# , , and the ring of Witt vectors

No.10:

LEMMA 10.1   Let be a commutative ring. Then:
1. For any , we have

2. If satisfies for some positive integer , then we have

3. Let be a positive integer. If satisfies

such that

then we have

such that

Recall that the ring of -adic Witt vectors is a quotient of the ring of universal Witt vectors. We have therefore a projection . But in the following we intentionally omit to write .

PROPOSITION 10.2   Let be a prime number. Let be a ring of characteristic. Then:
1. Every element of is written uniquely as

2. For any , we have

3. A map

is a ring homomorphism from to .
4. .
5. An element is invertible in if and only if is invertible in .

COROLLARY 10.3   If is a field of characteristic , then is a local ring with the residue field . If furthermore the field is perfect (that means, every element of has a -th root in ), then every non-zero element of may be writen as

(i.e. $x$:invertible)

Since any integral domain can be embedded into a perfect field, we deduce the following

COROLLARY 10.4   Let be an integral domain of characteristic . Then is an integral domain of characteristic 0 .

PROOF.. is always an injection when is.

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