 1 online resource (112 pages) - Memoirs of the American Mathematical Society Series ; v.265 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- Chapter 1. Review of , , and -- 1.1. Review of spectral categories -- 1.2. Review of the construction of , , and -- 1.3. Review of the invariance properties of -- 1.4. The Dennis-Waldhausen Morita Argument -- Chapter 2. and of simplicially enriched Waldhausen categories -- 2.1. Simplicially enriched Waldhausen categories -- 2.2. Spectral categories associated to simplicially enriched Waldhausen categories -- 2.3. The \Sdot and Moore nerve constructions -- 2.4. The Moore \Spdot construction -- 2.5. , , and the cyclotomic trace -- Chapter 3. -theory theorems in and -- 3.1. The Additivity Theorem -- 3.2. The Cofiber Theorem -- 3.3. The Localization Theorem -- 3.4. The Sphere Theorem -- 3.5. Proof of the Sphere Theorem -- Chapter 4. Localization sequences for and -- 4.1. The localization sequence for of a discrete valuation ring -- 4.2. The localization sequence for ( ) and related ring spectra -- 4.3. Proof of the Dévissage Theorem -- Chapter 5. Generalization to Waldhausen categories with factorization -- 5.1. Weakly exact functors -- 5.2. Embedding in simplicially tensored Waldhausen categories -- 5.3. Spectral categories and Waldhausen categories -- Bibliography -- Index -- Back Cover.
The authors develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. They resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized dévissage theorem. As applications, the authors prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding THH(ku), and more generally establish a framework for advancing the Rognes program for studying Waldhausen's chromatic filtration on A(*).
9781470461409
K-theory.
Algebraic topology.
Cobordism theory.
Homology theory.
Electronic books.
QA612.33 .B586 2020
512/.66