Demazure, lectures on p-divisible groups, III.3, the Witt rings over k
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Let be a file of prime characteristic . Let denote the Witt ring over Z?
The Witt ring over denoted by is defined by the coefficient extension . and
The phantom-components? reduce now to .
Since we can identify , see Definition Frobenius morphism, and and the Frobenius morphism becomes the endomorphism
This is a ring morphism since since commutes with products. Similar statements are true for and the affine -group defined in Artin-Hasse exponential series?.
The Verschiebung morphism? of is given by .
The Verschiebung morphism of is the translation? .
The Verschiebung morphism of is .
If , , then .
Let be perfect. Then
is a discrete valuation ring.
(Witt) Let be perfect, let be compete, noetherian local ring with residue field . Let be the canonical projection. There exists a unique ring morphism
which is compatible with the projections and .
If moreover is a discrete valuation ring with , then is a free finite -module of rank .
In particular if , then is an isomorphism.