Example 2

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

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

Let us make explicit the operations in $W_2(A)$ . We have

[a_0, a_1] + [b_0, b_1] = \left[ a_0 + b_0, a_1 + b_1 - \sum_{i=1}^{p-1} \frac{1}{p} \binom{p}{i} a_0^i b_0^{p-i} \right]

[a_0, a_1] \times [b_0, b_1] = [a_0 b_0, a_0^p b_1 + a_1 b_0^p + p \cdot a_1 b_1]

for $[a_0, a_1]$ and $[b_0, b_1]$ in $W_2(A)$ . The ghost components of $[a_0, a_1]$ are $a_0$ and $a_0^p + p \cdot a_1$ .

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

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