P-jets of finite algebras. II: \(p\)-typical Witt rings (Q374021)

Let \(p\) be a prime that is not \(2\) or \(3\) and let \(k\) be a perfect field of characteristic \(p\). For any \(W(k)\)-algebra \(C = W(k)[x]/(f)\) where \(f\) is a family of polynomials, we can define \(p\)-jet algebras of \(C\) NEWLINE\[NEWLINEJ^n(C) = \frac{W(k)[x,x',\dots,x^{(n)}]}{(f,\delta f,\dots, \delta^nf)}, \qquad J^{\infty}(C) = \frac{W(k)[x,x',x'',\dots]}{(f,\delta f, \delta^2 f, \dots )}NEWLINE\]NEWLINE where \(x,x',\dots\) are families of variables \(x=(x_{\alpha})_{\alpha \in \Omega}\), \(x' = (x'_{\alpha})_{\alpha \in \Omega}\), etc, indexed by the same set \(\Omega\), and \(\delta : W(k)[x,x',x'',\dots] \to W(k)[x,x',x'',\dots]\) is defined by \(\delta(F) = (\phi(F)-F^p)/p\) where \(\phi\) is the Frobenius map of \(W(k)\).NEWLINENEWLINELet \(C = W_m(W(k))\), we denote \(J^n(C)/p\) by \(\overline{J^n(C)}\). Let \(\pi = 1 -\delta(0,1,\dots) \in J^1(W_m(W(k)))\). The first main theorem of the subject paper gives a full description of the identity component of \(\overline{J^n(W_m(W(k)))}\) and shows that it is just the ring of fraction of \(\overline{J^n(W_m(W(k)))}\) with denominators powers of \(\pi\).NEWLINENEWLINEThe second main theorem gives a similar result for \(C = W_m(W_m'(W(k)))\), the iterated Witt rings.NEWLINENEWLINEThe third main theorem gives full description of the reduction mod \(p\) of the comonad map \(\Delta: W_{m+m'}(W(k)) \to W_m(W_m'(W(k)))\) NEWLINE\[NEWLINE\overline{J^{\infty}(\Delta)} : \overline{J^{\infty}(W_{m+m'}(W(k)))_{\pi}} \to \overline{J^{\infty}W_m(W_{m'}(W(k)))_{\Pi}}NEWLINE\]NEWLINE where the domain and the codomain are isomorphic to the identity component of \(\overline{J^{\infty}(W_{m+m'}(W(k)))_{\pi}}\) and \(\overline{J^{\infty}W_m(W_{m'}(W(k)))_{\Pi}}\) respectively by the first and second main theorems.NEWLINENEWLINEThe last main theorem states that a lower bound of the Krull dimension of the ring \((\overline{J^n(W_m(W(k)))})_{1-\pi}\) when \(n \geq 3\) and \(m \geq 2\) is \(2m-1\).