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Definition 28.3.1.

Let \(W\) be the ring scheme of \(p\)-typical Witt vectors. That is, for any ring \(A\text{,}\) there is a natural (in \(A\)) bijection between the underlying set of the ring \(W(A)\) and the set of morphisms \(\Spec A \to W\text{.}\)

Write \(W\) as \(\Spec \ZZ[x_0, x_1,\dots]\) in terms of the Witt coordinates, and let \(W_n = \Spec \ZZ[x_0,\dots,x_{n-1}]\) be the \(n\)-th truncation of \(W\text{.}\) For \(n \geq 1\text{,}\) let \(W_{\prim,n}\) be the completion of \(W_n\) along the locally closed subscheme defined by the conditions

\begin{equation*}
p = x_0 = 0, \qquad x_1 \neq 0.
\end{equation*}

Since \(W_n^\times\) acts on \(W_{\prim,n}\) by multiplication, we can form the quotient

\begin{equation*}
\CW_n = W_{\prim,n}/W_n^\times
\end{equation*}

in the category of sheaves on the category of \(p\)-adic formal schemes. Similarly, we may form the sheaf \(\CW = \lim_n \CW_n\text{,}\) called the Cartier-Witt stack.

By definition, for any oriented prism \((A, (d))\) we get a morphism \(\Spec A \to W_{\prim}\) under which the distinguished element \((x_0, x_1, \dots)\) of \(\calO(W_{\prim})\) pulls back to \(d\text{.}\) Consequently, for any prism \((A,I)\text{,}\) we get a morphism \(\Spec A \to \CW\text{.}\)