Schemes for extending the theory of scheme extensions

Christopher French
Grinnell College



Content: We say a group $K$ is an extension of $G$ by $N$ if $K$ contains a normal subgroup isomorphic to $N$ with quotient isomorphic to $G$; such an extension then determines a ``twisting" homomorphism $\phi:G\to \text{Aut}(N)$. If, on the other hand, we are given groups $N$ and $G$, and a homomorphism $\phi:G\to\text{Aut}(N)$, the obstruction to finding an associated extension $K$ is an element in a third cohomology group. When the obstruction vanishes, a second cohomology group then parametrizes the family of such extensions, up to isomorphism. In full generality, the theory of extensions of association schemes seems much harder. Some work has been done to develop a cohomological approach to restricted versions of this problem, most notably by Bang and Hirasaka. In this talk, I build on this work, presenting some ideas for classifying extensions of schemes in greater generality.

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