Structure and cohomology of moduli of formal modules

Abstract

Given a commutative ring $A$, a "formal $A$-module" is a formal group equipped with an action of $A$. There exists a classifying ring $L^A$ of formal $A$-modules. This paper proves structural results about $L^A$ and about the moduli stack $\mathcal{M}_{fmA}$ of formal $A$-modules. We use these structural results to aid in explicit calculations of flat cohomology groups of $\mathcal{M}_{fmA}^{2-buds}$, the moduli stack of formal $A$-module $2$-buds. For example, we find that a generator of the group $H^1_{fl}(\mathcal{M}_{fm\mathbb{Z}}; ω)$, which also generates (via the Adams-Novikov spectral sequence) the first stable homotopy group of spheres, also yields a generator of the $A$-module $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds}; ω)$ for any torsion-free Noetherian commutative ring $A$. We show that the order of the $A$-modules $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds}; ω)$ and $H^2_{fl}(\mathcal{M}_{fmA}^{2-buds}; ω\otimes ω)$ are each equal to $2^{N_1}$, where $N_1$ is the leading coefficient in the $2$-local zeta-function of $Spec A$. We also find that the cohomology of $\mathcal{M}_{fmA}^{2-buds}$ is closely connected to the delta-invariant and syzygetic ideals studied in commutative algebra: $H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; ω\otimes ω)$ is the delta-invariant of the largest ideal of $A$ which is in the kernel of every ring homomorphism $A\rightarrow \mathbb{F}_2$, and consequently $H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; ω\otimes ω)$ vanishes if and only if $A$ is a ring in which that ideal is syzygetic.