Perfect Field - Perfect Closure and Perfection

Perfect Closure and Perfection

One of the equivalent conditions says that, in characteristic p, a field adjoined with all pr-th roots (r≥1) is perfect; it is called the perfect closure of k and usually denoted by .

The prefecture closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable if and only if is reduced.

In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring A_p of characteristic p together with a ring homomorphism u : A → A_p such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : A_p → B such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves "adjoining p-th roots of elements of A", similar to the case of fields.

The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system

where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists of sequences (x₀, x₁, ... ) of elements of A such that for all i. The map θ : R(A) → A sends (x_i) to x₀.