Tensor-triangular fields: ruminations
type: Item
instance of: scholarly article
title: Tensor-triangular fields: ruminations
zbMATH Open document ID: 1409.18011
author: Paul Balmer
author: Henning Krause
author: Greg Stevenson
published in: Selecta Mathematica. New Series
publication date: 2019-02-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1707.02167
review text: The authors study a notion of field in the abstract setting of tensor-triangulated categories. They analyse various approaches, with the ultimate objective that such theory would apply to many contexts (homotopy theory, algebraic geometry, KK-theory of $\mathbb C^*$-algebras, and more). \par A \textit{big} tt-category is a rigidly-compactly generated tensor-triangulated category $\mathcal T$, which admits all coproducts, whose compact and rigid objects coincide, and such that the essentially small subcategory $\mathcal T^c$ of compact-rigids generates $\mathcal T$ as a localizing subcategory. \par Their basic model is inspired from commutative algebra, and the reduction of problems from global to local rings (by localization), and then to residue fields (by quotient). The localization process for tt-categories is well-understood, and so the authors concentrate on the reduction of a problem from a set of local data to that of a `residue field data', which they first need to define. Namely, given a local tt-category, $\mathcal T$, they want to find a tt-category $\mathcal F$ and a tt-functor $F:\mathcal T\to\mathcal F$, such that $\mathcal F$ is a `tt-field'. Drawing from examples in modular representation theory of finite groups, the authors come up with the following notion: A non-trivial big tt-category $\mathcal F$ is a tt-field if every object of $\mathcal F$ is a coproduct of compact-rigid objects of $\mathcal F^c$, and if every non-zero object $X$ of $\mathcal F$ is $\otimes$-faithful, i.e. the functor $X\otimes-:\mathcal F\to\mathcal F$ is faithful. \par The authors prove some basic properties of such tt-fields; in particular every tt-field has minimal spectrum. Thus they address the question of the existence of a coproduct-preserving tt-functor $F:\mathcal T\to\mathcal F$ with $\mathcal F$ conservative on $\mathcal T^c$. This question remains open in general, but they propose a palliative using abelian categories, in which there exists a restricted Yoneda fun…
reviewed by: Nadia Mazza
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 7024173
zbMATH Keywords: residue field
zbMATH Keywords: tt-geometry
zbMATH Keywords: module category
MaRDI profile type: Publication
OpenAlex ID: W3099738767
arXiv ID: 1707.02167
cites work: Representation Theory of Artin Algebras II
cites work: Spectra, spectra, spectra -- tensor triangular spectra versus Zariski spectra of endomorphism rings
cites work: Q3096591
cites work: Nilpotence theorems via homological residue fields
cites work: Restriction to finite-index subgroups as étale extensions in topology, KK-theory and geometry
cites work: Grothendieck–Neeman duality and the Wirthmüller isomorphism
cites work: Relative homological algebra and purity in triangulated categories
cites work: Generalized tensor idempotents and the telescope conjecture
cites work: The frame of smashing tensor-ideals
cites work: Modules of constant Jordan type
cites work: Des catégories abéliennes
cites work: Sur quelques points d'algèbre homologique
cites work: Primes and fields in stable motivic homotopy theory
cites work: Homotopy Invariant Algebraic Structures
cites work: Axiomatic stable homotopy theory
cites work: Endofiniteness in stable homotopy theory
cites work: Smashing subcategories and the telescope conjecture -- an algebraic approach
cites work: Residue fields for a class of rational \(\mathbf E_{\infty}\)-rings and applications
cites work: Q4215784
cites work: The chromatic tower for \(D(R)\). With an appendix by Marcel Bökstedt
cites work: The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel
cites work: Triangulated categories
cites work: Q4079762
cites work: A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry
Wikidata QID: Q128366676
DOI: 10.1007/S00029-019-0454-2
description: scientific article