Online seminar, U. Haifa Topology & Geometry seminar on December 6, 2020
A recording of the talk is available here.
Topology & Geometry Seminar
Speaker: Irakli Patchkoria, University of Aberdeen
Topic: Classification of module spectra and Franke’s algebraicity conjecture
Place: This is an online seminar. Please email David Blanc “blanc at math dot haifa dot ac dot il” for the Zoom ID and password.
Date: Sunday, December 6, 2020
This is all joint work with Piotr Pstrągowski. Given an E_1-ring R such that the graded homotopy ring pi_*R is q-sparse and the global projective dimension d of pi_*R is less than q, we show that the homotopy (q-d)-category of Mod(R) is equivalent to the homotopy (q-d)-category of differential graded modules over pi_*R. Thus for such E_1-rings the homotopy theory of their modules is algebraic up to the level (q-d). Examples include appropriate Morava K-theories, Johnson-Wilson theories, truncated Brown-Peterson theories and some variations of topological K-theory spectra. We also show that the result is optimal in the sense that (q-d) is the best possible level in general where algebraicity happens. At the end of the talk we will outline how the results for modules can be generalized to the settings where we do not have compact projective generators. This proves Franke’s algebraicity conjecture which provides a general result when certain nice homology theories provide algebraic models for homotopy theories.