Rational SU(3)-equivariant cohomology theories

TL;DR

This work completes the program of providing an algebraic framework for rational SU(3)-spectra by constructing the abelian category $\mathcal{A}(SU(3))$ and establishing a Quillen equivalence with DG-objects over this category. The authors achieve this by decomposing the Balmer spectrum $\mathfrak{X}_{SU(3)}=\mathrm{Sub}(SU(3))/SU(3)$ into 18 Zariski clopen blocks, each controlled by a subgroup and its Weyl-group data, then assembling the blockwise models into an overall algebraic model. The paper builds on a five-part sequence of prior works to address all 1D and 2D blocks and to classify new SU(3) subgroups not represented in $U(2)$, resulting in a complete 0-, 1-, and 2-dimensional block structure and explicit sheaf-of-rings models. The results provide a concrete, tractable algebraic handle on rational equivariant cohomology theories for SU(3) and its subgroups, enabling Quillen equivalences and potential generalizations via the Balmer-prismatic framework.

Abstract

We describe the spectral space of conjugacy classes of subgroups of SU(3), together with the additional structure of a sheaf of rings and a component structure. It is a disjoint union of 18 blocks each dominated by a subgroup. For each of these blocks we identify a sheaf of rings and component structure. Taken together, this gives an abelian category A(SU(3)) designed to reflect the structure of rational SU(3)-equivariant cohomology theories, and we assemble the results from elsewhere to show that the category of rational SU(3)-spectra is Quillen equivalent to the category of differential graded objects of A(SU(3)).

Theorems & Definitions (14)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Corollary 3.5
• Lemma 3.6
• Proposition 4.1
• Lemma 4.2
• Theorem 4.3
• Lemma 5.1