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- W4362642449 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 $H^1_{fl}(mathcal{M}_{fmmathbb{Z}}; omega)$, 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 $H^2_{fl}(mathcal{M}_{fmA}^{2-buds}; omegaotimes omega)$ 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: $H^0_{fl}(mathcal{M}_{fmA}^{2-buds}; omegaotimes omega)$ is the delta-invariant of the largest ideal of $A$ which is in the kernel of every ring homomorphism $Arightarrow mathbb{F}_2$, and consequently $H^0_{fl}(mathcal{M}_{fmA}^{2-buds}; omegaotimes omega)$ vanishes if and only if $A$ is a ring in which that ideal is syzygetic." @default.
- W4362642449 created "2023-04-07" @default.
- W4362642449 creator A5031763507 @default.
- W4362642449 date "2010-05-01" @default.
- W4362642449 modified "2023-09-30" @default.
- W4362642449 title "Structure and cohomology of moduli of formal modules" @default.
- W4362642449 doi "https://doi.org/10.48550/arxiv.1005.0119" @default.
- W4362642449 hasPublicationYear "2010" @default.
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