## Algebraic & Geometric Topology

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

Paul Balmer

#### 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 $A1$–homotopy theory.

#### Article information

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

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

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

#### 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

#### References

• P Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005) 149–168
• P Balmer, Supports and filtrations in algebraic geometry and modular representation theory, Amer. J. Math. 129 (2007) 1227–1250
• P Balmer, Niveau spectral sequences on singular schemes and failure of generalized Gersten conjecture, Proc. Amer. Math. Soc. 137 (2009) 99–106
• P Balmer, Picard groups in triangular geometry and applications to modular representation theory, Trans. Amer. Math. Soc. 362 (2010) 3677–3690
• P Balmer, D J Benson, J F Carlson, Gluing representations via idempotent modules and constructing endotrivial modules, J. Pure Appl. Algebra 213 (2009) 173–193
• P Balmer, G Favi, Gluing techniques in triangular geometry, Q. J. Math. 58 (2007) 415–441
• D J Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, second edition, Cambridge Studies in Advanced Math. 30, Cambridge Univ. Press (1998)
• D Benson, S B Iyengar, H Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. $(4)$ 41 (2008) 573–619
• A B Buan, H Krause, Ø Solberg, Support varieties: an ideal approach, Homology, Homotopy Appl. 9 (2007) 45–74
• J M Cohen, Coherent graded rings and the non-existence of spaces of finite stable homotopy type, Comment. Math. Helv. 44 (1969) 217–228
• I Dell'Ambrogio, Prime tensor ideals in some triangulated categories of C$^*$–algebras, PhD thesis, ETH Zürich (2008)
• M E Harris, Some results on coherent rings, Proc. Amer. Math. Soc. 17 (1966) 474–479
• R Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, New York (1977)
• M Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969) 43–60
• M J Hopkins, J H Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. $(2)$ 148 (1998) 1–49
• M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997) x+114
• A Krishna, Perfect complexes on Deligne–Mumford stacks and applications, J. K-Theory 4 (2009) 559–603
• T Y Lam, Introduction to quadratic forms over fields, Graduate Studies in Math. 67, Amer. Math. Soc. (2005)
• F Lorenz, J Leicht, Die Primideale des Wittschen Ringes, Invent. Math. 10 (1970) 82–88
• A Lorenzini, Graded rings, from: “The curves seminar at Queens, Vol. II (Kingston, Ont., 1981/1982)”, (A V Geramita, editor), Queen's Papers in Pure and Appl. Math. 61, Queen's Univ., Kingston, ON (1982) Exp. No. D, 41
• S Mac Lane, Categories for the working mathematician, second edition, Graduate Texts in Math. 5, Springer, New York (1998)
• H R Margolis, Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category, North-Holland Math. Library 29, North-Holland, Amsterdam (1983)
• R Meyer, Categorical aspects of bivariant $K$–theory, from: “$K$–theory and noncommutative geometry”, (G Cortiñas, J Cuntz, M Karoubi, R Nest, C A Weibel, editors), EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2008) 1–39
• F Morel, An introduction to $\mathbb A\sp 1$–homotopy theory, from: “Contemporary developments in algebraic $K$–theory”, (M Karoubi, A O Kuku, C Pedrini, editors), ICTP Lect. Notes XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004) 357–441
• F Morel, On the motivic $\pi\sb 0$ of the sphere spectrum, from: “Axiomatic, enriched and motivic homotopy theory”, (J P C Greenlees, editor), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht (2004) 219–260
• F Morel, The stable ${\mathbb A}\sp 1$–connectivity theorems, $K$–Theory 35 (2005) 1–68
• A Neeman, Triangulated categories, Annals of Math. Studies 148, Princeton Univ. Press (2001)
• N C Phillips, Equivariant $K$–theory and freeness of group actions on $C\sp *$–algebras, Lecture Notes in Math. 1274, Springer, Berlin (1987)
• D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Math. Studies 128, Princeton Univ. Press (1992) Appendix C by J Smith
• J Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989) 303–317
• G Segal, The representation ring of a compact Lie group, Inst. Hautes Études Sci. Publ. Math. (1968) 113–128
• J-L Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239, Soc. Math. France, Paris (1996) With a preface by L Illusie, Edited and with a note by G Maltsiniotis
• V Voevodsky, $\bold A\sp 1$–homotopy theory, from: “Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)”, Doc. Math. Extra Vol. I (1998) 579–604