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

Balmer, Paul

doi:10.2140/agt.2010.10.1521

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

Paul Balmer

Algebraic & Geometric Topology 10 (2010) 1521–1563

DOI: 10.2140/agt.2010.10.1521

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 $ℤ$ . We also give a first discussion of the spectrum in two new examples, namely equivariant $KK$ –theory and stable $A^{1}$ –homotopy theory.

Keywords

tensor triangular geometry, spectra

Mathematical Subject Classification 2000

Primary: 18E30

Secondary: 14F05, 19K35, 20C20, 55P42, 55U35

References

Publication

Received: 28 May 2009

Revised: 26 March 2010

Accepted: 28 May 2010

Published: 2 July 2010

Authors

Paul Balmer
Mathematics Department UCLA Box 951555 Los Angeles 90095-1555 United States
http://www.math.ucla.edu/~balmer

Volume 10, issue 3 (2010)