Structure and cohomology of moduli of formal modules

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 $H_{fl}^1({\mathcal{M}}_{fm\mathbb{Z}};\omega )$, 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}$_{fmA}^{2-buds}; \omega)$foranytorsion-freeNoetheriancommutativering$A$. We show that the order of the $A$-modules $H_{fl}^1({\mathcal{M}}_{fmA}^{2-buds};\omega )$ and $H_{fl}^2({\mathcal{M}}_{fmA}^{2-buds};\omega \otimes \omega )$ are each equal to $2^{N_1}$, where $N_1$ is the leading coefficient in the $2$-local zeta-function of $SpecA$. 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: $H_{fl}^0({\mathcal{M}}_{fmA}^{2-buds};\omega \otimes \omega )$ is the delta-invariant of the largest ideal of $A$ which is in the kernel of every ring homomorphism $A\to {\mathbb{F}}_2$, and consequently $H_{fl}^0({\mathcal{M}}_{fmA}^{2-buds};\omega \otimes \omega )$ vanishes if and only if $A$ is a ring in which that ideal is syzygetic.