MOTIVIC HOCHSCHILD HOMOLOGY
type: CreativeWork
description: Let S be a quasi-compact, quasi-separated scheme, such as the spectrum of a field k, and let SH(S) be Morel and Voevodski's stable motivic homotopy category over S. Let R be a motivic ring spectrum in SH(S); we define the motivic Hochschild homology of R via the derived tensor product: MHH(R)=R ∧_{R ∧ R^{op}} R. This spectrum can be considered the motivic analogue of topological Hochschild homology. The purpose of the first part of this thesis is to investigate the homotopy structure of motivic Hochschild homology of MZ/p- the Suslin-Voevodsky mod-p motivic cohomology ring spectrum for p any prime number. We will in particular consider the case S=Spec(k) is the spectrum of an algebraically closed field k of characteristic different from p. In this context, the homotopy groups of M Z/p are a polynomial ring over the field with p elements on a single generator τ in bidegree (0,-1). To do so, we introduce a tri-graded spectral sequence in motivic homotopy groups inspired by an analogue in stable homotopy theory defined by J. Greenlees in a 2016 paper. We will use our spectral sequence to investigate different spectra. As a motivation for the following computations, we first have a look at MHH(MZ/p) itself. Here we produce a first quadrant, multiplicative, tri-graded spectral sequence with E^2_{s,t,*}=π_{s,*}(MHH(MZ/p)) ⊗_{π_{0,*}(MZ/p)} π_{t,*}(A(p)) converging to π_{s+t,*}(MZ/p). A(p) here denotes the dual motivic Steenrod algebra. Notice that each page in our spectral sequences is tri-graded, meaning we can decompose the associated graded object as a direct sum of modules indexed on three natural numbers- the degrees. In this case, we will not provide a direct argument computing all the homotopy groups: we will simply use the spectral sequence to directly determine the structure in low degrees at the prime p=2. This allows us to perceive how abundant are elements of τ-torsion and motivates the following step: study a τ-inverted and mod-τ^{p-1} versions of MHH(MZ/p), …
abstract: Let S be a quasi-compact, quasi-separated scheme, such as the spectrum of a field k, and let SH(S) be Morel and Voevodski's stable motivic homotopy category over S. Let R be a motivic ring spectrum in SH(S); we define the motivic Hochschild homology of R via the derived tensor product: MHH(R)=R ∧_{R ∧ R^{op}} R. This spectrum can be considered the motivic analogue of topological Hochschild homology. The purpose of the first part of this thesis is to investigate the homotopy structure of motivic Hochschild homology of MZ/p- the Suslin-Voevodsky mod-p motivic cohomology ring spectrum for p any prime number. We will in particular consider the case S=Spec(k) is the spectrum of an algebraically closed field k of characteristic different from p. In this context, the homotopy groups of M Z/p are a polynomial ring over the field with p elements on a single generator τ in bidegree (0,-1). To do so, we introduce a tri-graded spectral sequence in motivic homotopy groups inspired by an analogue in stable homotopy theory defined by J. Greenlees in a 2016 paper. We will use our spectral sequence to investigate different spectra. As a motivation for the following computations, we first have a look at MHH(MZ/p) itself. Here we produce a first quadrant, multiplicative, tri-graded spectral sequence with E^2_{s,t,*}=π_{s,*}(MHH(MZ/p)) ⊗_{π_{0,*}(MZ/p)} π_{t,*}(A(p)) converging to π_{s+t,*}(MZ/p). A(p) here denotes the dual motivic Steenrod algebra. Notice that each page in our spectral sequences is tri-graded, meaning we can decompose the associated graded object as a direct sum of modules indexed on three natural numbers- the degrees. In this case, we will not provide a direct argument computing all the homotopy groups: we will simply use the spectral sequence to directly determine the structure in low degrees at the prime p=2. This allows us to perceive how abundant are elements of τ-torsion and motivates the following step: study a τ-inverted and mod-τ^{p-1} versions of MHH(MZ/p), …
author: MOCCHETTI, FEDERICO
language: eng
keywords: Settore MATH-02/B - Geometria, Settore MATH-02/A - Algebra
date published: 2025-02-10
date modified: 2026-07-27T20:19:12Z
provider: Università degli Studi di Milano
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Università degli Studi di Milano

PropertyValue
property ID: Handle
value: https://hdl.handle.net/2434/1141325

PropertyValue
property ID: DOI
value: 10.13130/mocchetti-federico_phd2025-02-10