In this talk, we give a variant of the proof of Kisin's connected component analysis of a certain flat deformation rings, which is one of the key technical steps in his proof of modularity lifting theorem for potentially Barsotti-Tate representations. The variant that will be presented at the talk does not use the ``Breuil-Kisin classification of finite flat group schemes,'' and it does not take a serious separate input to treat the case $p=2$. Note that to prove the classification of connected finite flat group schemes over $2$-adic dvr one needs Zink's theory of windows and displays, and Khare-Wintenberger's proof of Serre's conjecture uses the case $p=2$ of potentially Barsotti-Tate modularity lifting theorem.

The new ingredient in this variant is the Gal(Kbar/Kinfty)-deformation theory. Let K be a finite extension of $