nLab thick subcategory theorem

Idea

What is called the thick subcategory theorem gives 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 $n \in \mathbb{N}$, write $K(n)$ for the Morava K-theory spectrum.

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

Write $Spectra^{fin}_p \hookrightarrow Spectra$ 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 $Spectra^{fin}_p$ are precisely the full subcategories on the spectra of type $\geq n$, for some $n$. (e.g. Lurie, theorem 8).

In other words, the thick subcategories are the kernels of $K(n)_\ast$ (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

• Michael Hopkins, Jeff Smith, Nilpotence and stable homotopy theory II. Ann. of Math. (2),148(1):1–49, 1998

Further discussion is in

• Alain Jeanneret, Peter Landweber, Douglas Ravenel, A note on the thick subcategory theorem (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, lecture 26, Thick subcategories (pdf)

Some big-picture motivation is in

• Aaron Mazel-Gee, You could’ve invented $tmf$, April 2013 (pdf slides)