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 Ap of characteristic p together with a ring homomorphism u : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → 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 (xi) to x0.
Read more about this topic: Perfect Field
Famous quotes containing the words perfect and/or perfection:
“I sit in the top of the wood, my eyes closed.
Inaction, no falsifying dream
Between my hooked head and hooked feet:
Or in sleep rehearse perfect kills and eat.”
—Ted Hughes (b. 1930)
“What journeyings on foot and on horseback through the wilderness, to preach the gospel to these minks and muskrats! who first, no doubt, listened with their red ears out of a natural hospitality and courtesy, and afterward from curiosity or even interest, till at length there “were praying Indians,” and, as the General Court wrote to Cromwell, the “work is brought to this perfection that some of the Indians themselves can pray and prophesy in a comfortable manner.””
—Henry David Thoreau (1817–1862)