nLab
thick subcategory theorem

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

What is called the thick subcategory theorem gives a a characterization of all thick subcategories of the stable homotopy category of p-local finite spectra.

This is a consequence of the nilpotence theorem.

Statement

For nn \in \mathbb{N}, write K(n)K(n) for the Morava K-theory spectrum.

Say that a pp-local finite spectrum XX has type nn if K(n) *(X)0K(n)_\ast(X) \neq 0 but K(k) *(X)0K(k)_\ast(X) \simeq 0 for all k<nk \lt n. (e.g. Lurie, def. 6).

Write Spectra p finSpectraSpectra^{fin}_p \hookrightarrow Spectra be the full subcategory of the (infinity,1)-category of spectra on the p-local finite spectra.

The thick subcategory theorem says that thick subcategories of Spectra p finSpectra^{fin}_p are precisely the full subcategories on the spectra of type <n\lt n, for some nn. (e.g. Lurie, theorem 8).

In other words, the thick subcategories are the kernels of K(n) *K(n)_\ast (for any given prime).

Applications

The thick subcategory theorem serves to determine the prime spectrum of a symmetric monoidal stable (∞,1)-category of the stable homotopy theory.

(For an exposition see MazelGee 13, around slide 9).

References

The original article is

Further discussion is in

Some big-picture motivation is in

Last revised on March 25, 2014 at 01:31:26. See the history of this page for a list of all contributions to it.