  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
$H^1_{fl}(\mathcal{M}_{fm\mathbb{Z}}; \omega)$, which also generates (via the
Adams-Novikov spectral sequence) the first stable homotopy group of spheres,
also yields a generator of the $A$-module $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds};
\omega)$ for any torsion-free Noetherian commutative ring $A$. We show that the
order of the $A$-modules $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega)$ and
$H^2_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ are each equal to
$2^{N_1}$, where $N_1$ is the leading coefficient in the $2$-local
zeta-function of $Spec A$. 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:
$H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ 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
$H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ vanishes if and
only if $A$ is a ring in which that ideal is syzygetic.

Algebraic Topology;Algebraic Geometry