Chromatic Homotopy Theory

2 Thick Subcategories

As a consequence of the nilpotence theorem, we can calculate all the “prime ideals" in the category of spectra. However, one needs the periodicity theorem to see that these are distinct.

Definition 2.1. A thick subcategory of \(p\)-local finite spectra is a full subcategory closed under finite limits and colimits, and retracts.

Some notes about the definition: for something to be thick it suffices to check that it is closed under shifts and cofibres, and if \(X\vee Y\) is in the category, \(X\) is too. Furthermore note if \(X\) is in the category, then \(X \otimes Y\) is too. A consequence of the nilpotence theorem is the following characterization of thick subcategories:

Theorem 2.2 (Thick Subcategory Theorem). If \(C\) is thick, it consists of all \(p\)-local finite spectra of type \(\geq n\) for some \(0\leq n \leq \infty \).

Proof. Let \(X\) be an object of minimal type, \(n\), in \(C\), and form the fibre sequence \(W \to \SP \to X \otimes DX\) (\(DX\) is the Spanier-Whitehead dual). Observe that \(\SP \to X\otimes DX\) is nonzero in \(K(m)\) homology for \(m\geq n\) since the adjoint of the map is nonzero as X is type \(n\). Thus since \(K(m))\) is a field, it is injective on \(K(m)\) homology, so the map \(W \to \SP \) is zero in \(K(m)\) homology.

Now let \(Y\) be any other type \(\geq n\) finite \(p\)-local spectrum. The composite \(f:W \to \SP \to Y \otimes DY\) is zero in all \(K(m)\) homologies so must be nilpotent by the nilpotent theorem. Thus for some \(k\), \(f^k:W^k \otimes Y \to Y\) is zero. Since \(C\) is closed under retracts, it suffices to show the cofibre is in \(C\). But the cofibre has a finite filtration by the cofibres of \(f^i\), whose associated graded is the cofibre of \(f:W^i\otimes Y \to W^{i-1}\otimes Y\) which is \(W^{i-1}\otimes X\otimes DX \otimes Y\), which is in \(C\) since \(X\) is. □