Chromatic Homotopy and Telescopic Localization

2 Important Cohomology Theories and Theorems

Two important families of complex oriented cohomology theories are Morava E-theory and Morava K-theory.

The \(n^{th}\) Morava E-theory, denoted \(E_n\), is an \(\EE _{\infty }\)-ring spectrum that depends on a choice of perfect field \(k\) and formal group law on \(k\) of height \(n\). However, none of the choices will matter for anything said here about it. Its coefficient ring is \((E_n) _* = W(k)[[v_1,\dots ,v_{n-1}]][\beta ^{\pm 1}]\) where \(|\beta | = 2\), and \(W(k)\) denotes the Witt vectors of \(k\), and its formal group law is the universal deformation of the formal group law on \(k\), which was studied by Lubin and Tate.

One of its important properties is that \((E_n)_*(X)=0\) if and only if \(BP_*(X)\) is supported at height \(\geq n+1\) on \(\cM _{fg}\). Thus it detects information from height \(0\) to height \(n\).

The \(n^{th}\) Morava K-theory, denoted \(K(n)\), is an \(\EE _1\)-ring spectrum with coefficient group \(K(n)_* = \FF _p[v_n^{\pm 1}]\). Once again, there are different versions of it, but the different versions will not be relevant here. \(K(n)\) can be constructed from \(BP\) by iteratively taking cofibres by \(v_i\) for \(i \neq n\) and inverting \(v_n\). \(K(\infty )\) is just \(H\FF _p\). An important property of \(K(n)\) is that it is a field, since its homotopy groups are a graded field. This means that any module over \(K(n)\) is free.

\(K(n)\) detects information at height \(n\). For example, \((E_n)_*(X)=0\iff (\oplus _0^n K(n))_*X = 0\).

Next, we turn to some fundamental results in chromatic homotopy theory. The first is the nilpotence theorem, due to Devinatz, Hopkins, and Smith.

Theorem 2.1 (Nilpotence Theorem v1). Let \(R\) be a ring spectrum, and \(\alpha \in \pi _n(R)\) be an element sent to \(0\) in \(MU_n(R)\). Then \(\alpha \) is nilpotent.

This says that \(MU\) is able to detect nilpotence in rings. An equivalent version of the theorem is

Theorem 2.2 (Nilpotence Theorem v2). Let \(f:X \to Y\) be a map of finite spectra such that \(f\otimes MU\) is \(0\). Then \(f^{\otimes n}:X^{\otimes n} \to Y^{\otimes n}\) is null for \(n \gg 0\).

This formulation emphasizes the fact that \(MU\) detects nilpotence phenomena for finite spectra. When working \(p\)-locally, \(MU_*f = 0\) iff \(BP_*f = 0\) iff \(K(n)_*f=0\) for all \(n\).

Definition 2.3. A finite complex/spectrum \(X\) is type \(n\) if \(K(i)_*(X) = 0\) for all \(i <n\) and \(K(n)_*X \neq 0\).

Remark 2.4. Every nonzero \(p\)-local finite spectrum is type \(n\) for some \(n\). This is because for \(i\gg 0\), the \(K(i)\)-based Atiyah-Hirzebruch spectral sequence degenerates for degree reasons for a fixed finite spectrum \(X\), so \(K(i)_*(X) = (H\FF _p)_*(X)\otimes _{(H\FF _p)_*}K(i)_* \neq 0\).

Remark 2.5. For a finite spectrum \(X\), \(K(m)_*X = 0 \implies K(m-1)_*(X) = 0\). This is essentially because its \(MU\)-homology is a coherent sheaf over \(\cM _{fg}\), so has closed support. This shows that for a type \(n\) spectrum \(X\), \(K(m)_*(X) \neq 0\) for \(m\geq n\).

Definition 2.6. Let \(C\) be a stable \(\infty \)-category. A thick subcategory \(C' \to C\) is a stable subcategory closed under retracts.

Example 2.7. Let \(\Sp ^{\omega }_{(p)}\) be the category of \(p\)-local finite spectra, and let \(\Sp _{\geq n}\) be the category of type \(\geq n\) spectra. Then \(\Sp _{\geq n} \to \Sp ^{\omega }_{(p)}\) is a thick subcategory.

It turns out these are all the examples. This is the content of the following result, which is a corollary of the nilpotence theorem, due to Hopkins and Smith.

Theorem 2.8 (Thick subcategory Theorem). Let \(C \subset \Sp ^\omega _{(p)}\) be a nonzero thick subcategory. Then \(C = \Sp _{\geq n}\) for some \(n\).

It is true that \(\Sp _{\geq n}\) are distinct as \(n\) varies, but showing this requires a bit more work.

Definition 2.9. Let \(X\) be a finite complex/spectrum. A \(v_n\)-self map \(v_n:\Sigma ^d X \to X\) is a map that
- 1. induces \(0\) on \(K(m)_*\) for \(m \neq n\).
- 2. induces an isomorphism on \(K(n)_*\).

The use of \(v_n\) as a name for the \(v_n\)-self map is slightly misleading: a more appropriate name is \(v_n^k\), because when they exist, they can be chosen to induce multiplication by a power of \(v_n\) on \(K(n)_*\).

Using a construction due to Smith, Hopkins and Smith proved the following result:

Theorem 2.10 (Periodicity Theorem). Every type \(n\) spectrum admits a \(v_n\)-self map.

From this theorem, it is easy to see why \(\Sp _{\geq n}\) are distinct. For example, the sphere \(\SP \) is a type \(0\) but not type \(1\) spectrum. Given a type \(n\) but not type \(n+1\) spectrum, we can take the cofibre of a \(v_n\)-self map to obtain a type \(n+1\) but not type \(n+2\)-spectrum, thereby inductively distinguishing the categories \(\Sp _{\geq n}\).

\(v_n\)-self maps are well behaved. After replacing one with a sufficiently large power, we can assume

• the \(v_n\)-self map induces multiplication by \(v_n^i\) on \(K(n)_*\).
• the \(v_n\)-self map is central in \(\End _*(X) = \pi _*X\otimes DX\).

Given a map of finite type \(n\)-spectra \(f:X \to Y\) equipped with a \(v_n\)-self map, we can replace the \(v_n\)-self maps by an iterate to make the diagram below commute:

(-tikz- diagram)

In this sense, \(v_n\)-self maps are almost functorial. Note that if we take \(f\) above to be the identity, we see that \(v_n\)-self maps are also unique up to taking iterations.