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Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer$\pi\in K$, and put $K_\infty:=K(\sqrt[p^\infty]{\pi})$. We introduce a newtechnique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformationrings. We show the existence of deformation rings for $\Gal(\olK/K_\infty)$-representations ``of height $\leqslant h$'' for any positiveinteger $h$, and we use them to give a variant of Kisin's proof of connectedcomponent analysis of a certain flat deformation rings, which was used to proveKisin's modularity lifting theorem for potentially Barsotti-Taterepresentations. Our proof does not use the classification of finite flat groupschemes, so it avoids Zink's theory of windows and displays when $p=2$.  This $\Gal(\ol K/K_\infty)$-deformation theory has a good analogue inpositive characteristics analogue of crystalline representations in the senseof Genestier-Lafforgue. In particular, we obtain a positive characteristicanalogue of crystalline deformation rings, and can analyze their localstructure.

Galois deformation theory for norm fields and flat deformation rings