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Constructing K-Theory Spectra from Algebraic Structures with a Class of Acyclic Objects.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Constructing K-Theory Spectra from Algebraic Structures with a Class of Acyclic Objects./
作者:	Sarazola Duarte, Maria Eugenia.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
面頁冊數:	184 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-01, Section: B.
Contained By:	Dissertations Abstracts International83-01B.
標題:	Theoretical mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28416703
ISBN:	9798516921681

Constructing K-Theory Spectra from Algebraic Structures with a Class of Acyclic Objects.
Sarazola Duarte, Maria Eugenia.

Constructing K-Theory Spectra from Algebraic Structures with a Class of Acyclic Objects. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 184 p.

Source: Dissertations Abstracts International, Volume: 83-01, Section: B.

Thesis (Ph.D.)--Cornell University, 2021.

This item must not be sold to any third party vendors.

This thesis studies different ways to construct categories admitting an algebraic K-theory spectrum, focusing on categories that contain some flavor of underlying algebraic structure as well as relevant homotopical information.In Part I, published as Sarazola (2020), we show that under certain technical conditions, a cotorsion pair (C, C⊥) in an exact category E, together with a subcategory Z ⊆ E containing C⊥, determines a Waldhausen structure on C in which Z is the class of acyclic objects. This yields a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A ⊆ B to that of a cofiber. The novel approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, A need not be a Serre subcategory, which results in new examples.In Part II, joint work with Brandon Shapiro, we upgrade the K-theory of (A)CGW categories due to Campbell and Zakharevich by defining a new type of structures, called FCGWA categories, that incorporate the data of weak equivalences. FCGWA categories admit an S•-construction in the spirit of Waldhausen's, which produces a K-theory spectrum, and satisfies analogues of the Additivity and Fibration Theorems. Weak equivalences are determined by choosing a subcategory of acyclic objects satisfying minimal conditions, which results in a Localization Theorem that generalizes previous versions in the literature. Our main example is chain complexes of sets with quasi-isomorphisms; these satisfy a Gillet-Waldhausen Theorem, yielding an equivalent presentation of the K-theory of finite sets.

ISBN: 9798516921681Subjects--Topical Terms:

3173530
Theoretical mathematics.
Subjects--Index Terms:

Algebraic K-theory

Constructing K-Theory Spectra from Algebraic Structures with a Class of Acyclic Objects.
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506 $a This item must not be sold to any third party vendors.
520 $a This thesis studies different ways to construct categories admitting an algebraic K-theory spectrum, focusing on categories that contain some flavor of underlying algebraic structure as well as relevant homotopical information.In Part I, published as Sarazola (2020), we show that under certain technical conditions, a cotorsion pair (C, C⊥) in an exact category E, together with a subcategory Z ⊆ E containing C⊥, determines a Waldhausen structure on C in which Z is the class of acyclic objects. This yields a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A ⊆ B to that of a cofiber. The novel approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, A need not be a Serre subcategory, which results in new examples.In Part II, joint work with Brandon Shapiro, we upgrade the K-theory of (A)CGW categories due to Campbell and Zakharevich by defining a new type of structures, called FCGWA categories, that incorporate the data of weak equivalences. FCGWA categories admit an S•-construction in the spirit of Waldhausen's, which produces a K-theory spectrum, and satisfies analogues of the Additivity and Fibration Theorems. Weak equivalences are determined by choosing a subcategory of acyclic objects satisfying minimal conditions, which results in a Localization Theorem that generalizes previous versions in the literature. Our main example is chain complexes of sets with quasi-isomorphisms; these satisfy a Gillet-Waldhausen Theorem, yielding an equivalent presentation of the K-theory of finite sets.
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