# , , and the ring of Witt vectors

No.12:

DEFINITION 12.1   Let be a ring of characteristic . We call the ring

the ring of Witt vectors of length . Its elements are called Witt vectors of lenth .

Note that

may be considered as a set with an unusual ring structure.

PROPOSITION 12.2

PROOF.. Since is a unital commutative ring, there naturally exists a natural ring homomorphism

Let us first fix a positive integer and examine the kernel of a map

where is the natural projection. Since

we have

In other words, for some integer . On the other hand, we have

thus

This implies that and therefore we have an inclusion

which turns to be a bijection ( ).

We then take a projective limit of the both hand sides and obtain the resired isomorphism.