Algebraic chromatic homotopy theory for \(BP_\ast BP\)-comodules.

*(English)*Zbl 1412.55005The Adams-Novikov spectral sequence provides a close connection between stable homotopy theory and the homological algebra of \(BP_*BP\)-comodules. History has taught us that the cleanest way to do homological algebra is often to work directly on the derived level. The classical derived category \(\mathcal{D}_{BP_*BP}\) of the abelian category of \(BP_*BP\)-comodules has unpleasant properties (as e.g. the tensor unit \(BP_*\) is not compact in it). Thus the authors replace it by the stable category \(\mathrm{Stable}_{BP_*BP}\) of \(BP_*BP\)-comodules as introduced by Hovey. In their formulation it is defined as the ind-\(\infty\)-category on the dualizable objects in \(\mathcal{D}_{BP_*BP}\).

Besides basic formal properties about stable categories of Hopf algebroids, the results of the authors fall into two classes:

Firstly, they prove algebraic analogues of the Ravenel conjectures: An algebraic nilpotence theorem, an algebraic telescope conjecture and an algebraic chromatic convergence result. The nilpotence result has necessarily to be weaker than its topological counterpart though, essentially because the vanishing curve on the \(E_{\infty}\)-page of the Adams-Novikov spectral sequence is much better than on the \(E_2\)-term.

Secondly, they give cleaner proofs for some results in the homological algebra of \(BP_*BP\)-comodules, which simultaneously generalize them. We mention in particular their version of the chromatic spectral sequence, which is just a Bousfield-Kan spectral sequence associated to the tower of algebraic chromatic localizations. As a consequence of this spectral sequence, they prove in particular a nice comparison result between the \(BP\)-based Adams-Novikov spectral sequence for a bounded below spectrum whose \(BP\)-homology has finite projective dimension and its \(E_n\)-based one.

We remark that due to work of Isaksen, Gheorghe, Wang and Xu the \(\infty\)-category \(\mathrm{Stable}_{BP_*BP}\) embeds into motivic homotopy theory as modules over a certain motivic spectrum \(C\tau\). Thus, the algebraic analogues of the Ravenel conjectures in the present paper are promising to play a role in the exploration of chromatic homotopy theory in the motivic setting.

Besides basic formal properties about stable categories of Hopf algebroids, the results of the authors fall into two classes:

Firstly, they prove algebraic analogues of the Ravenel conjectures: An algebraic nilpotence theorem, an algebraic telescope conjecture and an algebraic chromatic convergence result. The nilpotence result has necessarily to be weaker than its topological counterpart though, essentially because the vanishing curve on the \(E_{\infty}\)-page of the Adams-Novikov spectral sequence is much better than on the \(E_2\)-term.

Secondly, they give cleaner proofs for some results in the homological algebra of \(BP_*BP\)-comodules, which simultaneously generalize them. We mention in particular their version of the chromatic spectral sequence, which is just a Bousfield-Kan spectral sequence associated to the tower of algebraic chromatic localizations. As a consequence of this spectral sequence, they prove in particular a nice comparison result between the \(BP\)-based Adams-Novikov spectral sequence for a bounded below spectrum whose \(BP\)-homology has finite projective dimension and its \(E_n\)-based one.

We remark that due to work of Isaksen, Gheorghe, Wang and Xu the \(\infty\)-category \(\mathrm{Stable}_{BP_*BP}\) embeds into motivic homotopy theory as modules over a certain motivic spectrum \(C\tau\). Thus, the algebraic analogues of the Ravenel conjectures in the present paper are promising to play a role in the exploration of chromatic homotopy theory in the motivic setting.

Reviewer: Lennart Meier (Bonn)