A characterization of ramification groups via jet algebras

We present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an \(R\) $R$ -algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.