text:
\section{Introduction}

A commutative unital ring $A$ is called $K_n$-regular if the canonical map $K_n(A)\to
K_n(A[X_1,\ldots,X_m])$ is an isomorphism for all positive integers $m$.
By~\cite[Cor.~2.1]{Vorst} a $K_{n+1}$-regular ring is also $K_n$-regular.
It is well known that a regular noetherian ring is $K_{n}$-regular  for all $n$.
In \cite{Vorst} Vorst conjectured the following partial converse.

\begin{conj*}[Vorst]
Let $k$ be a field, and let $A$ be essentially of finite type over $k$.
If $A$ is $K_{\dim(A)+1}$-regular, then $A$ is regular.
\end{conj*}

The case $\dim(A)=0$ is easy and the case $\dim(A)=1$ was shown by Vorst in \cite[Thm.~3.6]{Vorst}.
For fields $k$ of characteristic zero, Corti\~nas, Haesemeyer, and Weibel proved the conjecture in \cite[Thm.~0.1]{CHW-Vorst}.
Geisser and Hesselholt in~\cite{GH-vorst} proved the conjecture for $A$ of finite type
over a  perfect fields $k$ of positive characteristic assuming resolution
of singularities.

In order to formulate our results, we introduce the $p$-dimension of an $\mathbb{F}_p$-algebra $A$.
This number is defined as
\[
\pdim(A) = \sup \{  \pdim(k(\mathfrak{p})) + \height(\mathfrak{p}) \, | \, \mathfrak{p} \subset A \text{ prime ideal}\}
\]
where $\pdim(k(\mathfrak{p}))$ is the $p$-rank of the residue field $k(\mathfrak{p})$, see Section~\ref{section:p-dim} for details.
In general $\pdim(A)\geq \dim(A)$ and equality holds for instance if $A$ is of finite type over a perfect field.

\begin{introthm}\label{introthm:A}
Let $A$ be an excellent noetherian $\mathbb{F}_p$-algebra such that $[k(x):k(x)^p]<\infty$ for all points $x\in\Spec(A)$.
If $A$ is $K_{\pdim(A)+1}$-regular, then $A$ is regular.
\end{introthm}

In particular, this implies the result of Geisser and Hesselholt mentioned above without assuming resolution of singularities.
The theorem indicates that the condition that $A$ be essentially of finite type over a field is not necessary for the conjecture to hold.
In fact, using the result of Corti\~nas--Haesemeyer--Weibel, we show the following generalization in characteristic zero.

\begin{introthm}\label{introthm:B}
Let $A$ be an excellent noetherian ring of characteristic zero. If $A$ is
$K_{\dim(A)+1}$-regular, then $A$ is regular.
\end{introthm}

We also prove the following result for curves  in mixed characteristic. This seems to be
the first result of that form in mixed characteristic.

\begin{introthm}\label{introthm:mixed-char}
Let $A$ be an excellent noetherian ring with $\dim(A)\leq 1$ such that $A/\mathfrak{m}$ is perfect of characteristic $p> 2$ for every maximal ideal $\mathfrak{m} \subset A$.
If $A$ is $K_{2}$-regular, then $A$ is regular.
\end{introthm}

These results motivate the following question, already asked similarly by Vorst in~\cite{Vorst}.

\begin{introquestion}
  Let $A$ be an excellent noetherian ring which is $K_{\dim(A)+1}$-regular. Is $A$
  necessarily regular?
  \end{introquestion}

The proof of Theorem~\ref{introthm:mixed-char} is based on the $\dlog$-map to (absolute) de
Rham--Witt forms and  calculations of Hesselholt--Madsen~\cite{HesselholtMadsen-drW}.
The proof of Theorem~\ref{introthm:A} essentially follows the strategy of
Geisser--Hessel\-holt. We replace resolution of singuliarities by an argument involving
the Zariski--Riemann space of $\Spec(A)$. This forces us to study $K$-theory of valuations
rings. In fact, we use the following vanishing result for the $K$-theory of valuation rings.

\begin{introthm}\label{introthm:C}
Let $V$ be a valuation ring of characteristic $p$ with field of fractions $F$.
Then $K_{i}(V; {\mathbb Z}/p) = 0$ for $i > \pdim(F)$.
\end{introthm}

Using a recent result of Clausen, Mathew, and Morrow \cite{ClausenMathewMorrow}, this follows from an analogous vanishing of topological cyclic homology (see Proposition~\ref{prop:tc-vanishing-on-valuation-rings}).
The main new ingredient is a Cartier isomorphism for valuation rings, a proof of which was outlined in a letter of Gabber to the first author~\cite{Gabber_letter}.
We give a detailed account of his argument in the appendix.

This existence of a Cartier isomorphism is also used in a recent preprint of Kelly and Morrow \cite{KellyMorrow}, where they independently prove the finer result $K_i(V;{\mathbb Z}/p^r) \cong W_r\Omega^i_{V,\log}$ by similar methods.

\subsection*{Acknowledgement}
We thank Ofer Gabber for sending us an outline of the proof of the Cartier isomorphism
mentioned above and for helpful comments on our presentation of his results in the appendix.

\subsection*{Notation}
All rings in this text are assumed to be commutative and unital.
For a presheaf of spectra $E$ and an integer $p$ we write $E/p$ for the cofibre of the $p$-multiplication on $E$ and set
$E_i(X;{\mathbb Z}/p)=\pi_i(E/p(X))$.

\section{Preliminaries on the \textit{p}-dimension}
\label{section:p-dim}

Let $p$ be a prime number.
In this section we introduce the notion of $p$-dimension for $\mathbb{F}_{p}$-algebras.
Let $R$ be an $\mathbb{F}_{p}$-algebra, and let $A$ be an $R$-algebra. By $R[A^{p}] \subset A$ we denote the $R$-subalgebra of $A$ generated by the $p$-th powers of the elements in $A$. Recall the following definition from \cite[$0_{\text{IV}}$(21.1.9)]{EGA}.

\begin{defn}\label{sec2.defpbase}
A family  $(x_{i})_{i\in I}$ of elements of $A$ is called a \emph{$p$-basis of $A$ over $R$} if the family of monomials
\begin{equation}
	\label{eq:p-monomials}
\prod_{i} x_{i}^{n_{i}} \quad (0 \leq n_{i} < p, n_{i} = 0 \text{ for all but finitely many } i \in I)
\end{equation}
forms a basis of $A$ as an $R[A^{p}]$-module. It is called \emph{$p$-independent over $R$} if the family \eqref{eq:p-monomials} is linearly independent over $R[A^{p}]$. The monomials \eqref{eq:p-monomials} are called the \emph{$p$-monomials} of the family $(x_{i})_{i\in I}$.
A $p$-basis for $A$ over $\mathbb{F}_{p}$ is called an \emph{absolute $p$-basis} or simply a \emph{$p$-basis for $A$}.
\end{defn}

We record the following simple observation.
\begin{lemma}
	\label{lem:p-basis-localization}
Assume that the family $(x_{i})_{i\in I}$ forms a $p$-basis of $A$ over $R$. If $S \subset A$ is a multiplicative set, then $(\frac{x_{i}}{1})_{i\in I}$ forms a $p$-basis of the localization $S^{-1}A$ over $R$.
\end{lemma}
\begin{proof}
To see that the family of $p$-monomials associated with $(\frac{x_{i}}{1})_{i \in I}$ generates $S^{-1}A$ as an $R[(S^{-1}A)^{p}]$-module, note that we can  write any element of $S^{-1}A$ in the form $\frac{a}{s^{p}}$ with $a\in A$ and $s\in S$. It is also easy to see that the family  $(\frac{x_{i}}{1})_{i \in I}$ is $p$-independent over $R$.
\end{proof}

If the family  $(x_{i})_{i\in I}$ is a $p$-basis of $A$ over $R$, then it is also a differential basis, i.e.~the family $(dx_{i})_{i\in I}$ is a basis of the  $A$-module of K\"ahler differentials $\Omega_{A/R}$, see \cite[$0_{\text{IV}}$ Cor.~21.2.5]{EGA}. The converse holds if $R \to A$ is a field extension \cite[Thm.~26.5]{Matsumura}. In particular, if $k' \subset k$ is any  extension of fields of characteristic $p$, then $k$ admits a $p$-basis over $k'$ and  any two $p$-bases have the same cardinality. This cardinality is called the \emph{$p$-rank of $k$ over $k'$}. The $p$-rank of $k$ over $\mathbb{F}_{p}$ is simply called the \emph{$p$-rank of k} and denoted by $\pdim(k)$. So $\pdim(k) = \dim_{k} \Omega_{k}$, where $\Omega_{(-)}= \Omega_{(-)/\mathbb{F}_{p}}$ denotes the module of absolute K\"ahler differentials.

\begin{lemma}
	\label{lem:p-dim-field-extension}
Assume that $k \subset k'$ is a finitely generated field extension in characteristic $p$.
Then
\[
\pdim(k') = \pdim(k) + \trdeg_{k} k'.
\]
In particular, $\pdim(k) = \pdim(k')$ in case the extension is finite.
\end{lemma}
\begin{proof}
This follows from the exact sequence
\[
0 \to \Gamma_{k'/k/\mathbb{F}_{p}} \to \Omega_{k} \otimes_{k} k' \to \Omega_{k'} \to \Omega_{k'/k} \to 0,
\]
where the imperfection module $\Gamma_{k'/k/\mathbb{F}_{p}}$ is defined as the kernel of the map in the middle,
together with the Cartier equality \cite[Thm.~26.10]{Matsumura}
\[
\dim_{k'} \Omega_{k'/k} - \dim_{k'} \Gamma_{k'/k/\mathbb{F}_{p}} = \trdeg_{k'} k,
\]
which holds since the field extension $k \subset k'$ is finitely generated.
\end{proof}

\begin{defn}
	\label{def:p-dim}
For an $\mathbb{F}_p$-scheme $X$ the \emph{$p$-dimension} is defined as
\[
\pdim(X) = \sup \{  \pdim(k(x)) + \dim(\mathcal{O}_{X,x}) \, | \, x \in X \}
\]
where $k(x)$ is the residue field of $X$ at $x$. For an $\mathbb{F}_{p}$-algebra $A$ we set $\pdim(A) = \pdim(\Spec(A))$.
\end{defn}

Note that  $\dim( \mathcal{O}_{\Spec(A), \mathfrak{p}} ) = \height(\mathfrak{p})$ and
\[
\pdim(A) = \sup \{  \pdim(A/\mathfrak{p}') \, | \, \mathfrak{p}' \subset A \text{ minimal prime ideal} \}.
\]

In the following, we collect some elementary properties of the $p$-dimension.

\begin{lemma}
	\label{lem:p-dim-for-p-basis}
If the noetherian $\mathbb{F}_{p}$-algebra $A$ is reduced and has an absolute $p$-basis of cardinality $r$, then $\pdim(A) = r$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:p-basis-localization} it is enough to show that
\[
\pdim(k) + \dim(A) = r
\]
provided $A$ is moreover local with residue field $k$. Since $A$ is reduced, the Frobenius map $a \mapsto a^{p}$ is injective. According to \cite[$0_{\text{IV}}$ Thm.~21.2.7]{EGA} the $\mathbb{F}_{p}$-algebra $A$ is then formally smooth and hence regular by Theorem~22.5.8 there. Let $\mathfrak{m}$ denote the maximal ideal of $A$. The claim now follows from the fundamental exact sequence \cite[Thm.~25.2]{Matsumura}
\[
0 \to \mathfrak{m}/\mathfrak{m}^{2} \to \Omega_{A} \otimes_{A} k \to \Omega_{k} \to 0
\]
using $\dim_{k} \mathfrak{m}/\mathfrak{m}^{2} = \dim(A)$ by the regularity of $A$ and $\dim_{k}(  \Omega_{A} \otimes_{A} k ) = r$ by the remarks preceding Lemma~\ref{lem:p-dim-field-extension}.
\end{proof}

\begin{lemma}
	\label{lem:p-dim-finite-extension}
Let $A\subset B$ be a finite extension of integral  $\mathbb{F}_p$-algebras. Then $\pdim(A)=\pdim(B)$.
\end{lemma}
\begin{proof}
For a prime ideal $\mathfrak{q}\subset B$ and $\mathfrak{p}=\mathfrak{q}\cap A\subset A$, we have the equality $\pdim(k(\mathfrak{q}))=\pdim(k(\mathfrak{p}))$ by Lemma~\ref{lem:p-dim-field-extension} as the residue field extension $k(\mathfrak{p})\subset k(\mathfrak{q})$ is finite.
Moreover, $\height(\mathfrak{q})\leq\height(\mathfrak{p})$, since $A \subset B$ is integral,  and hence $\pdim(B)\leq \pdim(A)$.
On the other hand, for every prime ideal $\mathfrak{p}\subset A$ there exists a prime ideal $\mathfrak{q}\subset B$ with $\mathfrak{p}=\mathfrak{q}\cap A$ and $\height(\mathfrak{q})\geq\height(\mathfrak{p})$ by going-up \cite[Thm.~9.4]{Matsumura}.
Therefore $\pdim(B)\geq \pdim(A)$.
This shows the claim.
\end{proof}

\begin{lemma}\label{lem:p-dim-and-dimension}
Let $A$ be either an $\mathbb{F}_p$-algebra of finite type over a field $k$ or a complete local noetherian $\mathbb{F}_{p}$-algebra with residue field $k$. Then
\[
\pdim(A) = \pdim(k) + \dim(A)
\]
where $\dim$ denotes the Krull dimension.
\end{lemma}
\begin{proof}
We may assume that $\pdim(k) < \infty$, since otherwise both sides of the equation are $\infty$.
By the remark after Definition~\ref{def:p-dim}, we may assume that $A$ is integral. Write $d$ for the Krull dimension of $A$.
In case $A$ is of finite type over a field $k$, Noether normalization \cite[(14.G)]{MatsumuraCA} yields a finite injective map $k[x_{1}, \dots, x_{d}] \to A$.
If $A$ is complete,  $A$  contains a coefficient field \cite[Thm.~28.3]{Matsumura}, and a choice of a system of parameters yields a finite injective map $k[[x_{1}, \dots, x_{d}]] \to A$.
By Lemma~\ref{lem:p-dim-finite-extension} it now suffices to show that
\[
\pdim(k[x_{1}, \dots, x_{d}] ) = \pdim(k[[x_{1}, \dots, x_{d}]] ) = \pdim(k) + d.
\]
But in view of Lemma~\ref{lem:p-dim-for-p-basis} this follows from the fact that if $b_{1}, \dots, b_{r}$ form a $p$-basis of $k$, then $b_{1}, \dots, b_{r}, x_{1}, \dots, x_{d}$ form a $p$-basis of $k[x_{1}, \dots, x_{d}]$ and of $k[[x_{1}, \dots, x_{d}]]$, see \cite[Lemma~2.1.5]{GabberOrgogozo}.
\end{proof}

\begin{lemma}\label{lem:p-dim-and-transcendence-degree}
Let $A \subset B$ be an extension of finite type of integral noetherian $\mathbb{F}_{p}$-algebras. Then
$\pdim(B) \leq \pdim(A) + \trdeg_{\Frac(A)} \Frac(B)$.
\end{lemma}
\begin{proof}
Let $\mathfrak{q} \subset B$ be a prime ideal, and let $\mathfrak{p} = \mathfrak{q} \cap A \subset A$. The dimension inequality \cite[Thm.~15.5]{Matsumura} gives
\[
\height(\mathfrak{q}) + \trdeg_{k(\mathfrak{p})} k(\mathfrak{q}) \leq \height(\mathfrak{p}) + \trdeg_{\Frac(A)} \Frac(B).
\]
Since the field extension $k(\mathfrak{p}) \subset k(\mathfrak{q})$ is finitely generated, Lemma~\ref{lem:p-dim-field-extension} implies that
\[
\pdim(k(\mathfrak{q})) = \pdim(k(\mathfrak{p})) + \trdeg_{k(\mathfrak{p})} k(\mathfrak{q}).
\]
Taken together, these facts imply the claim.
\end{proof}

\section{Derived differential forms and valuation rings}\label{sec:deriform}

Let $A$ be an $\mathbb{F}_{p}$-algebra.  Since the differential of the (absolute) de Rham complex $\Omega^{*}_{A}$ is Frobenius-linear, the subgroups of cycles
$Z\Omega^{i}_{A} \subset \Omega^{i}_{A}$ and boundaries $B\Omega^{i}_{A} \subset \Omega^{i}_{A}$, and the cohomology groups $H^{i}(\Omega^{*}_{A})$ are canonically $A$-modules via the Frobenius $a \mapsto a^{p}$.
There is a unique $A$-linear multiplicative map
\[
\gamma_{A} \colon \Omega^{i}_{A} \to H^{i}(\Omega^{*}_{A})
\]
characterized by $\gamma_{A}(1) = 1$ and $\gamma_{A}(dx) = x^{p-1}dx$, see \cite[Proof of Thm.~7.2]{Katz-nilpotent}. The map $\gamma_{A}$ is usually denoted by $C^{-1}$ and is called the \emph{inverse Cartier operator}.
The classical theorem of Cartier \cite[Thm.~7.2]{Katz-nilpotent} says that $\gamma_{A}$ is
an isomorphism provided that $A$ is smooth over a perfect field. If $\gamma_{A}$ is an isomorphism,  the inverse induces a map $C\colon Z\Omega^{i}_{A} \to \Omega^{i}_{A}$ called the \emph{Cartier operator}.
In the appendix (Corollary \ref{cor:Cartier-iso-valuation-ring}(iii)) we explain a proof
of the following
result due to Gabber~\cite{Gabber_letter}.
\begin{thm}
	\label{thm:Cartier-iso-valuation-ring}
Let $V$ be a valuation ring of characteristic $p$. Then the inverse Cartier operator $\gamma_{V}$ is an isomorphism.
\end{thm}

In the following, we need some nonabelian derived functors, see \cite[\S5.5.8]{htt} and \cite[Ch.~25]{sag} for a  general treatment.
We denote by $\mathrm{Poly}_{\mathbb{F}_{p}}$  the category of polynomial $\mathbb{F}_{p}$-algebras in finitely many variables, and by $\mathrm{SCR}_{\mathbb{F}_{p}}$  the $\infty$-category  obtained from the category of simplicial commutative $\mathbb{F}_{p}$-algebras by inverting the quasi-isomorphisms. Then $\mathrm{Poly}_{\mathbb{F}_{p}}$ is a full subcategory of $\mathrm{SCR}_{\mathbb{F}_{p}}$.  Moreover, if $D$ is any $\infty$-category that admits sifted colimits, then any functor $F\colon \mathrm{Poly}_{\mathbb{F}_{p}} \to D$  admits an essentially unique extension $LF\colon \mathrm{SCR}_{\mathbb{F}_{p}} \to D$ which preserves sifted colimits. The functor $LF$ is called the derived functor of $F$.

If $F$ is a functor $\mathrm{SCR}_{\mathbb{F}_{p}} \to D$, we still denote by $LF$ the derived functor of the restriction of $F$ to $\mathrm{Poly}_{\mathbb{F}_{p}}$. There is a natural transformation $LF \to F$. If the functor $F$ commutes with filtered colimits, then $LF(A) \simeq \colim_{\Delta^{\mathrm{op}}} F(P)$ where $P \xrightarrow{\sim} A$ is a simplicial resolution by free $\mathbb{F}_{p}$-algebras.

\begin{ex}
For $F = \Omega_{(-)}$, viewed as functor on $\mathrm{Poly}_{\mathbb{F}_{p}}$ with values in the derived  $\infty$-category $D(\mathbb{F}_{p})$ of  $H\mathbb{F}_{p}$-modules,
we obtain the functor $L\Omega_{(-)}\colon \mathrm{SCR}_{\mathbb{F}_{p}} \to D(\mathbb{F}_{p})$. For any $\mathbb{F}_{p}$-algebra $A$, $L\Omega_{A}$ is equivalent to the underlying $H\mathbb{F}_{p}$-module of the cotangent complex $\mathbb{L}_{A/\mathbb{F}_{p}} \in D_{+}(A)$, see \cite[\S25.3]{sag}.

Moreover, $L\Omega^{i}_{A}$ is equivalent to the underlying $H\mathbb{F}_{p}$-module of the derived exterior power $L\bigwedge^{i}_{A}\mathbb{L}_{A/\mathbb{F}_{p}}$, see \cite[\S25.2]{sag}. This follows directly from the constructions and the fact that for a polynomial $\mathbb{F}_{p}$-algebra $P$ the $P$-module $\Omega_{P}$ is free.
\end{ex}

The following result is essentially due to Gabber and Ramero.
\begin{thm}
	\label{thm:Gabber-Ramero-Omega-i}
Let $V$ be a valuation ring of characteristic $p$, and let $i \geq 0$. Then the following hold.
\begin{enumerate}
\item $L\Omega^{i}_{V} \simeq \Omega^{i}_{V}[0]$
\item The $V$-module $\Omega^{i}_{V}$ is torsion-free or, equivalently, $\Omega^{i}_{V}$  flat.
\end{enumerate}
\end{thm}

\begin{proof}
  For $i=0$ both claims are clear; for $i=1$, assertion (1) follows from
  \cite[Thm.~6.5.12]{GabberRamero} and (2) is Corollary 6.5.21 there. An alternative
  argument due to Gabber is explained in the appendix,
  see Corollary~\ref{cor:Cartier-iso-valuation-ring}. Let now $i \geq 2$. Since
  $\mathbb{L}_{V} \simeq \Omega_{V}[0]$ and $\Omega_{V}$ is torsion-free and hence flat, it
  follows that
\[
L\Omega^{i}_{V} \simeq L\bigwedge\nolimits^{i}_{V}\mathbb{L}_{V} \simeq \bigwedge\nolimits^{i}_{V}\Omega_{V} = \Omega^{i}_{V}[0],
\]
see \cite[Prop.~25.2.3.4]{sag}.
It remains to prove (2) for $i \geq 2$. As $\Omega_{V}$ is  flat, it is isomorphic to a filtered colimit of free modules by Lazard's theorem. Since exterior powers of free modules are free, and since taking exterior powers commutes with filtered colimits, $\Omega^{i}_{V}$ is a filtered colimit of free modules and hence flat.
\end{proof}

We want to prove the analog of Theorem~\ref{thm:Gabber-Ramero-Omega-i} for the de Rham--Witt groups. For this we need some preparations.
\begin{lemma}
	\label{lem:derived-BOmega}
Let $V$ be a valuation ring of characteristic $p$. Then $LB\Omega^{i}_{V} \simeq B\Omega^{i}_{V}[0]$ and $LZ\Omega^{i}_{V} \simeq Z\Omega^{i}_{V}[0]$
\end{lemma}
\begin{proof}
By the Cartier isomorphism, we have the following short exact sequence of functors on $\mathrm{Poly}_{\mathbb{F}_{p}}$:
\[
0 \to B\Omega^{i} \to Z\Omega^{i} \xrightarrow{C} \Omega^{i} \to 0
\]
Taking derived functors and evaluating at $V$ we obtain the upper cofibre sequence in the following diagram, where the vertical maps are the canonical ones.
\begin{equation*}
\begin{tikzcd}
LB\Omega^{i}_{V} \arrow[r]\arrow[d] & LZ\Omega^{i}_{V} \arrow[r]\arrow[d] & L\Omega^{i}_{V} \arrow[d, "\simeq"]\\
B\Omega^{i}_{V}[0] \arrow[r]& Z\Omega^{i}_{V}[0] \arrow[r] & \Omega^{i}_{V}[0]
\end{tikzcd}
\end{equation*}
The lower line is a cofibre sequence by Theorem~\ref{thm:Cartier-iso-valuation-ring} and the right vertical map is an equivalence by Theorem~\ref{thm:Gabber-Ramero-Omega-i}. So we see that if we prove the assertion about $B\Omega^{i}$, then also the assertion about $Z\Omega^{i}$ follows. We now argue by induction on $i$. As $B\Omega^{0} = 0$, the assertion is clearly true in the case $i = 0$.
Similarly as above, the exact sequence of functors
\[
0 \to Z\Omega^{i} \to \Omega^{i} \xrightarrow{d} B\Omega^{i+1} \to 0
\]
gives rise to the following diagram of cofibre sequences
\[
\begin{tikzcd}
LZ\Omega^{i}_{V} \arrow[r]\arrow[d] & L\Omega^{i}_{V} \arrow[r]\arrow[d, "\simeq"] & LB\Omega^{i+1}_{V} \arrow[d] \\
Z\Omega^{i}_{V}[0] \arrow[r] & \Omega^{i}_{V}[0] \arrow[r] & B\Omega^{i+1}_{V}[0].
\end{tikzcd}
\]
By induction, the left vertical map is an equivalence, hence so is the right vertical map.
\end{proof}

We next recall the definition of the higher boundaries and cycles in the de Rham complex from \cite[I.2.2]{Illusie-drW}. Let  $A$ be an  $\mathbb{F}_{p}$-algebra for which the inverse Cartier operator $\gamma_{A}\colon \Omega^{i}_{A} \to H^{i}(\Omega^{*}_{A})$ is an isomorphism, for example a polynomial algebra or a valuation ring.
One defines the chain of subgroups
\begin{multline*}
0 = B_{0}\Omega^{i}_{A} \subset B_{1}\Omega^{i}_{A} \subset \cdots \subset B_{n}\Omega^{i}_{A} \subset \cdots \\
	\cdots \subset Z_{n}\Omega^{i}_{A} \subset \cdots \subset Z_{1}\Omega^{i}_{A} \subset Z_{0}\Omega^{i}_{A} = \Omega^{i}_{A}
\end{multline*}
inductively by setting $B_{1}\Omega^{i}_{A} = B\Omega^{i}_{A}$, $Z_{1}\Omega^{i}_{A} = Z\Omega^{i}_{A}$ and requiring that $\gamma$ induces isomorphisms
\begin{equation}
	\label{eq:def:higher-cycles-and-boundaries}
B_{n}\Omega^{i}_{A} \xrightarrow{\simeq} B_{n+1}\Omega^{i}_{A}/B\Omega^{i}_{A} \quad  \text{ and } \quad
Z_{n}\Omega^{i}_{A} \xrightarrow{\simeq} Z_{n+1}\Omega^{i}_{A}/B\Omega^{i}_{A}.
\end{equation}
Since the $B_{n}\Omega^{i}$ and $Z_{n}\Omega^{i}$ define functors on $\mathrm{Poly}_{\mathbb{F}_{p}}$, we get
 derived functors $LB_{n}\Omega^{i}$ and $LZ_{n}\Omega^{i}$ defined on $\mathrm{SCR}_{\mathbb{F}_{p}}$. In particular, we may evaluate them on any $\mathbb{F}_{p}$-algebra.

\begin{lemma}
	\label{lem:derived-higher-boundaries}
Let $V$ be a valuation ring of characteristic $p$. Then $LB_{n}\Omega^{i}_{V} \simeq B_{n}\Omega^{i}_{V}[0]$ and $LZ_{n}\Omega^{i}_{V} \simeq Z_{n}\Omega^{i}_{V}[0]$.
\end{lemma}
\begin{proof}
We argue by induction on $n$. The case $n=1$ is done in Lemma~\ref{lem:derived-BOmega}. By definition, we have the following exact sequence of functors on $\mathrm{Poly}_{\mathbb{F}_{p}}$:
\[
0 \to B\Omega^{i} \to B_{n+1}\Omega^{i} \xrightarrow{C} B_{n}\Omega^{i} \to 0
\]
Taking derived functors and evaluating at $V$ gives the inductive step, similarly as in the proof of Lemma~\ref{lem:derived-BOmega}. The proof for $Z_{n}\Omega^{i}$ is the same.
\end{proof}

Next we recall that to any $\mathbb{F}_{p}$-algebra $A$ one functorially associates its de Rham--Witt pro-complex $\{W_{n}\Omega^{*}_{A}\}_{n}$, see \cite{Illusie-drW}, \cite[Thm.~A]{HesselholtMadsen-drW}. The structure maps of the pro-system are denoted by $R$ and are called restriction maps. It follows directly from \cite[Thm.~I.1.3]{Illusie-drW} that the restriction maps are surjective.
We view $W_{n}\Omega^{i}$  as a functor on $\mathbb{F}_{p}$-algebras with values in $D({\mathbb Z})$. So we have its derived functor available.

\begin{prop}
	\label{prop:Witt-analog-of-Gabber-Ramero}
Let $V$ be a valuation ring of characteristic $p$ with field of fractions $F$, and let $n\geq 1$ and $i \geq 0$. Then the following hold.
	\begin{enumerate}
	\item $LW_{n}\Omega^{i}_{V} \simeq W_{n}\Omega^{i}_{V}[0]$
	\item The natural map $W_{n}\Omega^{i}_{V} \to W_{n}\Omega^{i}_{F}$ is injective.
	\end{enumerate}
\end{prop}
\begin{proof}
(1) As a first step, we  treat the case $i=0$. Note that $W_{n}\Omega^{0}$ is the ring $W_{n}$ of Witt vectors of length $n$. We claim that in fact $LW_{n}(A) \simeq W_{n}(A)[0]$ for any $\mathbb{F}_{p}$-algebra $A$. This is clear for $n=1$. For $n >1$ the claim follows by induction using the short exact sequence
\[
0 \to A \xrightarrow{V^{n}} W_{n+1}(A) \to W_{n}(A) \to 0
\]
which is natural in $A$.

We next prove that $\pi_{0}LW_{n}\Omega^{i}_{A} = W_{n}\Omega^{i}_{A}$ for any $\mathbb{F}_{p}$-algebra $A$.
Let $P \xrightarrow{\sim} A$ be a simplicial resolution of $A$ by free $\mathbb{F}_{p}$-algebras.
We have to  show that the sequence
\[
 W_{n}\Omega^{i}_{P_{1}} \xrightarrow{\partial_{0} - \partial_{1}} W_{n}\Omega^{i}_{P_{0}} \to W_{n}\Omega^{i}_{A} \to 0
 \]
is exact. According to \cite[Lemma~2.4]{GeisserHesselholtComplete}, the  right-hand map is surjective and its kernel is generated by elements of the form $x\cdot \omega$ and $dx\cdot\omega$ with $x \in \ker( W_{n}(P_{0}) \to W_{n}(A))$ and $\omega \in W_{n}\Omega^{i}_{P_{0}}$ or $\omega\in W_{n}\Omega^{i-1}_{P_{0}}$, respectively. By the first step of the proof we know that $W_{n}(P) \xrightarrow{\sim} W_{n}(A)$ is a simplicial resolution. In particular, for $x$ as above there exists an element $y \in W_{n}(P_{1})$ such that $\partial_{0}(y) = x$, $\partial_{1}(y) = 0$. If $s$ denotes the degeneracy map $W_{n}(P_{0}) \to W_{n}(P_{1})$, then $y\cdot s(\omega)$ respectively $dy\cdot s(\omega)$ is a preimage of $x\cdot\omega$ respectively $dx\cdot\omega$ under $\partial_{0} - \partial_{1}$, thus showing the desired exactness.

It remains to prove that $LW_{n}\Omega^{i}_{V}$ is concentrated in degree 0 for $i \geq 1$ and $n \geq 1$.
We argue by induction on $n$. Since $W_{1}\Omega^{i} \cong \Omega^{i}$ (see \cite[Thm.~I.1.3]{Illusie-drW}), the case $n=1$ follows from Theorem~\ref{thm:Gabber-Ramero-Omega-i}(1).
Assume our assertion  is proven for some $n\geq 1$.
Recall from \cite[I.3.1]{Illusie-drW} the canonical filtration on $W\Omega^{i}_{A} = \lim_{n} W_{n}\Omega^{i}_{A}$ given by
\[
\Fil^{n}W\Omega^{i}_{A} = \ker ( W\Omega^{i}_{A} \xrightarrow{\text{can.}} W_{n}\Omega^{i}_{A} )
\]
 for any smooth $\mathbb{F}_{p}$-algebra $A$. Its associated graded pieces sit in a short exact sequence
\begin{equation}
	\label{seq:gr-drW}
	0 \to \gr^{n}W\Omega^{i}_{A} \to W_{n+1}\Omega^{i}_{A} \xrightarrow{R} W_{n}\Omega^{i}_{A} \to 0.
\end{equation}
Viewed  as a short exact sequence of functors in the  smooth $\mathbb{F}_{p}$-algebra $A$, the latter
gives rise to a cofibre sequence of derived functors
\begin{equation}
	\label{seq:cofibre-sequence-1}
L\gr^{n}W\Omega^{i} \to LW_{n+1}\Omega^{i} \to LW_{n}\Omega^{i}.
\end{equation}
By induction, it now suffices to show that $L\gr^{n}W\Omega^{i}_{V}$ is concentrated in degree 0.
By a fundamental result of Illusie  \cite[Cor.~I.3.9]{Illusie-drW}, there is a natural short exact sequence
\begin{equation}
	\label{seq:Illusie-fundamental-sequence}
	0 \to \Omega^{i}_{A} / B_{n}\Omega^{i}_{A} \to \gr^{n}W\Omega^{i}_{A} \to \Omega^{i-1}_{A} / Z_{n}\Omega^{i-1}_{A} \to 0
\end{equation}
for any smooth $\mathbb{F}_{p}$-algebra $A$. By the same argument as before, we now finish the proof of (1) by noting that $L(\Omega^{i}/B_{n}\Omega^{i})_{V}$ and $L(\Omega^{i-1}/Z_{n}\Omega^{i-1})_{V}$ are concentrated in degree 0. Indeed, this follows immediately from Lemma~\ref{lem:derived-higher-boundaries}  together with Theorem~\ref{thm:Gabber-Ramero-Omega-i}(1).

\medskip

\noindent (2) We again argue by induction on $n$. The case $n=1$ is Theorem~\ref{thm:Gabber-Ramero-Omega-i}(2). Note that by part (1) applied to the trivial valuation ring  $F$, we also have $LW_{n}\Omega^{i}_{F} \simeq W_{n}\Omega^{i}_{F}$ for all $n \geq 1$ and $i \geq 0$ and Lemma~\ref{lem:derived-higher-boundaries} holds with $V$ replaced by $F$. Using the cofibre sequence \eqref{seq:cofibre-sequence-1} and the result of part (1), we see that for the inductive step it suffices to show that
\[
\pi_{0} L\gr^{n}W\Omega^{i}_{V} \to \pi_{0} L\gr_{n}W\Omega^{i}_{F}
\]
is injective for all $n \geq 0$. Then, using the cofibre sequence of derived functors obtained from \eqref{seq:Illusie-fundamental-sequence}, we  reduce to proving that the maps
\[
\Omega^{i}_{V}/B_{n}\Omega^{i}_{V} \to \Omega^{i}_{F}/B_{n}\Omega^{i}_{F} \quad \text{ and } \quad \Omega^{i}_{V}/Z_{n}\Omega^{i}_{V} \to \Omega^{i}_{F}/Z_{n}\Omega^{i}_{F}
\]
are injective for any $i \geq 0$ and $n \geq 1$.

We now prove the latter statement for the higher cycles by induction on $n$. To simplify notation, we drop the index $V$ or $F$, whenever a statement holds for both of them. In the  case $n=1$, the desired injectivity follows from the injectivity of $d\colon \Omega^{i}/Z\Omega^{i} \to \Omega^{i+1}$ and of $\Omega^{i+1}_{V} \to \Omega^{i+1}_{F}$, see Theorem~\ref{thm:Gabber-Ramero-Omega-i}(2). Assume, we have proven injectivity for some $n$. It follows from the definition of higher cycles \eqref{eq:def:higher-cycles-and-boundaries} that the inverse Cartier operator $\gamma$ induces an isomorphism
\[
\Omega^{i}/Z_{n}\Omega^{i} \xrightarrow{\cong} Z\Omega^{i}/Z_{n+1}\Omega^{i}.
\]
Combining this isomorphism with the exact sequence
\[
0 \to Z\Omega^{i}/Z_{n+1}\Omega^{i} \to \Omega^{i}/Z_{n+1}\Omega^{i} \to \Omega^{i}/Z\Omega^{i} \to 0
\]
and the injectivity for $n=1$ gives the inductive step.
The proof for higher boundaries is  the same, replacing in the above formulas $Z_{n}$ by $B_{n}$ and $Z_{n+1}$ by $B_{n+1}$, and using the injection $\Omega^{i}/B\Omega^{i} \hookrightarrow \Omega^{i}$ given by the Cartier operator for the case $n=1$.
\end{proof}

\begin{cor}
	\label{cor:dRW-p-torsion-free}
The $p$-multiplication $p\colon W_{n}\Omega^{i}_{V} \to W_{n}\Omega^{i}_{V}$ factors as
\[
W_{n}\Omega^{i}_{V} \xrightarrow{R} W_{n-1}\Omega^{i}_{V} \xrightarrow{\ul{p}} W_{n}\Omega^{i}_{V}
\]
and the induced map $\ul{p}$ is injective. In particular, the pro-group $\{ W_{n}\Omega^{i}_{V}\}_{n}$ is $p$-torsion-free.
\end{cor}
\begin{proof}
For $V$ replaced by any smooth $\mathbb{F}_{p}$-algebra the same assertion is proved in \cite[Prop.~I.3.4]{Illusie-drW} and remains true for ind-smooth $\mathbb{F}_{p}$-algebras.
Hence the $p$-multiplication on the derived functor $LW_{n}\Omega^{i}$ factors as
\[
LW_{n}\Omega^{i} \xrightarrow{R} LW_{n-1}\Omega^{i} \xrightarrow{\ul{p}} LW_{n}\Omega^{i}
\]
Evaluating on $V$ and using Proposition~\ref{prop:Witt-analog-of-Gabber-Ramero}(1) gives the desired factorisation.
Using part (2) of the proposition and Illusie's result for the ind-smooth $\mathbb{F}_{p}$-algebra $F$ proves the asserted injectivity.
\end{proof}

We will use the above results to prove a vanishing result for topological cyclic homology of valuation rings in characteristic $p$.
Recall that for any (simplicial) $\mathbb{F}_{p}$-algebra $A$ one defines the spectra
\[
\TR^{n}(A;p) = \THH(A)^{C_{p^{n-1}}}
\]
as the genuine fixed points of the  topological Hochschild homology spectrum $\THH(A)$.
There are natural  maps $R, F \colon \TR^{n}(A;p) \to \TR^{n-1}(A;p)$ called restriction and Frobenius, and one defines the spectrum $\TR(A;p)  = \lim_{R} \TR^{n}(A;p)$. The topological cyclic homology of $A$ then sits in a fibre sequence
\begin{equation}
	\label{seq:def-TC}
\TC(A) \to \TR(A;p) \xrightarrow{1-F} \TR(A;p).
\end{equation}
Hesselholt--Madsen prove in \cite[Prop.~5.4]{HesselholtMadsenWitt} that $\TR_{*}^{n}(\mathbb{F}_{p})$ is isomorphic to the polynomial ring ${\mathbb Z}/p^{n}[\sigma_{n}]$ with $\sigma_{n}$ of degree 2 and the restriction map sends $\sigma_{n}$ to $p\sigma_{n-1}$ up to a unit in ${\mathbb Z}/p^{n-1}$.
Hesselholt shows in \cite{HesselholtTypicalCurves} that for  any $\mathbb{F}_{p}$-algebra $A$ and every $n\geq 1$ one gets a naturally induced map of graded rings
\begin{equation}
	\label{eq:canonical-map-drW-TR}
W_{n}\Omega^{*}_{A} [\sigma_{n}] \to  \TR_{*}^{n}(A;p)
\end{equation}
and that \eqref{eq:canonical-map-drW-TR} is an isomorphism provided that $A$ is smooth over $\mathbb{F}_{p}$, see \cite[Thm.~B]{HesselholtTypicalCurves}. Since both sides of \eqref{eq:canonical-map-drW-TR} commute with filtered colimits, \eqref{eq:canonical-map-drW-TR} is an isomorphism if $A$ is only assumed to be ind-smooth over $\mathbb{F}_{p}$.

\begin{lemma}
	\label{lem:dRW-TR-iso}
Let $V$ be a valuation ring of characteristic $p$. Then the map
\[
W_{n}\Omega^{*}_{V} [\sigma_{n}] \to  \TR_{*}^{n}(V;p)
\]
from \eqref{eq:canonical-map-drW-TR} is an isomorphism of graded rings. In particular, the natural map of pro-groups
$\{W_{n}\Omega^{i}_{V} \}_{n} \to \{ \TR^{n}_{i}(V;p)\}_{n}$ is an isomorphism.
\end{lemma}
\begin{proof}
Let $P \xrightarrow{\sim} V$ be a simplicial resolution by free $\mathbb{F}_{p}$-algebras. As the spectrum valued functor $\TR^{n}(-;p)$ on  $\mathrm{SCR}_{\mathbb{F}_{p}}$ commutes with sifted colimits (this follows inductively from the basic cofibre sequence \cite[(1.3.10)]{HesselholtTypicalCurves}), we have an equivalence
\(
\colim_{\Delta^{\mathrm{op}}} \TR^{n}(P;p) \simeq \TR^{n}(V;p)
\)
and hence a convergent spectral sequence
\[
E^{1}_{rs} = \TR^{n}_{s}(P_{r};p) \Longrightarrow \TR^{n}_{r+s}(V;p).
\]
By Hesselholt's result above,  we have  $W_{n}\Omega^{s}_{P_{r}}[\sigma_{n}] \cong E^{1}_{rs}$ via the canonical map, and hence
$\pi_{r}(LW_{n}\Omega^{s}_{V}[\sigma_{n}]) \cong E^{2}_{rs}$ by the definition of derived functors.
So Proposition~\ref{prop:Witt-analog-of-Gabber-Ramero} implies $E^{2}_{rs} = 0$ for $r > 0$ and $W_{n}\Omega^{s}_{V}[\sigma_{n}] \cong E^{2}_{0s}$. This gives the first claim. The second statement follows immediately from the fact that the transition map sends $\sigma_{n}$ to $p\sigma_{n-1}$ up to a unit.
\end{proof}

\begin{rmk}
With a  similar approach, the result of Lemma~\ref{lem:dRW-TR-iso} has been shown by Kelly and Morrow \cite[\S 2.3]{KellyMorrow} for Cartier-smooth $\mathbb{F}_{p}$-algebras.
\end{rmk}

\begin{prop}\label{prop:tc-vanishing-on-valuation-rings}
Let $V$ be a valuation ring of characteristic $p$ with field of fractions $F$. Then $TC_{i}(V; {\mathbb Z}/p) = 0$ for $i > \pdim(F)$.
\end{prop}
\begin{proof}
In view of the fibre sequence \eqref{seq:def-TC} and the Milnor sequence for $\TR$, it suffices to show that the pro-group $\{ \TR^{n}_{i}(V;p,{\mathbb Z}/p) \}_{n}$ vanishes for $i > \pdim(F)$. From Corollary~\ref{cor:dRW-p-torsion-free} and Lemma~\ref{lem:dRW-TR-iso} we deduce an isomorphism $\{ W_{n}\Omega^{i}_{V} / p \}_{n} \cong \{ \TR^{n}_{i}(V;p,{\mathbb Z}/p) \}_{n}$. Since the map $W_{n}\Omega^{i}_{V} \to W_{n}\Omega^{i}_{F}$ is injective  by Proposition~\ref{prop:Witt-analog-of-Gabber-Ramero}, it now suffices to observe that $W_{n}\Omega^{i}_{F} = 0$ for $i>\pdim(F)$. Indeed, since $F$ is ind-smooth over $\mathbb{F}_{p}$ and $\Omega^{i}_{F}$ vanishes for $i>\pdim(F)$ this follows easily by induction using the exact sequences~\eqref{seq:gr-drW} and \eqref{seq:Illusie-fundamental-sequence} (note that $\Omega^{i}_{F} = Z\Omega^{i}_{F} = \dots = Z_{n}\Omega^{i}_{F}$ for $i \geq \pdim(F)$).
\end{proof}

The following consequence is Theorem~\ref{introthm:C} from the introduction.
\begin{cor}\label{cor:vanishing-on-valuation-rings}
Let $V$ be a valuation ring of characteristic $p$ with field of fractions $F$. Then $K_{i}(V; {\mathbb Z}/p) = 0$ for $i > \pdim(F)$.
\end{cor}
\begin{proof}
As $V$ is a local ring, the cyclotomic trace $K_{i}(V;{\mathbb Z}/p)\to TC_{i}(V;{\mathbb Z}/p)$ is injective by \cite[Thm.~D]{ClausenMathewMorrow} and the statement follows from the previous proposition.
\end{proof}

\begin{rmk}
In case $V=F$ is a field Corollary~\ref{cor:vanishing-on-valuation-rings} was proved
 for $\pdim (K)=0 $ in~\cite[Cor.~5.5]{Kratzer} and in~\cite{GeisserLevine} in general.
\end{rmk}

\section{The main result}

\begin{lemma}\label{lem:non-vanishing}
Let $k$ be a field of characteristic $p$, and let $B$ be the $k$-algebra $B=k[x_{1}, \dots, x_{r}]/(x_{1}, \dots, x_{r})^{2}$.
Let $y_{1}, \dots, y_{s} \in k$ be $p$-independent. Then the  symbol $\{y_{1}, \dots, y_{s}, 1+x_{1}, \dots, 1+x_{r}\} \in K_{s+r}(B)$ does not vanish in $K_{s+r}(B)/pK_{s+r}(B)$.
\end{lemma}
\begin{proof}
It suffices to check that  the image of the symbol under the Dennis trace map $K_{s+r}(B) \to \HH_{s+r}(B/\mathbb{F}_{p})$ does not vanish.
Note that $B \cong B_{0} \otimes_{\mathbb{F}_{p}} k$ where $B_{0} = \mathbb{F}_{p}[x_{1}, \dots, x_{r}]/(x_{1}, \dots, x_{r})^{2}$.
By the K\"unneth formula for Hochschild homology \cite[Prop.~9.4.1]{Weibel-homo} and the Hochschild--Kostant--Rosenberg theorem \cite[Thm.~9.4.7]{Weibel-homo} we have a natural isomorphism of graded rings
\[
\HH_{*}(B/\mathbb{F}_{p}) \cong \Omega^{*}_{k}  \otimes_{\mathbb{F}_{p}} \HH_{*}(B_{0}/\mathbb{F}_{p}).
\]
By \cite[Thm.~2.1]{GH-vorst} the Dennis trace maps the symbol $\{1+x_{1}, \dots, 1+x_{r}\} \in K_{r}(B_{0})$ to a non-zero element of $\HH_{r}(B_{0}/\mathbb{F}_{p})$.
On the other hand, the Dennis trace is a map of graded rings, and the symbol $\{y_{1}, \dots, y_{s}\} \in K_{s}(k)$ is mapped to
\[
d\log(y_{1}) \dots d\log(y_{s}) = (y_{1}\cdots y_{s})^{-1} dy_{1} \dots dy_{s} \in \Omega^{s}_{k},
\]
see the proof of \textit{loc.~cit.} and the references given there. The latter element does not vanish since $y_{1}, \dots, y_{s}$ are $p$-independent,
finishing the proof.
\end{proof}

Recall that the \emph{homotopy $K$-theory} of a scheme $X$ is defined as
\[
KH(X) = \colim_{\Delta^\mathrm{op}} K(X\times\Delta^\bullet)
\]
where $\Delta^\bullet$ is the cosimplicial scheme with $\Delta^p=\operatorname{Spec}(\mathbb{Z}[T_0,\ldots,T_p]/(\sum T_j-1))$ and $K(-)$ denotes the non-connective $K$-theory spectrum.

\begin{prop}\label{prop:vanishing}
Let $X$ be a noetherian $\mathbb{F}_p$-scheme. Then $KH_i(X;{\mathbb Z}/p) = 0$ for $i>\pdim(X)$.
\end{prop}
\begin{proof}
 We can assume that $d=\dim(X)\le \pdim(X)$ is finite and we use induction on $d$.
A zero-dimensional noetherian scheme is a finite disjoint union of schemes
$\Spec(A)$ where $A$ is artinian local with residue field $k$.
As homotopy $K$-theory is nilinvariant $KH_i(A;\mathbb Z /p)\cong KH_i(k;\mathbb Z /p)$ and
this group vanishes for $i>\pdim(k)$ by Corollary~\ref{cor:vanishing-on-valuation-rings}.

We proceed with the inductive step for $d>0$.
Again by Zariski descent we may assume $X=\Spec(A)$ where $A$ is a noetherian $\mathbb{F}_p$-algebra of finite Krull dimension $d$.
By cdh-descent \cite[Thm.~3.9]{Cisi13} and nil-invariance of $KH$ we may assume that $X$ is integral.
Let $p\colon X'\to X$ be a modification, i.e.~a proper birational morphism with $X'$ integral.
There exists a closed subscheme $Y$ of $X$ with $\dim(Y)<\dim(X)$ such that $p$ is an isomorphism outside $Y$.
We obtain an abstract blow-up square
\[
\begin{tikzcd}
X' \arrow[d, "p"'] & \arrow[l]\arrow[d] Y'\\
X & \arrow[l] Y
\end{tikzcd}
\]
inducing a long exact  sequence
\begin{multline*}
\ldots\to KH_{i+1}(Y';{\mathbb Z}/p) \to KH_i(X;{\mathbb Z}/p)\to\\\to  KH_i(Y;{\mathbb Z}/p)\oplus KH_i(X';{\mathbb Z}/p)\to KH_{i}(Y';{\mathbb Z}/p)\to\dots
\end{multline*}
Let $i>\pdim(X)$.
Since $\pdim(X)\geq \pdim(Y)$, the group $KH_i(Y;{\mathbb Z}/p)$ vanishes by the inductive hypothesis.
Lemma~\ref{lem:p-dim-and-transcendence-degree} implies that $\pdim(X)\geq\pdim(X')$.
As $\pdim(X')\geq\pdim(Y')$, the inductive hypothesis yields $KH_{i+1}(Y';{\mathbb Z}/p)=KH_{i}(Y';{\mathbb Z}/p)=0$.
Hence, a modification $X'\to X$ induces an isomorphism $KH_i(X;{\mathbb Z}/p) \cong KH_i(X';{\mathbb Z}/p)$ for all integers $i>\pdim(X)$ and consequently an isomorphism
\begin{equation}\label{eqn:colimit-isomorphism}
 KH_i(X;{\mathbb Z}/p)  ~\cong \colim_{\substack{X'\to X \\ \text{modification}}} KH_i(X';{\mathbb Z}/p).
\end{equation}
There is a convergent Zariski descent spectral sequence
\[
E_2^{pq}= H^p (X', a_\mathrm{Zar} KH_{-q}(-;{\mathbb Z}/p)) \Longrightarrow KH_{-p-q}(X';{\mathbb Z}/p)
\]
where $a_\mathrm{Zar}$ denotes Zariski sheafification.
Taking the filtered colimit as above of these spectral sequences yields a convergent spectral sequence
\[
\colim_{\substack{X'\to X \\ \text{modification}}}  H^p ( X' , a_\mathrm{Zar} KH_{-q}(-;{\mathbb Z}/p) ) \Longrightarrow KH_{-p-q}(X;{\mathbb Z}/p)
\]
where we have used \eqref{eqn:colimit-isomorphism} to identify the right-hand side.
We want to show that the left-hand side vanishes for all $p\geq 0$ and $-q < -\pdim(X)$.
Consider the Zariski--Riemann space
\[
ZR(X) = \lim_{\substack{X'\to X \\ \text{modification}}} X'
\]
where the limit is formed in the category of locally ringed spaces (see~\cite[Def.~E.2.3]{FK}).
By \cite[Prop.~3.1.19]{FK} we can rewrite the left-hand side of the above spectral sequence as $H^p ( ZR(X) , \mathcal{F}_{-q})$ where $\mathcal{F}_q$ is the colimit of the sheaves $a_\mathrm{Zar} KH_{q}(-;{\mathbb Z}/p)$ on $X'_\mathrm{Zar}$.
Consider an integer $q > \pdim(X)$.
It suffices to show that the sheaf $\mathcal{F}_q$ vanishes on the topological space $ZR(X)$.
This can be testet on stalks.
Hence, by \cite[Cor.~E.2.13]{FK} we must show that $\mathcal{F}(V)$ vanishes for every valuation ring $V$ of $F$ where $F$ denotes the function field of $X$.
As (homotopy) $K$-theory commutes with filtered colimits, we have $\mathcal{F}(V)\cong KH_q(V;\mathbb Z / p)$.
Now $\pdim(F)\leq \pdim(X)$ implies $q>\pdim(F)$. So the vanishing of $KH_q(V;\mathbb Z /
p)$ follows from Corollary~\ref{cor:vanishing-on-valuation-rings} and the following
Lemma~\ref{lem:kkhvalring}.
\end{proof}

\begin{lem}\label{lem:kkhvalring}
For a valuation ring $V$ and $m\ge 0$ the canonical map $K(V ) \to K(V[ X_1,\ldots , X_m
])$ is an equivalence. In
particular we get an equivalence $K(V)\simeq KH(V)$.
\end{lem}
\begin{proof}

We have to show that any element $\xi\in NK_i(V[X_1,\ldots , X_m]) $ vanishes. As $V$ is a
filtered colimit of noetherian integral subrings and as $K$-theory commutes with filtered
colimits of rings we can assume that there exists a noetherian ring $A\subset V$ and an
element $\xi_A\in NK_i (A [X_1,\ldots , X_m])$ mapping to $\xi$.
By~\cite[Prop.~6.4]{KerzStrunkTamme} there exists a projective birational morphism $X'\to
\Spec (A)$ such that $\xi_A$ maps to $0$ in $  NK_i (X'\times \mathbb A^m)$. As the
morphism $\Spec(V)\to \Spec(A)$ factors through $X'$ by the valuative criterion for
properness, we see that $\xi=0$.
\end{proof}

\begin{rmk}
  Lemma~\ref{lem:kkhvalring} was suggested by Christian Dahlhausen.
Kelly and Morrow also prove Lemma~\ref{lem:kkhvalring}  by a different method, see \cite[Thm.~3.3]{KellyMorrow}.
\end{rmk}

Let $n$ be an integer.
Recall that a ring $A$ is called $K_n$-regular if the canonical map $K_n(A)\to K_n(A[X_1,\ldots,X_m])$ is an isomorphism for all positive integers $m$, or equivalently, if $N^pK_n(A)=0$ (see \cite[Def.~III.3.4]{Kbook}) for all positive integers $p$.
Vorst and van der Kallen proved that $K_n$-regularity implies $K_{n-1}$-regularity \cite[Cor.~2.1]{Vorst}.
Together with the spectral sequence
\[
E^1_{pq} = N^pK_q(A)  \Longrightarrow KH_{p+q}(A)
\]
from \cite[Thm.~IV.12.3]{Kbook} this implies that, if $A$ is $K_n$-regular, the canonical map
\[
 K_i(A) \to KH_i(A)
\]
is an isomorphism for all integers $i\leq n$ and surjective for $i=n+1$. The next result is Theorem~\ref{introthm:A} from the introduction.

\begin{thm}\label{thm:main}
Let $A$ be an excellent $\mathbb{F}_p$-algebra such that $[k(x):k(x)^p]<\infty$ for all points $x\in\Spec(A)$.
If $A$ is $K_{\pdim(A)+1}$-regular, then $A$ is regular.
\end{thm}
\begin{proof}
First observe that we can assume without loss of generality that $A$ has finite Krull dimension.
Indeed, we have to show that the finite dimensional ring $A_\mathfrak{p}$ is a regular local ring for all prime ideals
$\mathfrak{p}\subset A$. But by \cite[Cor.~1.9]{Vorst} the localization $A_\mathfrak{p}$ is
$K_{\pdim(A_\mathfrak{p})+1}$-regular.

We show the statement by induction on the finite Krull dimension $d$ of $A$.
For $d=0$ the noetherian ring $A$ is regular if it is reduced and this is immediately implied by $K_1$-regularity.
We proceed with the inductive step for $d>0$. By the above observation
we can assume that $A$ is local.

Next, we want to reduce to the case of complete local $\mathbb{F}_p$-algebras.
Let $A\to \widehat A$ be the completion at the maximal ideal $\mathfrak{m}$.
As $A$ is excellent, this map is regular and it suffices to show that $\widehat A$ is a regular ring.
For an integer $q\geq 1$ and $i=\pdim(A)+1$ consider the commutative diagram
\begin{equation}\label{thm.comdiag}
\begin{tikzcd}[column sep=small]
N^qK_{i+1}(X) \arrow[d] \arrow[r]    & N^qK_i(\text{$A$ on
  $\mathfrak{m}$}) \arrow[r]\arrow[d,"\cong"'] & N^qK_i(A) \arrow[r]\arrow[d]&
N^qK_i(X) \arrow[d]  \\
 N^qK_{i+1}(\widehat X)  \arrow[r] & N^qK_i(\text{$\widehat A$ on $\mathfrak{m}$}) \arrow[r]          & N^qK_i(\widehat A) \arrow[r]& N^qK_i(\widehat X)
\end{tikzcd}
\end{equation}
with exact rows, where $X= \Spec(A)\setminus \{\mathfrak{m}\}$ and $\widehat X = \Spec(\widehat  A)\setminus \{\mathfrak{m}\}$.
By Thomason--Trobaugh excision \cite[Prop.~3.19]{Thomason-Trobaugh} for $N^qK$, the second vertical map is an isomorphism.
The groups in the corners involving $X$ and $\widehat X$ vanish as these schemes are regular by the inductive hypothesis.
This implies that the ring $\widehat A$ is $K_{\pdim(A)+1}$-regular.
As $\pdim(\widehat A)\leq \pdim(A)$ by Lemma~\ref{lem:p-dim-and-dimension}, we can now assume that $A$ is a complete $\mathbb{F}_p$-algebra.

Let $k$ denote the residue field of $A$ and set $e=\pdim(A)$.
We have
\begin{equation}\label{eqn:dimension-in-the-main-theorem}
e = \pdim(k) + d
\end{equation}
by Lemma~\ref{lem:p-dim-and-dimension} which is a finite number as $\pdim(k)$ is finite by assumption.
Since $A$ is $K_{e+1}$-regular, the fibre of $K\to KH$ is $(e+1)$-connected by the
discussion preceding Theorem~\ref{thm:main}.
Hence the fibre of $K/p\to KH/p$ is $(e+1)$-connected as well and in particular
\[
 K_{e+1}(A;{\mathbb Z}/p) \to KH_{e+1}(A;{\mathbb Z}/p)
\]
is injective.
The group on the right-hand side vanishes by Proposition~\ref{prop:vanishing}, and consequently $K_{e+1}(A;{\mathbb Z}/p)=0$.

Consider a minimal set of generators $x_1,\ldots,x_r$ for the maximal ideal $\mathfrak{m}$ of $A$.
We have $r=\dim_k\mathfrak{m}/\mathfrak{m}^2\geq d=\dim (A)$ and $A$ is regular if and only if equality holds.
By Cohen's theorem~\cite[Thm.~28.3]{MatsumuraCA} the equicharacteristic complete local ring $A$ has a coefficient field, i.e.~the projection $A\to A/\mathfrak{m}=k$ admits a split $k\to A$.
Hence also $k\to A/\mathfrak{m}^2\to k$ is the identity and there is a surjection $k[X_1,\ldots,
X_r]\to A/\mathfrak{m}^2$ which induces an isomorphism $B=k[X_1,\ldots, X_r]/(X_{1}, \dots, X_{r})^{2} \cong A/\mathfrak{m}^2$.

Finally we will show that $K_{i}(A;{\mathbb Z}/p)\neq 0$ for all $i\in\{1,\ldots,\pdim(k)+r\}$.
This implies $\pdim(k)+r\leq e$ by the vanishing result from before and
 equation~\eqref{eqn:dimension-in-the-main-theorem} then gives $r\leq d$, whence the regularity of $A$.
In order to show the non-vanishing, let $y_{1}, \dots, y_{s} \in k$ be $p$-independent elements and consider the symbol $\bar\xi\in K_{s+r}(B)$ which is given by the image of $\{y_{1}, \dots, y_{s}, 1+X_{1}, \dots, 1+X_{r}\}$.
This has a preimage $\xi\in K_{s+r}(A)$ as $A\to B$ is surjective.
Consider the diagram
\[
\begin{tikzcd}
\xi\in K_{s+r}(A) \arrow[r]\arrow[d] & K_{s+r}(A)/p\arrow[d]              \\
\bar\xi\in K_{s+r}(B) \arrow[r]          & K_{s+r}(B)/p
\end{tikzcd}
\]
The image of $\xi$ in the lower right corner is non-trivial by Lemma~\ref{lem:non-vanishing}.
Hence the image of the element $\xi$ in $K_{s+r}(A)/p$ is non-trivial.
As the canonical map $K_{s+r}(A)/p\to K_{s+r}(A;{\mathbb Z}/p)$ is injective, we obtain $K_{s+r}(A;{\mathbb Z}/p)\neq 0$.
\end{proof}

In conjunction with Lemma~\ref{lem:p-dim-and-dimension} we obtain:

\begin{cor}
Let $k$ be a perfect field of positive characteristic and let $A$ be a $k$-algebra of finite type.
Suppose that $A$ is $K_{\dim(A)+1}$-regular. Then $A$ is a regular ring.
\end{cor}

We close this section by proving that in characteristic zero Vorst's conjecture can be
generalized from affine algebras over fields to all excellent rings. Recall that the Vorst
conjecture in characteristic
is shown in~\cite{CHW-Vorst}. Our generalization is
based on the Hironaka--Artin algebraization of isolated singularities. The following result is Theorem~\ref{introthm:B} from the introduction.

\begin{thm}\label{thm:char0}
Let $A$ be an excellent noetherian ring of characteristic zero. If $A$ is
$K_{\dim(A)+1}$-regular, then $A$ is regular.
\end{thm}

\begin{proof}
  As in the proof of Theorem~\ref{thm:main} we can assume without loss of generality that
  $A$ has finite Krull dimension and that it is local. So we prove Theorem~\ref{thm:char0}
  by induction on $d=\dim(A)$. By the induction assumption the localization $A_g$ is regular for any $g\in
  A\setminus A^\times$, so $A$ has at most an isolated singularity at
  its maximal ideal $\mathfrak{m}$. Arguing as in diagram~\eqref{thm.comdiag} we can also assume
  that $A$ is complete.

  Let $k\subset A$ be a field of coefficients for $A$~\cite[Thm.~28.3]{Matsumura}.
  Then by~\cite[$0_{\mathrm{IV}}$ Prop.~22.7.2]{EGA} the $k$-algebra $A$ satisfies the assumptions
  of  Hironaka--Artin algebraization \cite[Thm.~3.8]{Artin} so that
 $A$ is the completion of a
  $k$-algebra $R$ of finite type at a maximal ideal $\mathfrak{m}$. We can assume that $R$ is regular away from the ideal $\mathfrak{m}$ and that $\dim(R)=\dim(A)$. Let $X$ be
  $\Spec R\setminus \{\mathfrak{m} \}$, $\widehat X =  \Spec (A) \setminus {\mathfrak{m}}$, and let $q\ge 1$. In the commutative diagram with exact rows
\begin{equation}\label{thm.comdiag2}
\begin{tikzcd}[column sep=small]
N^qK_{i+1}(X) \arrow[d] \arrow[r]    & N^qK_i(\text{$R$ on
  $\mathfrak{m}$}) \arrow[r]\arrow[d,"\cong"'] & N^qK_i(R) \arrow[r]\arrow[d]&
N^qK_i(X) \arrow[d]  \\
 N^qK_{i+1}(\widehat X)  \arrow[r] & N^qK_i(\text{$A$ on $\mathfrak{m}$}) \arrow[r]          & N^qK_i( A) \arrow[r]& N^qK_i(\widehat X)
\end{tikzcd}
\end{equation}
the groups in the corners involving $X$ and $\widehat X$ vanish since these schemes are regular. The second vertical map
is an isomorphism by Thomason--Trobaugh excision \cite[Prop.~3.19]{Thomason-Trobaugh} for
$N^qK$. So for $i\le \dim(R)+1 = \dim(A)+1$ the vanishing of $N^qK_i(A)$ implies the
vanishing of  $N^qK_i(R)$. By~\cite[Thm.~0.1]{CHW-Vorst} the ring $R$ is regular and so is its completion $A$.
\end{proof}

\section{A one-dimensional analog of Vorst's conjecture in mixed  characteristic}

In this section we prove Theorem~\ref{introthm:mixed-char} (see
Theorem~\ref{thm:charmixed} below). The proof is based on
calculations involving de Rham--Witt forms in mixed
charactersitic, here the key non-vanishing result is
Proposition~\ref{sec6:nonvanishing}.

\subsection{de Rham--Witt computations}

Let $p$ be an odd prime.
For any commutative ring $R$ we denote by $R[\epsilon]$ the ring of dual numbers over $R$, i.e.~$R[\epsilon] \cong R[t]/(t^{2})$ with $\epsilon$ corresponding to the class of $t$.
Recall from \cite[Lemma~4.1.1]{HesselholtMadsen-drW} that every element of the ring $W_{n}(R[t])$ of $p$-typical Witt vectors length $n$ may be written uniquely as a finite sum
\begin{equation}
	\label{eq:HM-formula-Witt}
\sum_{j\in {\mathbb N}_{0}} a_{0,j}^{(n)} [t]_{n}^{j} + \sum_{s=1}^{n-1}\sum_{j \in I_{p}} V^{s}(a_{s,j}^{(n-s)}[t]_{n-s}^{j})
\end{equation}
with $a_{s,j}^{(n-s)} \in W_{n-s}(R)$. Here $I_{p}$ is the set of natural numbers not divisible by $p$
and for any ring $A$ the symbol $[\cdot]_{n} \colon A \to W_{n}(A)$ denotes the Teichm\"uller map.
The index $n$ will usually be clear from the context, and we often drop it from the notation to increase readability.
With the same proof as in \cite{HesselholtMadsen-drW} one obtains the following lemma.
\begin{lemma}
	\label{lem:Witt-vectors-dual-numbers}
Any element in $W_{n}(R[\epsilon])$ may be written uniquely in the form
\[
a_{0,0}^{(n)} + a_{0,1}^{(n)}[\epsilon]_{n} + \sum_{s=1}^{n-1} V^{s}(a_{s,1}^{(n-s)}[\epsilon]_{n-s})
\]
with $a_{s,j}^{(n-s)} \in W_{n-s}(R)$.
The kernel of the canonical surjection $W_{n}(R[t]) \to W_{n}(R[\epsilon])$ consists of those elements \eqref{eq:HM-formula-Witt} for which $a^{(n-s)}_{s,j} = 0$ whenever $j =0$ or $j=1$.
\end{lemma}

We now assume that $R$ is a ${\mathbb Z}_{(p)}$-algebra.

\begin{lemma}
	\label{lem:deRham-Witt-dual-numbers}
Any element in $W_{n}\Omega^{q}_{R[\epsilon]}$ can be written uniquely as a finite  sum of the form
\begin{equation}
	\label{eq:elements-in-drW-of-dual-numbers}
a_{0,0}^{(n)} + a_{0,1}^{(n)}[\epsilon]_{n} + b_{0}^{(n)} d[\epsilon]_{n}
	+ \sum_{s=1}^{n-1} \left( V^{s}(a_{s,1}^{(n-s)}[\epsilon]_{n-s} ) + dV^{s}( b_{s}^{(n-s)}[\epsilon]_{n-s} ) \right)
\end{equation}
where $a_{s,i}^{(n-s)} \in W_{n-s}\Omega^{q}_{R}$ and $b_{s}^{(n-s)} \in W_{n-s}\Omega^{q-1}_{R}$.
In other words, as abelian group
\begin{equation}
	\label{eq:deRham-Witt-direct-sum-decomposition}
W_{n}\Omega^{q}_{R[\epsilon]} \cong
	W_{n}\Omega^{q}_{R} \oplus W_{n}\Omega^{q}_{R} \oplus W_{n}\Omega^{q-1}_{R}
	\oplus \bigoplus_{s=1}^{n-1} \left( W_{n-s}\Omega^{q}_{R} \oplus W_{n-s}\Omega^{q-1}_{R} \right).
\end{equation}
The structure maps of the pro-system $W_{\bullet}\Omega^{q}_{R[\epsilon]}$ are induced by the ones of $W_{\bullet}\Omega^{q}_{R}$.
On the first summand $W_{n}\Omega^{q}_{R}$ the structure maps $d$, $F$, and $V$ are given by the underlying maps of $W_{\bullet}\Omega^{*}_{R}$, on the other summands they are given in Table~\ref{table:de-rham-witt}.

The product is given as follows.  On the first summand, the product is the one from $W_{n}\Omega^{*}_{R}$. The product of two summands from \eqref{eq:elements-in-drW-of-dual-numbers} each of which has an $[\epsilon]$ vanishes. Finally, for $a' \in W_{n}\Omega^{q'}_{R}$ the left multiplication with $a'$ is given in Table~\ref{table:product-dRW}.
\end{lemma}

\begin{table}[t]
	\renewcommand{\arraystretch}{1.3}
\begin{small}
\begin{tabular}{cccc}
\toprule
	$x$			& $d(x)$ 							& $F(x)$ 				& $V(x)$ \\
\midrule
$a[\epsilon]$		& $(da)[\epsilon] + (-1)^{q}d[\epsilon]$	& $0$				& $V(a[\epsilon])$ \\
$bd[\epsilon]$		& $(db)d[\epsilon]$ 					& $0$				& $(-1)^{q-1}pdV(b[\epsilon]) + (-1)^{q}V((db)[\epsilon])$ \\
$V^{s}(a[\epsilon])$	& $dV^{s}(a[\epsilon])$				& $pV^{s-1}(a[\epsilon])$ 	& $V^{s+1}(a[\epsilon])$ \\
$dV^{s}(b[\epsilon])$	& $0$							& $dV^{s-1}(b[\epsilon])$	& $pdV^{s+1}(b[\epsilon])$ \\
\bottomrule
\end{tabular}
\end{small}
\caption{Structure maps on $W_{n}\Omega^{q}_{R[\epsilon]}$}
\label{table:de-rham-witt}
\end{table}

\begin{table}[t]
	\renewcommand{\arraystretch}{1.3}
\begin{small}
\begin{tabular}{ccccc}
\toprule
$x$			& $a[\epsilon]$ 		& $bd[\epsilon]$ 	& $V^{s}(a[\epsilon])$  			& $dV^{s}(b[\epsilon])$ \\
\midrule
$a'\cdot x$	& $(a'a)[\epsilon]$ 	& $(a'b)d[\epsilon]$	& $ V^{s}( F^{s}(a')a [\epsilon])$ 	&  $(-1)^{q+qq'} V^{s}(bF^{s}(da')[\epsilon]) $   \\
			&				&				&							& $+ (-1)^{qq'}dV^{s}( bF^{s}(a')[\epsilon]$ \\
\bottomrule
\end{tabular}
\end{small}
\caption{Multiplication by $a' \in W_{n}\Omega^{q'}_{R}$ on $W_{n}\Omega^{q}_{R[\epsilon]}$}
\label{table:product-dRW}
\end{table}

\begin{proof}
By \cite[Lemma~2.4]{GeisserHesselholtComplete} the canonical map $W_{n}\Omega^{*}_{R[t]} \to W_{n}\Omega^{*}_{R[\epsilon]}$ is surjective with kernel the dg-ideal generated by the kernel of the map $W_{n}(R[t]) \to W_{n}(R[\epsilon])$.
Using the description of this ideal in Lemma~\ref{lem:Witt-vectors-dual-numbers},
the lemma now follows directly from the corresponding description of the de Rham--Witt complex of the polynomial ring $R[t]$ in \cite[Thm.~4.2.8]{HesselholtMadsen-drW}.
\end{proof}

For ease of notation, we introduce the following abbreviation. For any ring $k/\mathbb F_p$ we set
\[
B(k) = W_{2}(k)[\epsilon]/(p\epsilon).
\]
For example, $B(\mathbb{F}_{p}) = {\mathbb Z}[\epsilon]/(p, \epsilon)^{2}$.

\begin{prop}
	\label{prop:non-vanishing-in-dRW}
For $n \geq 2$, the element $d[1+p]_{n}d[1+\epsilon]_{n}$ does not vanish in $W_{n}\Omega^{2}_{B(\mathbb{F}_{p})}/p$.
\end{prop}

\begin{proof}
We write $\omega_{n,R}^{*}$ for the quotient of $W_{n}\Omega^{*}_{R[\epsilon]}$
by the dg-ideal given by  $ \bigoplus_{s=1}^{n-1} \left( W_{n-s}\Omega^{*}_{R} \oplus W_{n-s}\Omega^{*-1}_{R} \right)$  in \eqref{eq:deRham-Witt-direct-sum-decomposition}.

\begin{claim}
	\label{claim:d-epsilon}
We have $d[1+\epsilon]_{n} = d[\epsilon]_{n}$ in $\omega^{1}_{n,R}$.
\end{claim}
\begin{proof}
In $W_{n}(R[\epsilon])$ we have $[1+\epsilon]_{n} = 1 + [\epsilon]_{n} + V(x)$ for some element $x \in W_{n-1}(R[\epsilon])$. From Lemma~\ref{lem:Witt-vectors-dual-numbers} we see that $V(x)$ is of the form $V(x_{0}) + \sum_{s=1}^{n-1} V^{s}(x_{s,1}[\epsilon]_{n-s})$ with $x_{0} \in W_{n-1}(R)$ and $x_{s,1} \in W_{n-s}(R)$. Since the ring homomorphism $W_{n}(R[\epsilon]) \to W_{n}(R)$ that is induced by $\epsilon \mapsto 0$ must  send $V(x)$ to $0$, we find that in fact $V(x_{0}) = 0$. Thus
\[
[1+\epsilon]_{n} = 1 + [\epsilon]_{n} + \sum_{s=1}^{n-1} V^{s}(x_{s,1}[\epsilon]_{n-s}).
\]
Applying $d$, the claim follows.
\end{proof}

\begin{claim}
	\label{claim:vanishing-of-[p]}
Multiplication by $[p]_{n}$ is the zero map on $W_{n}\Omega^{1}_{{\mathbb Z}_{(p)}}$ and hence also on $W_{n}\Omega^{1}_{R}$ where $R$ is any quotient of ${\mathbb Z}_{(p)}$.
\end{claim}
\begin{proof}
By \cite[Ex.~1.2.4]{HesselholtMadsen-drW} we have  isomorphisms
\begin{equation}
	\label{eq:drW-isos-Zp}
W_{n}({\mathbb Z}_{(p)}) \cong \bigoplus_{i=0}^{n-1} {\mathbb Z}_{(p)} \cdot V^{i}(1) \quad \text{ and }  \quad
W_{n}\Omega^{1}_{{\mathbb Z}_{(p)}} \cong \bigoplus_{i=1}^{n-1} {\mathbb Z}/p^{i}{\mathbb Z} \cdot dV^{i}(1)
\end{equation}
and for $i,j \in \{ 0, \dots, n-1\}$ we have  $V^{i}(1)dV^{j}(1) = p^{i}dV^{j}(1)$ if $i < j $  and $=0$ else.
For every element $x \in {\mathbb Z}_{(p)}$ we have
\begin{equation}
	\label{eq:coordinates}
[x]_{n} = x\cdot [1]_{n} + \sum_{i=1}^{n-1} p^{-i}(x^{p^{i}} - x^{p^{i-1}}) \cdot V^{i}( [1]_{n-i})
\end{equation}
as one checks by computing the ghost components of both sides.
By these formulas,  the action of $[p]_{n}$ on $dV^{j}(1)$ is given by
\[
[p]_{n}\cdot dV^{j}(1) = p dV^{j}(1) + \sum_{i=1}^{j-1} (p^{p^{i}} - p^{p^{i-1}}) dV^{j}(1) = p^{p^{j-1}} dV^{j}(1) = 0
\]
as $p^{j-1} \geq j$ for every $j \geq 1$. This proves the claim.
\end{proof}

\begin{claim}
	\label{claim:retract}
For $R = {\mathbb Z}_{(p)}$ or some quotient of it, the projection $W_{n}\Omega^{2}_{R[\epsilon]} \to \omega_{n,R}^{2}$ factors through the
 canonical surjection $W_{n}\Omega^{2}_{R[\epsilon]} \to W_{n}\Omega^{2}_{R[\epsilon]/(p\epsilon)}$.
\end{claim}
Note that for $R = {\mathbb Z}/(p^{2})$, $R[\epsilon]/(p\epsilon) = B(\mathbb{F}_{p})$.
\begin{proof}
The kernel of the surjection $W_{n}\Omega^{*}_{R[\epsilon]} \to W_{n}\Omega^{*}_{R[\epsilon]/(p\epsilon)}$ is the dg-ideal generated by the kernel of the map $W_{n}(R[\epsilon]) \to W_{n}(R[\epsilon]/(p\epsilon))$. An element of this kernel is of the form $\sum_{i=0}^{n-1} V^{i}( [x_{i}p\epsilon]_{n-i} )$ with $x_{i} \in R$.
The dg-ideal generated by the elements $V^{i}( [x_{i}p\epsilon]_{n-i})$ for $i > 0$ lies in the kernel of $W_{n}\Omega^{*}_{R[\epsilon]} \to \omega^{*}_{n,R}$ by definition.

It remains to show that the  dg-ideal generated by $[p\epsilon]_{n}$  in  $\omega^{*}_{n,R}$ vanishes in degree 2.
We have $d[p\epsilon] = [p]d[\epsilon] + [\epsilon]d[p]$.
Hence the vanishing of $W_{n}\Omega^{2}_{R}$ \cite[Ex.~1.2.4]{HesselholtMadsen-drW}, Claim~\ref{claim:vanishing-of-[p]}, and Lemma~\ref{lem:deRham-Witt-dual-numbers} together imply that the product of $d[p\epsilon]$ with any 1-form in $\omega^{1}_{n,R}$ vanishes.
Since every 2-form in $\omega^{*}_{n,R}$ is a multiple of $d[\epsilon]$, also the product of $[p\epsilon]$ with any element of $\omega^{2}_{n,R}$ vanishes. This finishes the proof of the claim.
\end{proof}

\begin{claim}
	\label{claim:mod-p-iso}
The natural map $W_{n}\Omega^{*}_{{\mathbb Z}_{(p)}} / p \to W_{n}\Omega^{*}_{{\mathbb Z}/(p^{2})}/p$ is an isomorphism.
\end{claim}
\begin{proof}
The kernel of $W_{n}\Omega^{*}_{{\mathbb Z}_{(p)}} \to W_{n}\Omega^{*}_{{\mathbb Z}/(p^{2})}$ is the dg-ideal generated  by the kernel $W_{n}(p^{2}{\mathbb Z}_{(p)})$ of the canonical map  $W_{n}({\mathbb Z}_{(p)}) \to W_{n}({\mathbb Z}/(p^{2}))$. It is enough to show that  $W_{n}(p^{2}{\mathbb Z}_{(p)}) \subset pW_{n}({\mathbb Z}_{(p)})$.
The ideal $W_{n}(p^{2}{\mathbb Z}_{(p)})$ is additively generated by elements of the form $V^{i}([xp^{2}]_{n-i})$.
Set $m = n-i$. In the expression \eqref{eq:coordinates} of $[xp^{2}]_{m}$ the coefficient of $[1]_{m}$ is $xp^{2}$. For $i > 0$, the coefficient of $V^{i}([1]_{m-i})$ is divisible by $p^{-i} (p^{2})^{p^{i-1}} = p^{2p^{i-1} - i}$. Since $2p^{i-1} - i \geq 1$ for all $i \geq 1$, it is divisible by $p$. Thus $[xp^{2}]_{m}$ is a multiple of $p$ and so is any of its Verschiebungen.
\end{proof}

\begin{claim}
	\label{claim:non-vanishing}
The element $d[1+p]_{2} \in W_{2}\Omega^{1}_{{\mathbb Z}_{(p)}}$ is not divisible by $p$.
\end{claim}
\begin{proof}
Using the expression  \eqref{eq:coordinates} we have
\[
d[1+p]_{2} = p^{-1}( (1+p)^{p} - (1+p) ) dV(1).
\]
But $p^{-1}( (1+p)^{p} - (1+p) )
= \sum_{k=1}^{p}  \binom{p}{k}p^{k-1} - 1 \equiv -1 \pmod p$. In view of the second isomorphism in~\eqref{eq:drW-isos-Zp} this finishes the proof.
\end{proof}

We now finish the proof of Proposition~\ref{prop:non-vanishing-in-dRW}.
In order to show that  $d[1+p]_{n}d[1+\epsilon]_{n}$ does not vanish in $W_{n}\Omega^{2}_{B(\mathbb{F}_{p})}/p$, it suffices by Claim~\ref{claim:retract} to show that the image of $d[1+p]_{n}d[1+\epsilon]_{n}$ in $\omega^{2}_{n,{\mathbb Z}/(p^{2})}/p$ does not vanish.
By Claim~\ref{claim:d-epsilon} this image coincides with the image of $d[1+p]_{n}d[\epsilon]_{n}$. By Lemma~\ref{lem:deRham-Witt-dual-numbers} it suffices to show that $d[1+p]_{n}$ does not vanish in $W_{n}\Omega^{1}_{{\mathbb Z}/(p^{2})} / p$; equivalently, $d[1+p]_{n} \not= 0$ in $W_{n}\Omega^{1}_{{\mathbb Z}_{(p)}}/p$ by Claim~\ref{claim:mod-p-iso}. It is clearly enough to show this non-vanishing for $n=2$, which is done in Claim~\ref{claim:non-vanishing}.
\end{proof}

\begin{prop}\label{sec6:nonvanishing}
Let $k$ be a $\mathbb{F}_{p}$-algebra. The canonical map $W_{n}\Omega^{q}_{B(\mathbb{F}_{p})}/p \to W_{n}\Omega^{q}_{B(k)}/p$ is injective for every $n$ and $q$. In particular,
 $d[1+p]_{n}d[1+\epsilon]_{n}$ does not vanish in $W_{n}\Omega^{2}_{B(k)}/p$ for $n\geq 2$.
\end{prop}

\begin{proof}
The second part of Proposition~\ref{sec6:nonvanishing} follows from the first part and
Proposition~\ref{prop:non-vanishing-in-dRW}.

Write $F$ for the functor $k \mapsto W_{n}\Omega^{q}_{B(k)}/p$ from rings over $\mathbb F_p$ to abelian groups.
This functor commutes with filtered colimits.
Recall from \cite[Thm.~2.4]{vdKallen} or \cite[Prop.~6.9, Thm.~9.2]{Borger} that if $R \to S$ is an \'etale covering, i.e.~\'etale and faithfully flat, then so is $W_{n}(R) \to W_{n}(S)$.
Assume that  $k \to \ell$ is an \'etale covering. Then  $B(k) \to B(\ell)$ is also an \'etale covering by base change.
It follows from~\cite[Thm.~C]{Hesselholt-big} that the canonical map
\[
W_{n}(B(\ell)) \otimes_{W_{n}(B(k))} W_{n}\Omega^{q}_{B(k)}/p   \to   W_{n}\Omega^{q}_{B(\ell)}/p
\]
is an isomorphism. Using the result of van der Kallen and Borger for the map $B(k) \to B(\ell)$, we get that  $W_{n}(B(k)) \to W_{n}(B(\ell))$ is an \'etale covering.
Together these results imply  that $F(k) \to F(\ell)$ is injective.
In particular, $F(k) \to F(\ell)$ is injective for a finite separable field extension $k \to \ell$. Since $F$ preserves filtered colimits,  the map $F(\mathbb{F}_{p}) \to F(\bar \mathbb{F}_{p})$ is injective, where $\bar\mathbb{F}_{p}$ is an algebraic closure of $\mathbb{F}_{p}$.

To prove injectivity for a general  $\mathbb{F}_{p}$-algebra $k$, it is now enough to show that $F(\bar \mathbb{F}_{p}) \to F( k \otimes_{\mathbb{F}_{p}} \bar\mathbb{F}_{p})$ is injective.
We can thus assume that $k$ is an $\bar\mathbb{F}_{p}$-algebra. Write $k$ as a filtered colimit of finitely generated $\bar\mathbb{F}_{p}$-algebras $A_{i}$. As $\bar \mathbb{F}_{p}$ is algebraically closed, each map $\bar\mathbb{F}_{p} \to A_{i}$ has a section, and hence $F(\bar\mathbb{F}_{p}) \to F(A_{i})$ is injective. As a filtered colimit of injective maps, $F(\bar\mathbb{F}_{p}) \to F(k)$ is then also injective.
\end{proof}

\subsection{The mixed characteristic result}

The next proposition is a special case of~\cite[Prop.~B.1.1.]{GeisserHesselholtCycles}.

\begin{prop}\label{prop:dlog}
Let $p\neq 2$ be a prime and $A$ a ${\mathbb Z}_{(p)}$-algebra.
The natural map
\[
 \begin{array}{rcl}
 \dlog[-]_n\colon A^\times &\to     & W_n\Omega^1_A  \vspace{1ex}   \\
                 x        &\mapsto & [x]_n^{-1} d[x]_n
 \end{array}
\]
satisfies the Steinberg relation $\dlog[x]_n \dlog[1-x]_n=0$ and hence induces a natural map
\[
 \dlog[-]_n\colon K_2^M(A) \to W_n\Omega^2_A
\]
where the Milnor $K$-group $K_2^M(A)$ is defined as $A^\times\otimes A^\times$ modulo the subgroup generated by $a\otimes(1-a)$ for units $a$ and $1-a$.
\end{prop}

The following result is Theorem~\ref{introthm:mixed-char} from the introduction.

\begin{thm}\label{thm:charmixed}
Let $A$ be an excellent noetherian ring with $\dim(A)\leq 1$ such that $A/\mathfrak{m}$ is perfect of characteristic $p> 2$ for every maximal ideal $\mathfrak{m} \subset A$.
If $A$ is $K_{2}$-regular, then $A$ is regular.
\end{thm}

\begin{proof}
Let $A$ be a ring satisfying the assumptions of the theorem.
We must show that $A$ is regular.
As in the proof of Theorem~\ref{thm:main} we can assume without loss of generality that $A$ is local with maximal ideal $\mathfrak{m}$.

Once we prove that the strict henselization $A^\mathrm{sh}$ is regular, we can deduce the regularity of $A$, see \cite[\href{https://stacks.math.columbia.edu/tag/06LN}{Lemma 06LN}]{stacks-project}.
As the canonical map $A\to A^\mathrm{sh}$ is a filtered colimit of \'etale morphisms by \cite[\href{https://stacks.math.columbia.edu/tag/04GN}{Lemma 04GN}]{stacks-project}, van der Kallen's result on the $NK_n$-groups \cite[Thm.~3.2]{vdKallen} implies that $A^\mathrm{sh}$ is still $K_2$-regular.
Moreover, $A^\mathrm{sh}$ is still excellent by \cite[Cor.~5.6]{Greco}.
Hence we can assume without loss of generality that $A$ is strict henselian and excellent.

Arguing exactly as in the proof of Theorem~\ref{thm:main}, we reduce to the case of $A$ being a complete local ring with $\dim(A)\leq 1$ and maximal ideal $\mathfrak{m}$, algebraically closed residue field $k=A/\mathfrak{m}$ of characteristic $p> 2$ and quotient field $F$ of characteristic zero.

\begin{claim}\label{sec6.claimzero}
$K_2(A)/p =0. $
\end{claim}
\begin{proof}[Proof of Claim~\ref{sec6.claimzero}]
The $K_1$-regularity implies that $A$ is reduced.
Hence the normalization $A\to\tilde A$ is finite and we obtain an abstract blow-up square
\[
\begin{tikzcd}
\Spec(\tilde A) \arrow[d] & Y'    \arrow[d]\arrow[l]          \\
\Spec(A)           & \Spec(k)\arrow[l]
\end{tikzcd}
\]
with $Y'_\mathrm{red}=\Spec(k)\times\ldots\times \Spec(k)$ as $k$ is algebraically closed.
Note that $\tilde A$ is regular.
Descent for $KH(-;{\mathbb Z}/p)$, see \cite[Thm.~3.9]{Cisi13}, yields an an exact sequence
\[
\ldots\to KH_3(Y';{\mathbb Z}/p) \to KH_2(A;{\mathbb Z}/p) \to  KH_2(\tilde A;{\mathbb Z}/p) \oplus KH_2(k;{\mathbb Z}/p) \to\ldots
\]
The group $KH_3(Y';{\mathbb Z}/p)= KH_3(Y'_\mathrm{red};{\mathbb Z}/p)= K_3(Y';{\mathbb Z}/p)$ and the group $KH_2(k;{\mathbb Z}/p)= K_2(k;{\mathbb Z}/p)$ vanish by Corollary~\ref{cor:vanishing-on-valuation-rings}.

We have an injection $K_2(A)/p = KH_2(A)/p \hookrightarrow KH_2(A;{\mathbb Z}/p)$ where the first isomorphism uses the $K_2$-regularity of $A$, see \cite[Cor.~IV.12.3.2]{Kbook}.
Hence, the above exact sequence implies that the composite map
\[
 K_2(A)/p \hookrightarrow KH_2(A;{\mathbb Z}/p) \hookrightarrow KH_2(\tilde A;{\mathbb Z}/p) = K_2(\tilde A)/p
\]
is injective and it suffices to show that $K_2(\tilde A)/p=0$.

Consider the diagram
\[
\begin{tikzcd}
&K_2(\tilde A)/p \arrow[d, hook] \arrow[r]&  K_2(F)/p   \arrow[d, hook]         \\
K_3(k;{\mathbb Z}/p)\arrow[r] &K_2(\tilde A;{\mathbb Z}/p)\arrow[r]           & K_2( F;{\mathbb Z}/p)
\end{tikzcd}
\]
where the bottom horizontal line is part of the exact localization sequence of the
discrete valuation ring $\tilde A$.
As $K_3(k;{\mathbb Z}/p)=0$ by Corollary~\ref{cor:vanishing-on-valuation-rings}, we deduce
that the top horizontal map is injective. So to finish the proof of the claim it suffices to show that $K_2(F)/p= 0$.

By the Merkurjev--Suslin theorem \cite{MerkurjevSuslin}, the norm residue homomorphism
\[
K_2(F)/p \xrightarrow{\cong} H^2(F,\mu_p^{\otimes 2})
\]
is an isomorphism.
The group on the right-hand side vanishes as the field $F$ has property ($C_1$) by
\cite[II.3.3.(c)]{SerreGalois} invoking Lang's theorem and hence has cohomological dimension $\leq 1$ by \cite[II.3.2,Cor.~to~Prop.~8]{SerreGalois}.
Hence we get $K_2(A)/p=0$ as desired.
This shows the claim.
\end{proof}

Consider a minimal set of generators $x_1,\ldots,x_r$ for the maximal ideal $\mathfrak{m}$ of $A$.
The ring $A$ is regular if and only if $r=1$.
We consider two cases.

Suppose first that $p\in \mathfrak{m}^2$ and consider the $\mathbb{F}_p$-algebra $\bar A:=A/p$.
Then the images $\bar x_1,\ldots,\bar x_r$ are still a minimal set of generators for the maximal ideal $\bar\mathfrak{m}$ of $\bar A$.
Set $\bar B:= \bar A /\bar \mathfrak{m}^2$.
Arguing analogously as in the proof of Theorem~\ref{thm:main}, the image of the symbol $\{1+x_1,\ldots,1+x_r\}\in K_{i}(\bar B)$ in $K_{i}(\bar B)/p$ is non-zero and hence $K_{i}(A)/p\neq 0$ for all $i\in\{1,\ldots,r\}$.
This implies $r=1$ by Claim~\ref{sec6.claimzero} and $A$ is regular.

For the second case suppose that $p\notin \mathfrak{m}^2$. Then we can assume without loss of
generality  that $x_1=p$.
We argue by contradiction and suppose that $r\geq 2$.
By Cohen's theorem~\cite[Thm.~28.3]{MatsumuraCA} the complete local ring $A$ has a coefficient ring $W(k)$.
Hence, the canonical map $X_i\mapsto x_i$ induces an isomorphism $W(k)[X_2,\ldots, X_r]/(p,X_2,\ldots, X_r)^2\cong A/\mathfrak{m}^2$.
This ring canonically surjects onto the ring $W_2(k)[X]/(p,X)^2\cong
W_{2}(k)[\epsilon]/(p\epsilon)$ which we denote by $B(k)$.
Analogously to the first case, it suffices to show that the image of the symbol $\{1+p, 1+\epsilon\}$ in $K_2(B(k))/p$ does not vanish.
By \cite{vanderKallen} and as the residue field $k$ is infinite, the canonical map
$K^M_2(B(k))\to K_2(B(k))$ is an isomorphism.
The $\dlog$-map from Proposition~\ref{prop:dlog} sends the symbol $\{1+p, 1+\epsilon\}$ to the element $[1+p]_{n}^{-1}d[1+p]_{n}[1+\epsilon]_{n}^{-1}d[1+\epsilon]_{n}$ in $W_n\Omega^2_{B(k)}/p$ which does not vanish by Proposition~\ref{sec6:nonvanishing}.
Hence $K_2(A)/p \neq 0$ which contradicts Claim~\ref{sec6.claimzero}.
This implies that $A$ is regular.
\end{proof}

redpajama_set_name: RedPajamaArXiv


In Andrew Haigh's Lean on Pete, the image and idea of sprinting permeates much of the film. The movie opens on its teenage protagonist, Charly Thompson (Charlie Plummer), waking up in more of a shack than a house, and running throughout the underwhelming scenery of an almost dilapidated town in Portland, Oregon. Charly burns out in an empty parking lot, with not much to see or indicate a sense of an endpoint to his run. Later on in the film, the audience meets the racehorse Lean on Pete, a horse who has been run to the bone by his owner and is losing steam before his time.
This idea of burning out after a short sprint comes to stand in metaphorically in the film for the obstacles faced through the coming-of-age, when youth are forced to face the adult world much too quickly before they are able to catch their emotions up to the intense events surrounding them. There are countless comparisons made throughout the film of Charly to the horse, at times as the underdog, with him being a built like a rail and wanting to try out for the football team, and other times dealing with the thought that he is one of the world's cast-offs after he has to leave on a long journey in search of the only family he may have left.
It is the quiet moments between Charly and the horse, walking across the desert and alone late at night, that allow Charly to catch up to the world and feel the weight of the road ahead of him. These moments are powerful and illuminating into the poetry of the character piece. It is when Haigh takes us out of these moments and into the frantic subplot characters of the wild desert landscape that the film loses its hold on the audience.
Lean on Pete follows Charly as a teen in a new town with his drunk and deadbeat, but loving father (Travis Fimmel). Needing to find something to do over the summer, Charly finagles a job with a local racehorse owner Del (Steve Buscemi) where he meets the overworked and under-appreciated horse Lean on Pete. After his father's recklessness catches up to him and gets him killed, Charly steals Pete, who is off to the slaughter, in an act of desperation and takes him across the desert to try and locate his estranged aunt, the only person he can remember that ever showed him true love and care. The rest of the film follows Charly on a coming-of-age Odyssey of sorts, meeting various local color along the way.
Del, one of his jockeys Bonnie (Chloë Sevigny), and this local color trip up Charly, but also the story. The first half of the film, containing a confusingly structured first act, is populated by Charly's growing relationship with Del as well as Pete. Once Charly takes off, though, Del is far in Charly's rearview, never to be heard of again, ripping the audience from the emotional stock put into this relationship. From there, the people he meets and the events that occur seem to be ripped from a completely different film and exist in short and quick vignettes that string together in a dragging second half.
Charlie Plummer though, the film's star, carries the film through a quiet reserve of strong acting talent. Many scenes rely solely on a look or demeanor from him, and Plummer takes the image and atmosphere laden film into a deeply emotional character arc that holds the story together. While the second half of the film fails in terms of structure and pacing, Plummer still ekes out every moment of possible emotion and provides the audience with catharsis and closure by the story's end.
Andrew Haigh's Lean on Pete will be a hit for any fan of coming-of-age slow burns. Don't go into this film for the action, fiery emotion, or conventional construction, else you'll be sorely disappointed. It is apparent throughout much of the film that it is an adaptation of a novel, where if Haigh had made some necessary cuts, the narrative would have flowed more fluidly onscreen. The atmosphere that Haigh does create, though, clearly has the filmmaker's personal and elemental signature on it, as well as Plummer's growing sense of humanity and depth within his performances.


Mobile Auto Detail Doctor began with a vision to provide a superior cleaning service, brought to our client's door! Imagine walking out of your office at the end of the day to a freshly detailed car! Whether it's your home or office, we come to you with our fully-equipped eco-friendly mobile unit, outfitted with everything needed to provide your vehicle the finest professional auto detail service. We look forward to serving you and becoming your detailer of choice!
The MADD team is friendly and cleans efficiently while you save time to do what you want. Our team will have you car or commercial vehicle looking brand new upon your return.
We currently serve Edmonton, Fort McMurray, Grand Prairie, Fox Creek, Winnipeg, Fort Nelson, Regina, Moosomin and so much more. We are regularly adding new locations as we expand to meet the demand of our customers.
As Edmonton's premier mobile auto detail company, our mission is to provide a superior cleaning experience at a location of your choosing while embodying our company spirit of excellence. We believe we are the best at what we do and loo forward to sharing our work with you.


I made masala puri for dinner (recipe follows) and had no mixture at home to go with it. So instead of venturing out in the rainy Bangalore night to go buy mixture, I decided to just make something by myself at home. I'd never made it before, but the avalakki mixture seemed to be very simple to make. So I did give it a try and as I had expected, it is very easy to make. It was perfectly crispy and went along well with the masala puri.
Poha/Flattened rice deep fried and flavoured with spices and peanuts.
Add the peanuts and fry until browned. Remove from oil and keep aside.
Add the curry leaves and fry until crisp. Remove and add to the peanuts.
Add the avalakki in batches and fry until they puff up. Remove and add to the peanuts.
Add turmeric powder, red chilli powder and salt.


Dikembe Mutombo collapses to the floor with joy after Denver had secured the greatest upset in NBA playoff history.
Nobody gave the Denver Nuggets much of a chance at upsetting the Seattle SuperSonics in their First Round playoff series in 1994. The Sonics had just posted a franchise-record 63 victories, while the Nuggets had slipped into the playoffs with a ho-hum 42-40 record. Besides, no No. 8 seed had ever beaten a No. 1 seed since the NBA adopted its current playoff format in 1984.
When the Sonics handily won the first two games in Seattle, a sweep seemed a foregone conclusion. Dan Issel, the Hall of Fame center who was coaching the Nuggets, made the most of the situation. He told his players they had nothing to lose as the series shifted to Denver. "To be honest, we just wanted to get some playoff experience this year," he told the media, further easing the burden on his team.
It turned out to be quite an experience. The loose and lively Nuggets pounded the Sonics 110-93 in Game 3, then pulled out a 94-85 overtime win in Game 4. Suddenly the series was tied and going back to Seattle, where the Sonics won 39 of 43 outings in the 1993-94 regular season and playoffs. But that statistic was irrelevant when compared to the attitudes of the two teams going into the showdown.
"I can't deny the butterflies felt like rocks," said Sonics coach George Karl. Issel, meanwhile, looked at his team and suggested, "I don't think our kids knew they were supposed to be nervous."
Denver center Dikembe Mutombo put it this way: "I don't like to be rude, but these are the playoffs. Nobody invites you into their house. You just have to go in and get comfortable."
Not even Seattle's tenacious full-court press could rattle the Nuggets, who played like battle-tested veterans and committed just 11 turnovers. On offense they slowed the pace of the game and worked the ball into their big men, Mutombo, LaPhonso Ellis and Brian Williams. On defense they focused on stopping Seattle's stars. Gary Payton was ineffective after an early foot injury while Shawn Kemp could manage only 19 points, including just six in the second half.
Mutombo dominated the lane defensively, reserve guard Robert Pack scored 10 of his 23 points in the fourth quarter, Ellis tallied 19 and Williams posted 17 points and 19 rebounds. Seattle's Kendall Gill forced overtime with a last-second layup at the end of regulation, but he only postponed the unthinkable. Denver edged Seattle 98-94 in overtime. When the final buzzer sounded, Mutombo grabbed the last rebound and fell to the floor, a look of sheer joy on his face as his teammates piled upon him in celebration.
No. 8 had beaten No. 1. Though the Nuggets would lose to Utah in seven games in the next round, they had carved their place in history.


text:
Q: Javascript how to run async in another thread? I am trying to make an asynchronous method, run the function in a separate thread.
But tests show that async still runs in the same thread as the tab in the browser.
Here is a simple test:
async function asyncLoader() {
    while(true) { }
}

var res = await asyncLoader();

After running the code, the tab in the browser dies, although it had to run in a separate thread, as is done in C #
How do threads work in javascript? Is it possible to run a separate thread besides Worker?

A:
How do threads work in javascript?

JavaScript has no native notion of threads.

Is it possible to run a separate thread besides Worker?

Browsers provide Web Workers. Node.js provides Worker Threads. I'm not aware of any other options.

redpajama_set_name: RedPajamaStackExchange


text:
Seven Brides for Seven Brothers Tickets
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redpajama_set_name: RedPajamaCommonCrawl


It's hard to get appropriately geared up for Mardi Gras when you're hours away from dear New Orleans. Like most, we won't make the early March trip down to the shore for Mardi Gras. But we're not fussy. A bit of gumbo and a case of Ibita beer after work on Tuesday should suffice just fine for yet another year. (That is, until someone forms a Little Rock second line, in which case, call us immediately, please.) But this year, 101 Runners, a "heavy percussive funk band" of elaborately colorful Mardi Gras Indians in gorgeous, traditional suits, will take to Arkansas for a three-day goodwill tour to spread the celebration to their neighbors to the north.
Fronted by Big Chief Monk Boudreaux, formerly of the legendary Wild Magnolias funk group and — don't forget — the Biggest of Big Chiefs, 101 Runners have been Crescent City favorites for years, laying down driving, tribal beats and hypnotic voodoo call-and-response both on stage and the streets.
Thursday, Feb. 24, the Indians take to Fayetteville for a show at George's Majestic Lounge before heading down to Hot Springs on Friday for a night at the Arlington Hotel and, finally, to Little Rock for this show, at Dreamland Ballroom. $5 of the ticket price goes to the Friends of the Ballroom fund, dedicated to restoring the historic ballroom.


Bangor born bass player and composer, Huw V William's released his debut album, Hon, at the end of February.
The title, which is also the name of the band, comes from a poem by Welsh poet T. H Parry Williams, where he questions "What do I care about Wales?". Usually the origin of a bands leader is mentioned to give some vague background but throughout this project Williams constantly provides reminders on his North Walean roots. The fact that it was recorded in Wales and features on one tune, Mugs, the 'Caerhun Hooligan Choir', shows a composer that is proud to show off his heritage through music.
As well as Williams on bass, Hon features; Laura Jurd on trumpet, Alam Nathoo on saxophone, Elliot Galvin on accordion/ keyboard and Pete Ibbetson on drums. The album harnesses the collaborative power of music. Even though the records are composed by Williams , it is the improvisation and band dynamic that makes this such an exciting listen.
A lot of the time when trying to cover many different genres with one band tracks can start to have a cheesy medley feel. However listening to Hon the constant switches from jazz to a more Latin feel in Skardu's Missing appear to come from a genuine love of multiple genres and an inability to be boxed into one particular category. In the case of Slumps an almost West Side Story, Berstein-esque themes come out, making it feel like a whole film score condensed into one tune.
During the bands live gigs Williams talks of his inspiration behind the tunes. For example Mugs, which he says was written as a reaction to him trying to sell mugs with his face to an unwilling public. Taking into account this (slightly flimsy) talk of how he came to write the tunes, the more you listen to this band the more you realise how central the music is and everything other than that is insignificant. There is no pretence to these musicians, no wordy inspirations or explanations. This is why the album works so well for the listener because it is all about good, pure, lively music making. Although make sure to sit near the volume button, it gets loud.


text:
Home'It's been a big help'; Garcia teams up with Baffert barn at Oaklawn
'It's been a big help'; Garcia teams up with Baffert barn at Oaklawn
Three projected starters for upcoming stakes events, including two recent arrivals from Southern California, breezed over a fast track Friday morning at Oaklawn Park. Eight Rings, in company, recorded a five-furlong bullet out of the gate in 59 seconds for Hall of Fame trainer Bob Baffert, who is pointing the Grade 1 winner for the $100,000 Bachelor Stakes…
Eight Rings Fires Bullet at Oaklawn Park
HOT SPRINGS, AR – Trainer Bob Baffert's Grade 1-winning colt Eight Rings fired a bullet best-of-29 5-furlong work from the gate in :59.00 at Oaklawn Park on Friday as he gears up for a start in the $100,000 Bachelor Stakes on April 25. Eight Rings was making his first timed trip over the Oaklawn Park dirt since arriving Tuesday from his base at Santa Anita Park…
Once again, Baffert loaded for Kentucky Derby
Two months out from the Kentucky Derby on May 2, no trainer holds a stronger hand with 3-year-olds than does Bob Baffert, who has four members of the Derby Watch top 20, two of whom are scheduled to run Saturday in the Grade 2, $400,000 San Felipe Stakes at Santa Anita, with at least one more scheduled to run the following Saturday in the Grade 2, $1 million Rebel at Oaklawn…


text:
/ Resources
/ Case Studies And White Papers
Case Studies and White Papers - Coil coatings
Real-world case studies and white papers for best practices in durability and lasting protection. For additional stories and information, check out our blog.
David L. Lawrence Convention Centre, Pittsburgh, Pennsylvania
The David L. Lawrence Convention Center opened in Pittsburgh in 2003 as the world's first certified green convention center. Today, it remains one of the largest LEED® Gold-certified buildings in the world. Used inside and outside, PPG coatings and paints were instrumental to the environmental performance of the building. On the exterior metal building panels and skylights, various shades of PPG DURANAR® coatings provide extended durability, which reduced the need for repainting and other kinds of maintenance.
Categories: Building and Construction, Coil Coatings
Bulletin Building, Philadelphia, Pennsylvania
Designed in 1954 as the home to the historic Philadelphia Bulletin newspaper, the Bulletin Building has stood for decades as one of the iconic structures in the city's booming University City neighborhood. But after the newspaper folded in 1982, the building underwent a series of transformations. Most recently, KieranTimberlake Architects retrofitted the 282,000 square-foot facility into a state-of-the-art laboratory and retail space. As part of the historic renovation, PPG partnered with Spectrum Metal Finishing to develop and apply a customized PPG CORAFLON® powder coating for the structure's new signature feature, a multistory glass façade on the east elevation. With a glossy "Galaxy Red" PPG Coraflon powder coating formulated exclusively for the project, the new-look Bulletin Building is the main attraction in the city's new Schuylkill Yards development.
Categories: Building and Construction, Coil Coatings, Extrusion Coatings
Architects specify rainscreen systems to fulfill two primary functions: the first is to keep moisture from entering a building; the second is to prevent heat from escaping. Thanks to a collaboration between Morphosis Architects, A. Zahner Company and PPG, the rainscreen system on The Bloomberg Center at Cornell Tech does more than serve those purposes – it also works as a color-shifting wall of art. Featuring iradescent, color-shifting PPG DURANAR® VARI-COOL® coatings, the panels were designed by Morphosis with A. Zahner Company to create an artful façade that harmonizes with the local landscape.
Vehicle and Equipment


The perfect lip balm is both thin enough to sink into your lips and thick enough to sit on top of them in a protective, watertight layer. In other words, it doesn't exist—but you can fake it with a little bit of clever layering.
If your lips are painfully dry right now, try rubbing a tiny dab of your regular facial moisturizer or eye cream on your bare lips, then slathering a thick balm on top. Lotions and creams contain more water and moisturizing ingredients than lip balm, but more importantly, they're formulated to actually absorb into your skin. Layering something something waxy and occlusive on top slows down the evaporation process, keeping that moisture right where you need it: on your dry, cracked, miserable lips. This technique is a godsend for anyone with stubborn chapped lips, but it's especially handy for people who can't use lanolin. Waxy sheep sebum heals dry skin like nothing else on earth, but it's a fairly common allergen; trying to find an effective lip balm without it is both harder and more expensive than layering two products you already own.
Any moisturizer you like will work here, as long as it doesn't contain active ingredients. Sure, the concentrations of retinol, chemical exfoliants (AHAs, BHAs, or otherwise), and vitamin C derivatives found in moisturizers are tiny, but it's not worth the risk. Don't put your favorite retinol cream on your lips—use a facial oil, hydrating serum, eye cream, hand cream, or even body lotion instead. Your lips will thank you.


text:
I'm not ready to make nice…
September 11, 2013 Jo Tracey Scorpio, Venus
Over the last few weeks I've been accessorising a little more. Dare I say it, a little softer, a little more girlie. I've even been wearing pink and coral colours on my nails. My earrings have been lighter and more discreet, and my bracelets in the singular. It hasn't felt like me.
Last night I sat down and painted my nails a dark purple. This morning my earrings are chunky, and my bracelets are multiple again. This is more like it.
I know Venus is in rulership in Libra, but for me it doesn't feel quite right. It's a little too nice, and that makes me uncomfortable.
In a few hours Venus will be in Scorpio, a sign where she is in her detriment. Detriment- how bad does that sound? If Venus is the significator in a horary chart, it's not at all good, but otherwise? Essentially it just means that she's being asked to operate in a way that is different to how she likes to perform.
Venus, in the signs that she's at her strongest, doesn't have to do a lot other than look good, appreciate stuff, and bring people together. It's all a little easy.
In Scorpio, though, she's edgier, darker, and way more intense.
Scorpio's natural secrecy, paranoia and reluctance to trust makes it difficult for love to be given and received freely.
In Scorpio she goes as deeply as it's possible to go to really connect with another. In Scorpio we're talking both physical and emotional penetration. Sure it's confrontational, but it's also more than a little hot…did that come out loud?
Those of us with Venus in Scorpio natally, or even one of the other combos such as Pluto in the 7th, or Venus conjunct, square, opposition or inconjunct Pluto, or even Venus in the 8th, will get this. Indeed, you may actively seek this in your own relationships.
Others will feel uncomfortable- this is not a pretty girl feminine energy. It's so much more than that. It's primal.
It's how you feel when you're crushing on someone, when you feel like you could disappear into the other. It's how you feel when you're so deeply into it that you could inhale the other.
It's Heathcliff and Cathy stuff- raw, wild, honest, devouring and all consuming.
Venus in Scorpio has been to the Underworld, she knows what's down there, what's in all of us…and she's not afraid of the dark, nor is she ready to make nice just to keep the peace.
Venus in Scorpio
Something about love, Shakespeare and sonnets…and a Bali sunset.
Pleasure & Pain (re-visited)
astrologyJo TraceyPluto in the 7th houseScorpio womenVenus conjunct PlutoVenus in ScorpioVenus in the 8th houseVenus inconjunct PlutoVenus opposition PlutoVenus square Pluto
Previous Post:Bad Romance…
Next Post:Something about the big picture…


According to reports, Coventry City have signed Reece Charles-Cook.
The former Arsenal youngster who has been on loan at Bury, was a free agent.
Charles-Cook has signed a one year deal with the option of a second year.


Watec WAT910HXRC. As a result of popular request, Watec has added a convenient remote control unit to the new WAT-910HX super low light black and white camera. The remotre allows you to access all points of the OSD via a miniature remote control paddle.
This is a superb low light mono camera. At a standard 0.0001 Lux and 0.000005 Lux (integrated frame technology)this camera is designed to adjust via frame loss compensation based on the lighting requirements of your enviroment.


text:
The Studio Visit... Linder
If you're not familiar with Kirk Millar's work, then it's time to get acquainted. The Arizona-born, New York-based designer is one half of Linder and the genius at the helm of the brand's menswear line. Part of fashion's new wave, Millar and Sam Linder—the other half of Linder—are tearing up the rulebook, challenging tradition and confronting 'maleness' norms manifested within gay culture with urbane-bohemian, real wardrobe vibes.
We took a peek behind the seams of Linder's (stunning) Gramercy townhouse HQ to meet Kirk and find out more about growing up in a town called Cottonwood, riding horses on the family ranch and meeting Sam…
Where did you grow up and how has your background influenced your work?
I'm from a town called Cottonwood in Arizona, near Sedona. There are red rock mountains on one end of the valley and a forested mountain on the other with nothing but open skies in between. My Dad would work in Western boots and Wranglers every day, and we rode horses at a family ranch that had large pastures of cows, irrigation ditches, and Golden Retrievers running around. My Great Great Aunt would have me over every week and we would watch the great epics of old Hollywood and drink English tea in her fine china. This old world glamour made my imagination soar.
How did you become a designer and how did your partnership with Sam arise?
Sam and I became friends after meeting at a retail store I'd been working in. There was a great creative connection between us. We're both creative directors; I work on our Menswear while Sam does the Womenswear. We both photograph our own collections, and have that freedom to create our own worlds within our lanes.
How did you develop your aesthetic?
I don't believe you can even choose. It's a result of all these different micro decisions that come naturally to you. Your perspective cannot be something you're fully aware of - it emerges to me each season.
Where do you begin when starting a design?
It begins with a narrative that's floating in my mind. I do a ton of research and fill in the blanks as I go. Every weekend I spend digging through as much vintage as possible, both physically and online, plus a ton of visual references. It all gets poured into the collection; collaged, evaluated, and hopefully gels together to form something new.
What are your primary creative influences?
Flea markets, antique stores, second hand shops, etc. I love the idea of the human spirit and centuries of traditions having a role in the creation of these objects that have lasting impact on our psyches.
One reviewer described your work as "artfully awkward". How would you describe it? What are you trying to express?
Emotions and the interior life are things I'm deeply interested in, and I tend to incorporate much of my own into the work. I think it's artful because our agenda is not about commercial success, it's about artistic expression. I think the awkwardness is partially from the fact that our interior selves are not smooth and pretty. They're awkward and abject, but ultimately beautiful.
What's your daily process in the studio?
I come in and right away surf the internet for references. These then becomes mockups, and then we start exchanging ideas as a team. We then move into the period of photos, presentation, etc., and that's a very electric time. So engaging, but it takes a "typical" day off the table!
What are the most important qualities for a designer to have?
Taste, curiosity, intelligence, and conviction.
What things do you always have in your studio?
A design studio is stuffed with bits and bobs… Bolts of fabric, trims, camera equipment, computers...
What do you always have on hand whilst you work?
Coffee, and a fridge full of film.
Do you listen to music in the studio? If so, what kind of music?
I tend to listen to Lana Del Rey, classical music such as Peter Gynt, soundtracks from different films… As a romantic person I think the general desire is to listen to music that's emotive and that spans a very wide range of moods.
Where do you go when you need a break from work?
Vintage shopping, or hunting down a new piece of furniture. I'm absolutely obsessed with finding pretty china.
What's your favourite thing about living and working in New York?
Developing creatively. It's a constant process, both good and bad. Secondly those around me; my creative community.
Your favourite place in the city?
My apartment, where I'm surrounded by things I find beautiful. Or the backyard at our office, which is like a little oasis to pop out to.
Who do you most admire?
I admire the process of maturation as I've moved into my thirties. I don't think we talk about it enough, especially in fashion. It's all about youth and progress and change. I think we undervalue taking our own lives and our emotional wellbeing seriously. Creatively, I admire people who put the work first before the praise and schmoozing that seems so desirable.
What's the best way to cure a creative block?
Watching a documentary on a fellow creative; their process, their difficulties. It's a reminder that people who make strong work don't come by it easily. I think it goes back to that idea of the desert. When you're wandering around a creative desert, you find water eventually.
Where do you go for a good idea?
Consuming other mediums helps. I'm usually in no short supply of things I could catch up on!
What's your greatest ambition?
To live a principled, beautiful life that remains creative and full of experience.


Patricia is our full-time dental hygienist who has been working in periodontics for almost 30 years. She has been with Dr. Strauss since 1997, and has a loyal following of patients. She graduated from Forsyth School of Dental Hygiene in 1987 with an Associate of Science degree from Northeastern University, and she continues to learn and grow by attending yearly lectures at conferences and taking continuing education credit courses. In fact, she is licensed to administer local anesthesia! Patricia is very thorough, knowledgeable, and takes excellent care of her patients. She lives in Roslindale with her beautiful daughter and wonderful husband, and when not elbow deep in plaque, she enjoys spending time with her family, reading, and staying actively involved in her church.