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The Balmer spectrum of certain Deligne–Mumford stacks
Part of
Eike Lau
Journal:

Compositio Mathematica / Volume 159 / Issue 6 / June 2023

Published online by Cambridge University Press:

24 May 2023, pp. 1314-1346

Print publication:

June 2023
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We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.

On the Balmer spectrum for compact Lie groups
Part of
- Homotopy theory
Tobias Barthel, J. P. C. Greenlees, Markus Hausmann
Journal:

Compositio Mathematica / Volume 156 / Issue 1 / January 2020

Published online by Cambridge University Press:

14 November 2019, pp. 39-76

Print publication:

January 2020
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We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

3 - Finite Spectra
Eric Peterson, Harvard University, Massachusetts
Book:

Formal Geometry and Bordism Operations

Published online:

23 November 2018

Print publication:

06 December 2018, pp 97-164
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Summary

This chapter introduces the basic results of chromatic homotopy theory while adhering to the language of algebraic geometry. Our approach centers on the study of descent in the setting of stable homotopy theory; Quillen’s theorem from the preceding chapter gives an example of a ring spectrum whose unit map is of effective descent and whose algebraic properties are amenable to algebro–geometric study. Following these ideas to their conclusion leads us in turn to the study of the moduli of formal groups, and we dedicate several sections to the description of the geometry of this moduli stack. We produce reflections in the stable homotopy category of our main algebraic results, emphasizing especially the periodicity theorems of Devinatz, Hopkins, and Smith, which simultaneously give shape to the structure of the “prime ideals” of the stable homotopy category as well as organize the stable stems into identifiable families.