Remark 2

Tag 0CUN, permanent. Cite it as [Bourbaki, Tag 0CUN].

Commutative Algebra, chapter IX, § 1. Vecteurs de Witt

Let A be a ring of characteristic p. According to Prop. 5, one has the formulas

F_m^n[a_0, ..., a_{n+m-1}] = [a_0^{p^n}, ..., a_{m-1}^{p^n}]

p^n.[a_0, ..., a_{n+m-1}] = [\underbrace{0, ..., 0}_{n \text{ fois}}, a_0^{p^n}, ..., a_{m-1}^{p^n}]

for every Witt vector $[a_0, ..., a_{n+m-1}]$ of length $n + m$ .

In fact, the mapping $F : W(A) \to W(A)$ allows one, by passing to the quotients by $V_m(A)$ , to define a mapping $\overline{F}_m : W_m(A) \to W_m(A)$ . One has the formula

\overline{F}_m[a_0, ..., a_{m-1}] = [a_0^p, ..., a_{m-1}^p].

The mappings $V_m^1 \circ \overline{F}_m$ and $\overline{F}_{m+1} \circ V_m^1$ from $W_m(A)$ into $W_{m+1}(A)$ are equal and are obtained, by passing to the quotient, from multiplication by $p$ in $W_{m+1}(A)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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