Algebraic & Geometric Topology

Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings

Paul Balmer

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Abstract

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra has a triangular spectrum much bigger than the Zariski spectrum of . We also give a first discussion of the spectrum in two new examples, namely equivariant KK–theory and stable A1–homotopy theory.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1521-1563.

Dates
Received: 28 May 2009
Revised: 26 March 2010
Accepted: 28 May 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715144

Digital Object Identifier
doi:10.2140/agt.2010.10.1521

Mathematical Reviews number (MathSciNet)
MR2661535

Zentralblatt MATH identifier
1204.18005

Subjects
Primary: 18E30: Derived categories, triangulated categories
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 20C20: Modular representations and characters 55P42: Stable homotopy theory, spectra 55U35: Abstract and axiomatic homotopy theory

Keywords
tensor triangular geometry spectra

Citation

Balmer, Paul. Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol. 10 (2010), no. 3, 1521--1563. doi:10.2140/agt.2010.10.1521. https://projecteuclid.org/euclid.agt/1513715144


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