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K-Theoretic Invariants in Algebraic ...

Nanavaty, Anubhav,

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K-Theoretic Invariants in Algebraic Geometry /

Record Type:	Electronic resources : Monograph/item
Title/Author:	K-Theoretic Invariants in Algebraic Geometry // Anubhav Nanavaty.
Author:	Nanavaty, Anubhav,
Description:	1 electronic resource (237 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 87-01, Section: B.
Contained By:	Dissertations Abstracts International87-01B.
Subject:	Mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=32044739
ISBN:	9798288803765

K-Theoretic Invariants in Algebraic Geometry /
Nanavaty, Anubhav,

K-Theoretic Invariants in Algebraic Geometry /Anubhav Nanavaty. - 1 electronic resource (237 pages)

Source: Dissertations Abstracts International, Volume: 87-01, Section: B.

The aim of this thesis is to study and extract additive invariants of solutions to polynomials and their symmetries, and vector spaces over a field. There are 4 chapters: the first three chapters contains results in different pre-prints that are currently submitted for publication, while the last chapter contains mostly expository material.Results in first chapter are joint with Oliver Braunling and Michael Groechenig. We study K(Vark), the Zakharevich K-theory of varieties, which is a spectral enrichment of the usual Grothendieck ring of varieties K0(Vark). We then construct maps of spectra out of K(Vark), the most important of which is the map K(Vark) → K(Cgm), where Cgm is Voevodsky's DGcategory of motives. The goal of constructing these maps was to use them to study non-trivial elements in the higher homotopy groups of K(Vark), which was successfully done in later work of Braunling and Groechenig.In the second chapter, we lift a number of motivic measures, such as the Gillet-Soule measure and the compactly supported \uD835\uDD381-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soule weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a k-variety X, the collection of all Gillet-Soule weight complexes of X form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the K-theory of a Waldhausen category is equivalent to the K-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soule weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over k. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of K(Vk), the Zakharevich K-theory of varieties.In Chapter 3, we study the space of non-singular symmetric symmetries. We show that the space of symmetric matrices of a fixed rank k over a field K of characteristic not equal to 2 is split Tate. We do this by promoting the point-counting strategy of MacWilliams over finite fields to a filtration of the locus of rank ≤ k symmetric matrices that is independent of the field. This filtration immediately allows for a computation of their isomorphism classes in the Grothendieck ring of varieties in terms of the Lefschetz motive. We then promote this computation to prove that the space of symmetric matrices of a fixed rank k are split Tate in Voevodsky's category of motives in characteristic 0 and Kelly's category of motives in characteristic p.Lastly, in chapter 4, we provide an expository introduction to Euclidean Scissors Congruence, the study of polytopes in Euclidean space up to 'cut and paste' relations. We first re-frame questions in scissors congruence as those in group homology. We then use this perspective to review the proof of the Dehn-Sydler-Jessen Theorem as found in the works of Dupont and Sah.

English

ISBN: 9798288803765Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Campbell-Wolfson-Zakharevich

K-Theoretic Invariants in Algebraic Geometry /
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520 $a The aim of this thesis is to study and extract additive invariants of solutions to polynomials and their symmetries, and vector spaces over a field. There are 4 chapters: the first three chapters contains results in different pre-prints that are currently submitted for publication, while the last chapter contains mostly expository material.Results in first chapter are joint with Oliver Braunling and Michael Groechenig. We study K(Vark), the Zakharevich K-theory of varieties, which is a spectral enrichment of the usual Grothendieck ring of varieties K0(Vark). We then construct maps of spectra out of K(Vark), the most important of which is the map K(Vark) → K(Cgm), where Cgm is Voevodsky's DGcategory of motives. The goal of constructing these maps was to use them to study non-trivial elements in the higher homotopy groups of K(Vark), which was successfully done in later work of Braunling and Groechenig.In the second chapter, we lift a number of motivic measures, such as the Gillet-Soule measure and the compactly supported \uD835\uDD381-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soule weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a k-variety X, the collection of all Gillet-Soule weight complexes of X form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the K-theory of a Waldhausen category is equivalent to the K-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soule weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over k. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of K(Vk), the Zakharevich K-theory of varieties.In Chapter 3, we study the space of non-singular symmetric symmetries. We show that the space of symmetric matrices of a fixed rank k over a field K of characteristic not equal to 2 is split Tate. We do this by promoting the point-counting strategy of MacWilliams over finite fields to a filtration of the locus of rank ≤ k symmetric matrices that is independent of the field. This filtration immediately allows for a computation of their isomorphism classes in the Grothendieck ring of varieties in terms of the Lefschetz motive. We then promote this computation to prove that the space of symmetric matrices of a fixed rank k are split Tate in Voevodsky's category of motives in characteristic 0 and Kelly's category of motives in characteristic p.Lastly, in chapter 4, we provide an expository introduction to Euclidean Scissors Congruence, the study of polytopes in Euclidean space up to 'cut and paste' relations. We first re-frame questions in scissors congruence as those in group homology. We then use this perspective to review the proof of the Dehn-Sydler-Jessen Theorem as found in the works of Dupont and Sah.
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653 $a Symmetric matrices
653 $a Isomorphism classe
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690 $a 0642
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