Structural properties of equivariant and motivic stable homotopy categories
Final Report Abstract
Algebraic topology is about studying geometric objects (curves, surfaces, ...) by assigning suitable algebraic invariants (numbers, groups, vector spaces, ...) to them and then to use algebraic methods to solve geometric problems. Most important algebraic invariants arise through generalized cohomology theories, which in turn are representable in the stable homotopy category SH. Hence, doing algebraic topology often means to study SH or variants like the stable homotopy category of p-local finite spectra SH^fin_(p), all of which are examples of tensor triangulated categories. Thick ideals in the stable homotopy category of p-local finite spectra, SH^fin_(p) , are given by SH^fin_(p) = C_0 > C_1 > ... > C_n > ... > {0} and each thick ideal is characterized by the vanishing of a particular Morava K-theory, that is, C_n = {X in SH^fin_(p) with K(p,n-1)_*(X) = 0.
In equivariant stable homotopy theory for a finite group G, unpublished work of Strickland from 2010 contains a partial classification of thick ideals in the category SH(G)_f < SH(G), which is the full subcategory of compact objects in the G-equivariant stable homotopy category. In our project, carried out by Ruth Joachimi as a PhD thesis, we study thick ideals in SH(k), k a subfield of the complex numbers C, and related motivic categories, like SH(k)^f_(p), the p-localisation of the full subcategory of all compact objects, and SH(k)^fin_(p), the category of p-localised finite cell spectra. We use different approaches. The first one is to use the comparison functors SH -> SH(k) -> SH for k a subfield of C, and also SH(Z/2) -> SH(k) -> SH(Z/2) for k a subfield of the reals. The second approach is to use methods of classical nilpotence theory. They do not carry over directly, as MGL does not detect nilpotence, and motivic Morava K-theories are more complicated than the topological ones. They, too, describe a certain family of thick ideals, but not all of them. The third approach is to find different lifts of topological type-n spectra to the motivic world and to study whether they generate the same thick ideals or different ones. In SH^fin_(p) , the thick ideals are ordered linearly by inclusion, due to the fact that K(n + 1)_*(X) = 0 implies K(n)_*(X) = 0 for X in SH^fin.
This raises the question whether this implication also holds in SH(k)^f . For p > 2, we prove that the analog statement holds for motivic Morava K-theories over C if X is a finite cellular motivic spectrum. That is, it holds for X in SH(C)^fin < SH(C)^f . On the way, we prove a couple of interesting facts concerning the motivic versions of BP, K(n) and related theories. We prove that the analog of the decomposition of Bousfield classes
Publications
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“Thick ideals in equivariant and motivic stable homotopy categories”, Doktorarbeit an der BUW 2015
Ruth Joachimi
