Invariant prime ideals in equivariant Lazard rings

TL;DR

This work classifies invariant prime ideals of the $A$-equivariant Lazard ring $L_A$ for abelian compact Lie groups $A$, revealing that $ ext{Spec}^{ ext{inv}}(L_A)$ is a disjoint union over closed subgroups $B riangleleft A$ of copies of $ ext{Spec}^{ ext{inv}}(L)$. The authors define a canonical family of invariant primes $I^A_{B,p,n}$ via restriction to $L_B$ and localization at the geometric fixed points $ ext{Phi}^B L$, and prove a precise height-containment criterion: $I^A_{B,n} subseteq I^A_{B',n'}$ unless $B' riangleleft B$, $ ext{rank}_p( ext{π}_0(B/B'))$ is finite, and $n'\ge n+ ext{rank}_p( ext{π}_0(B/B'))$. They construct algebraic analogues of the classical $v_n$ elements, denoted $ar{ extbf{v}}_n$, whose behavior tracks chromatic height shifts in the $p$-toral and elementary abelian settings, and show that these elements (along with Euler classes) generate invariant primes in key cases. A detailed analysis of the Zariski topology and a comparison with the Balmer spectrum of compact $A$-spectra is given, via a universal $MU_A$-based support theory, culminating in a homeomorphism between $| ext{M}_{FG}^A|$ and $ ext{Spec}( ext{Sp}_A^c)$ and a parallel description of supports through equivariant formal groups. The results provide a robust algebraic framework for equivariant chromatic phenomena and establish a foundational link between formal group theory and equivariant stable homotopy theory. The methods illuminate how geometric fixed points govern height filtration and offer a blueprint for extending chromatic techniques to the equivariant setting.

Abstract

Let $A$ be an abelian compact Lie group. In this paper we compute the spectrum of invariant prime ideals of the $A$-equivariant Lazard ring, or equivalently the spectrum of points of the moduli stack of $A$-equivariant formal groups. We further show that this spectrum is homeomorphic to the Balmer spectrum of compact $A$-spectra, with the comparison map induced by equivariant complex bordism homology.

Figures (4)

• Figure 1: A schematic picture of $\widehat{\mathbb{G}}_m^{C_2}$ from \ref{['exa:Gm']}
• Figure 2: Decompositions of $\mathcal{M}_{FG}^{C_4}$ into open and closed substacks, using misty rose for open and lavender for closed. Every ellipse stands for the image of an immersion $\mathcal{M}_{FG}\to \mathcal{M}_{FG}^{C_4}$, as in \ref{['rem:stratification']}.
• Figure 3: Decompositions of $\mathcal{M}_{FG}^{C_2\times C_2}$ into open and closed substacks, $\Delta$ being the diagonal subgroup. Every circle stands for the image of an immersion $\mathcal{M}_{FG}\to \mathcal{M}_{FG}^{C_2\times C_2}$, as in \ref{['rem:stratification']}.
• Figure 4: A picture of $\mathop{\mathrm{Spec}}\nolimits^{\mathop{\mathrm{inv}}\nolimits}(L_{C_p})$, localized at $p$, including different choices of generators, with $i_j\colon C_p\to C_p^k$ denoting the $j$th canonical inclusion (generators arising from the further inclusions $C_p\to C_p^k$ are omitted). The yellow area depicts the closure of the point $I_{C_p,1}$.

Theorems & Definitions (143)

• Theorem 1.1
• Theorem 1.2: \ref{['thm:ClassificationOfInvariantPrimes']}
• Theorem 1.3: \ref{['thm:inclusions']}
• Theorem 1.4: \ref{['thm:generators1']}
• Theorem 1.5: \ref{['thm:zariski']}
• Theorem 1.6: \ref{['sec:universalsupport']}
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4