Constructing K-theory spectra from algebraic structures with a class of acyclic objects
| dc.contributor.author | Sarazola, Maru | |
| dc.contributor.chair | Zakharevich, Inna I. | |
| dc.contributor.committeeMember | Aguiar, Marcelo | |
| dc.contributor.committeeMember | Holm, Tara | |
| dc.date.accessioned | 2021-09-09T17:41:01Z | |
| dc.date.available | 2021-09-09T17:41:01Z | |
| dc.date.issued | 2021-05 | |
| dc.description | 183 pages | |
| dc.description.abstract | 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 [20], we show that under certain technical conditions, a cotorsion pair $(C,C^\bot)$ in an exact category E, together with a subcategory $Z\subseteq E$ containing $C^\bot$, 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 $A\subseteq B$ 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 $S_\bullet$-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. | |
| dc.identifier.doi | https://doi.org/10.7298/rw3q-1q83 | |
| dc.identifier.other | Sarazola_cornellgrad_0058F_12461 | |
| dc.identifier.other | http://dissertations.umi.com/cornellgrad:12461 | |
| dc.identifier.uri | https://hdl.handle.net/1813/109797 | |
| dc.language.iso | en | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.subject | algebraic K-theory | |
| dc.subject | cotorsion | |
| dc.subject | double categories | |
| dc.subject | exact categories | |
| dc.subject | K-theory | |
| dc.subject | localization | |
| dc.title | Constructing K-theory spectra from algebraic structures with a class of acyclic objects | |
| dc.type | dissertation or thesis | |
| dcterms.license | https://hdl.handle.net/1813/59810 | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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