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Motivic homotopy of group scheme actions

To smooth schemes equipped with a smooth affine group scheme action, we associate an equivariant motivic homotopy category. Underlying our construction is the choice of an `equivariant Nisnevich topology' induced by a complete, regular, and bounded $cd$-structure. We show equivariant $K$-theory of smooth schemes is represented in the equivariant motivic homotopy category. This is used to characterize equivariantly contractible smooth affine curves and equivariant vector bundles on such curves. Generalizations of the purity and blow-up theorems in motivic homotopy theory are shown for actions of finite cyclic groups.