Nilpotence and stable homotopy theory. II.

*(English)*Zbl 0927.55015Algebraic topology studies a homotopy category \(H\) by applying a functor \(F:H\to A\) with \(A\) a comparatively simple algebraic category. The dream of an algebraic topologist is to select \(F\) so that the algebraic structure of \(A\) reflects the homotopy structure of \(H\). In the paper [E. S. Devinatz and the authors, Ann. Math., II. Ser. 128, 207-241 (1988; Zbl 0673.55008)] and the paper under review functors constructed from complex bordism are used to detect fundamental structures in the stable homotopy category.

The statements of the theorems of these two papers originated in 1977 with seven conjectures by D. C. Ravenel which were published in [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003)]. Ravenel conjectured that the algebraic structure exhibited by the chromatic spectral sequence reflects a corresponding structure in the stable homotopy category. Six of these conjectures were essentially proved in 1986 by Devinatz, Hopkins and Smith while Ravenel found a counterexample to his telescope conjecture in 1990 [Lond. Math. Soc. Lect. Note Ser. 176, 1-21 (1992; Zbl 0751.55006)].

The nilpotence theorem of paper I solves Ravenel’s first conjecture. Let \(F\) be a finite spectrum, and let \(X\) be a ring spectrum. Then \(f:\Sigma^kF\to F\) is composition nilpotent or \(f:S^m \to X\) is nilpotent in \(\pi_*X\) if and only if \(MU_*f=0\). Let \(K(n)\) be the \(n\)th Morava \(K\)-theory at a prime \(p\). In this paper the nilpotence theorem is refined for the \(p\)-local category of finite spectra \(C_0\): either of these two maps \(f\) is nilpotent if and only if \(K(N)*(f)\) is nilpotent for \(0\leq n\leq \infty\). Let \(C_n\) denote the full subcategory of \(C_0\) consisting of \(K(n-1)\) acyclic spectra. The nilpotence theorem is used to prove the thick subcategory theorem: every thick subcategory of \(C_0\) equals some \(C_n\). This theorem is used to show that a \(p\)-local finite spectrum \(X\) is in \(C_n\) if and only if it admits a \(v_n\)-self map. That is, there is a self map \(v:\Sigma^{p^N2 (p^n-1)} X\to X\) for \(N\) large such that \(K(m)_*(v)\) is multiplication by \(v_n^{p^N}\) if \(m=n\) and \(K(m)_* (v)=0\) for \(m\neq n\). Let \(\langle X\rangle\) denote the Bousfield class of \(X\). Let \(\text{Cl}(X)\) denote the set of \((n,p)\), with \(n\) a natural number and \(p\) prime, such that \(K(n)_*(X)\neq 0\) at the prime \(p\). Let \(X\) and \(Y\) be finite spectra. Ravenel’s class invariance conjecture follows easily: \(\langle X\rangle\leq\langle Y\rangle\) if and only if \(\text{Cl}(X) \subset \text{Cl}(Y)\).

The reader is referred to D. C. Ravenel’s book [Nilpotence and periodicity in stable homotopy theory, Ann. Math. Stud. 128 (1992; Zbl 0774.55001)] for a leisurely exposition of the results of these papers.

The statements of the theorems of these two papers originated in 1977 with seven conjectures by D. C. Ravenel which were published in [Am. J. Math. 106, 351-414 (1984; Zbl 0586.55003)]. Ravenel conjectured that the algebraic structure exhibited by the chromatic spectral sequence reflects a corresponding structure in the stable homotopy category. Six of these conjectures were essentially proved in 1986 by Devinatz, Hopkins and Smith while Ravenel found a counterexample to his telescope conjecture in 1990 [Lond. Math. Soc. Lect. Note Ser. 176, 1-21 (1992; Zbl 0751.55006)].

The nilpotence theorem of paper I solves Ravenel’s first conjecture. Let \(F\) be a finite spectrum, and let \(X\) be a ring spectrum. Then \(f:\Sigma^kF\to F\) is composition nilpotent or \(f:S^m \to X\) is nilpotent in \(\pi_*X\) if and only if \(MU_*f=0\). Let \(K(n)\) be the \(n\)th Morava \(K\)-theory at a prime \(p\). In this paper the nilpotence theorem is refined for the \(p\)-local category of finite spectra \(C_0\): either of these two maps \(f\) is nilpotent if and only if \(K(N)*(f)\) is nilpotent for \(0\leq n\leq \infty\). Let \(C_n\) denote the full subcategory of \(C_0\) consisting of \(K(n-1)\) acyclic spectra. The nilpotence theorem is used to prove the thick subcategory theorem: every thick subcategory of \(C_0\) equals some \(C_n\). This theorem is used to show that a \(p\)-local finite spectrum \(X\) is in \(C_n\) if and only if it admits a \(v_n\)-self map. That is, there is a self map \(v:\Sigma^{p^N2 (p^n-1)} X\to X\) for \(N\) large such that \(K(m)_*(v)\) is multiplication by \(v_n^{p^N}\) if \(m=n\) and \(K(m)_* (v)=0\) for \(m\neq n\). Let \(\langle X\rangle\) denote the Bousfield class of \(X\). Let \(\text{Cl}(X)\) denote the set of \((n,p)\), with \(n\) a natural number and \(p\) prime, such that \(K(n)_*(X)\neq 0\) at the prime \(p\). Let \(X\) and \(Y\) be finite spectra. Ravenel’s class invariance conjecture follows easily: \(\langle X\rangle\leq\langle Y\rangle\) if and only if \(\text{Cl}(X) \subset \text{Cl}(Y)\).

The reader is referred to D. C. Ravenel’s book [Nilpotence and periodicity in stable homotopy theory, Ann. Math. Stud. 128 (1992; Zbl 0774.55001)] for a leisurely exposition of the results of these papers.

Reviewer: Stanley O.Kochman (Downsview)