Irakli Patchkoria

On the algebraic classification of module spectra

Using methods developed by Franke in [7], we obtain algebraic classification results for modules over certain symmetric ring spectra (S ‐algebras). In particular, for any symmetric ring spectrum R whose graded homotopy ring R has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number N 4, we prove that the homotopy category of R‐modules is equivalent to the derived category of the homotopy ring R. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of R‐modules. The main examples of ring spectra to which our result applies are the p ‐local real connective K ‐theory spectrum ko.p/ , the Johnson‐Wilson spectrum E.2/, and the truncated Brown‐Peterson spectrum BPh1i, all for an odd prime p . We also show that the equivalences for all these examples are exotic in the sense that they do not come from a zigzag of Quillen equivalences. 18E30, 55P42, 55P43; 18G55

On exotic equivalences and a theorem of Franke

Using Frankes methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum

Rigidity in equivariant stable homotopy theory

R

Stable finiteness properties of infinite discrete groups

whose graded homotopy ring

Topological Hochschild homology and the cyclic bar construction in symmetric spectra

\pi_*R

The derived category of complex periodic K -theory localized at an odd prime

is concentrated in dimensions divisible by a natural number

Cubical Resolutions and Derived Functors

N \geq 5

Cubical approach to derived functors

and has homological dimension at most three, the homotopy category of

Rigidity and exotic models for

R

v_1

-modules is equivalent to the derived category of