text:
\begin{document}

\title[Relative global dimensions and stable homotopy categories]
{Relative global dimensions and stable homotopy categories}

\author[L. Liang]{Li Liang}

\address{L.L. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China}

\email{lliangnju@gmail.com}

\urladdr{https://sites.google.com/site/lliangnju}

\author[J. Wang]{Junpeng Wang}

\address{J.W. College of Economics, Northwest Normal University, Lanzhou 730070, China}

\email{wangjunpeng1218@163.com}

\thanks{L. Liang was partly supported by the National Natural Science Foundation of China (Grant No. 11761045), the Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University, and the Natural Science Foundation of Gansu Province (Grant No. 18JR3RA113); J. Wang was partly supported by the Innovation Foundation of Higher Education of Gansu Province (Grant No. 2020B-087) and a grant from Northwest Normal University (Grant No. NWNU-LKQN2019-14).}

\keywords{global Gorenstein AC-dimension, Gorenstein module, stable category, compactly generatedness}

\subjclass[2010]{18G25; 18G20}

\begin{abstract}
In this paper we study the finiteness of global Gorenstein AC-homological dimensions for rings, and answer the questions posed by Becerril, Mendoza, P\'{e}rez and Santiago. As an application, we show that any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category, which improves the known result by Beligiannis.
\end{abstract}

\maketitle

\thispagestyle{empty}
\section{Introduction}
\noindent
Throughout this work, all rings are assumed to be associative. Let $R$ be a ring; we adopt the convention that an $R$-module is a left $R$-module, and we refer to right $R$-modules as modules over the opposite ring $\Rop$.

Building from Auslander and Bridger's work \cite{MAsMBr69} on modules of finite G-dimension, Enochs, Jenda and Torrecillas \cite{EEnOJn95b,EJT-93} introduced and studied Gorenstein projective, Gorenstein injective and Gorenstein flat modules, and developed ``Gorenstein homological algebra". Such a relative homological algebra theory has been developed rapidly
during the past several years and becomes a rich theory; we refer the reader to, for example, \cite{DB09,BM10,IEm12,EEnOJn95b,EJT-93,Gil17,Wang17} for related works.

For a quasi-Frobenius ring $R$, the category ${\sf Mod}$ of $R$-modules is
a Frobenius category with projective-injective objects all projective (or injective) $R$-modules. So the stable category $\underline{{\sf Mod}}$ modulo projectives is a triangulated category. Furthermore, it is compactly generated; see Krause \cite[sec. 1.5]{Krause00}. It is well known that over an arbitrary ring $R$ the subcategory ${\sf GP}$ (resp., ${\sf GI}$) of Gorenstein projective (resp., Gorenstein injective) $R$-modules is a Frobenius category with projective-injective objects projective (resp. injective) $R$-modules. Also, from a theorem by Christensen, Estrada and Thompson \thmcite[4.5]{CET19} the subcategory ${\sf GF}\cap{\sf Cot}$ of Gorenstein flat and cotorsion $R$-modules is a Frobenius category with projective-injective objects all flat and cotorsion $R$-modules. Hence, the stable categories $\underline{\sf GP}$, $\underline{\sf GI}$ and $\underline{{\sf GF}\cap{\sf Cot}}$ are triangulated categories. It is a natural question when these stable categories are compactly generated. It follows from Beligiannis \cite[lem. 6.6 and thm. 6.7]{Be01} that if $R$ is a right coherent and left perfect or left Morita ring with $\Ggd R<\infty$ then $\underline{\sf GP}\simeq\underline{\sf GI}$ are compactly generated. The same conclusion holds if $R$ is Iwanaga-Gorenstein; see Hovey \thmcite[9.4]{Ho02} or Chen \thmcite[4.1]{Chen11}. One of the main results in this paper is the next improved result; see Corollaries \ref{main-pd and fd-coherent 1} and \ref{main-pd and fd-coherent 2}.

\begin{thm}\label{thm2}
Let $R$ be a ring with $\Ggd R<\infty$.
\begin{prt}
\item If $R$ is right coherent, then $\underline{\sf GP}\simeq\underline{\sf GI}\simeq \underline{\sf GF\cap\sf{Cot}}$ are compactly generated.
\item If $R$ is left coherent, then $\underline{\sf GP}\simeq\underline{\sf GI}$ are compactly generated.
\end{prt}
\end{thm}
\noindent
Here $\Ggd R$ is the global Gorenstein dimension, which is defined as $\Ggd R=\sup\{\Gpd_R M\ |\ M\ \text{is an}\ R\text{-module}\}$. We notice that $R$ satisfies $\Ggd R<\infty$ if and only if $R$ is left Gorenstein\footnote{From Beligiannis \cite{Beligiannis2000}, a ring $R$ is called left Gorenstein if any projective $R$-module has finite injective dimension and any injective $R$-module has finite projective dimension.} ; see \ref{spli}. So as an immediate consequence of Theorem \ref{thm2}, by using \lemcite[6.6]{Be01}, we get the next result that improves \thmcite[6.7]{Be01} by removing the assumption that the ring $R$ should be left perfect or left Morita.

\begin{cor}\label{cor2}
Any left (or right) coherent and left Gorenstein ring has a projective and injective stable homotopy category.
\end{cor}
\noindent
We refer the reader to \cite[def. 6.2]{Be01} for the definition of projective/injective stable homotopy category.

An example is given to show that coherent rings of finite global Gorenstein dimension (or equivalently, left Gorenstein) may not be Iwanaga-Gorenstein nor perfect nor Morita; see Example \ref{exa-2}.

According to \thmcite[6.9]{Beligiannis2000} and \thmcite[4.1]{IEm12}, the finiteness of $\Ggd R$ can be characterized by the existence of the triangulated equivalences $\underline{\sf GP}\simeq \catD{R}/\catK{\sf Prj}$ and/or $\underline{\sf GI}\simeq \catD{R}/\catK{\sf Inj}$, where $\catD{R}$ denotes the bounded derived category of $R$, and $\catK{\sf Prj}$ (resp., $\catK{\sf Inj}$) denotes the bounded homotopy category of projective (resp., injective) $R$-modules. The Verdier quotient triangulated category $\catD{R}/\catK{\sf Prj}$ was first studied by Buchweitz \cite{ROB86} under the name of ¡°stable derived category¡±; it is named by ``singularity category" to emphasize certain homological singularity of the ring $R$ reflected by this quotient category (see Orlov \cite{Orlov04} and Chen \cite{Chen11}). As another immediate consequence of Theorem \ref{thm2}, we see that the singularity categories $\catD{R}/\catK{\sf Prj}\simeq\catD{R}/\catK{\sf Inj}$ are compactly generated over left (or right) coherent rings of finite global Gorenstein dimension (or equivalently, left Gorenstein); see Corollaries \ref{main-pd and fd-coherent 1} and \ref{main-pd and fd-coherent 2}.
\begin{equation*}
  \ast \ \ \ast \ \ \ast
\end{equation*}
\noindent
We prove Theorem \ref{thm2} above by using the finiteness of global Gorenstein AC-homological dimensions.

Gorenstein AC-projective (resp., Gorenstein AC-injective) dimension is defined in terms of resolutions by Gorenstein AC-projective (resp., Gorenstein AC-injective) modules that were initially introduced by Bravo, Gillespie and Hovey \cite{BGH14} as a natural way to extend the notion of Gorenstein projective (resp., Gorenstein injective) modules. We let $\GPacgd R$ and $\GIacgd R$ denote the global Gorenstein AC-projective and global Gorenstein AC-injective dimension of $R$, respectively. That is, $\GPacgd R=\{\Gacpd_R M\ |\ M\ \text{is an}\ R\text{-module}\}$ and $\GIacgd R=\{\Gacid_R M\ |\ M\ \text{is an}\ R\text{-module}\}$. Recently, Becerril, Mendoza, P\'{e}rez and Santiago \cite[6.15]{BMPS19} asked under which conditions on $R$ the following statements are true:
\begin{prt}
\item[$\bullet$] All $R$-modules have finite Gorenstein AC-projective dimension.
\item[$\bullet$] Any $R$-module has finite Gorenstein AC-projective dimension if and only if it has finite Gorenstein AC-injective dimension.
\end{prt}

In Section \ref{Gacgd} we focus on the above two questions. Our main results in this section are the next two theorems, where the first one is used in the proof of Theorem \ref{thm2}.

\begin{thm}
Let $R$ be a ring with $\Ggd R<\infty$.
\begin{prt}
\item If $R$ is right coherent, then $\GPacgd R<\infty$.
\item If $R$ is left coherent, then $\GIacgd R<\infty$.
\end{prt}
\end{thm}
\noindent
This result is proved in Theorem \ref{exa-pd}. The converses of the above statements are not true in general; see Example \ref{exa-1}.

\begin{thm}
If $R$ is a commutative ring, then $\GIacgd R=\GPacgd R$.
\end{thm}
\noindent
This result is proved in Corollary \ref{gacgdim}. However, to the best of our knowledge, we don't know whether the equality $\GIacgd R=\GPacgd R$ holds for an arbitrary ring $R$.

\section{Preliminaries}\label{pre}
\noindent
We begin with some notation and terminology for use throughout this paper.

\begin{ipg}
By an $R$-complex $M$ we mean a complex of $R$-modules as follows:
  \begin{equation*}
    \cdots \lra M_{i+1} \xra{\dif[i+1]{M}} M_i \xra{\dif[i]{M}}
    M_{i-1} \lra \cdots\: .
  \end{equation*}
We frequently (and without warning) identify $R$-modules with $R$-complexes concentrated in degree $0$. For an $R$-complex $M$, we set $\sup M=\sup\{i\in\ZZ\ |\ M_{i}\neq 0\}$ and $\inf M=\inf\{i\in\ZZ\ |\ M_{i}\neq 0\}$. An $R$-complex $M$ is called \emph{bounded} if $\sup M<\infty$ and $\inf M > -\infty$. The symbol $\H[n]{M}$ denotes the $n$th \emph{homology} of $M$, i.e., $\Ker\dif[n]{M}/\im\dif[n+1]{M}$. An $R$-complex $M$ is called \emph{homology bounded} if $\sup\H{M}<\infty$ and $\inf\H{M}> -\infty$. For an $R$-complex $M$, the symbol $M_{\leq n}$ denotes the subcomplex of $M$ with $(M_{\leq n})_i=M_i$ for $i\leq n$ and $(M_{\leq n})_i=0$ for $i > n$, and the symbol $M_{\ge n}$ denotes the quotient complex of $M$ with $(M_{\ge n})_i=M_i$ for $i\ge n$ and $(M_{\ge n})_i=0$ for $i < n$.

We denote by $\catD{R}$ the bounded derived category of $R$-modules, by $\sf{Prj}$ (resp., $\sf{Inj}$, $\sf{Flat}$, and $\sf{Cot}$) the subcategory of projective (resp., injective, flat, and cotorsion) $R$-modules, and by $\catK{\sf Prj}$ (resp., $\catK{\sf Inj}$, and $\catK{\sf FlatCot}$) the bounded homotopy category of projective (resp., injective, and flat and cotorsion) $R$-modules.
\end{ipg}

\begin{ipg}
An $R$-module $M$ is called \emph{Gorenstein projective} \cite{EEnOJn95b} if there exists an exact sequence $\cdots\to P_1\to P_0\to P_{-1}\to\cdots$ of projective $R$-modules such that $M \cong \Coker(P_{1}\to P_{0})$, and it remains exact after applying the functor $\Hom_{R}(-,P)$ for each projective $R$-module $P$. Dually, one has the definition of Gorenstein injective $R$-modules. An $R$-module $M$ is called \emph{Gorenstein flat} \cite{EJT-93} if there exists an exact sequence $\cdots\to F_1\to F_0\to F_{-1}\to \cdots$ of flat $R$-modules such that
$M\cong\Coker(F_1\to F_0)$, and it remains exact after applying the functor $I\otimes_R-$ for each injective $\Rop$-module $I$. We let $\sf GP$ (resp., $\sf GI$, and $\sf GF$) denote the subcategory of Gorenstein projective (resp., Gorenstein injective, and Gorenstein flat) $R$-modules.

The Gorenstein projective dimension of an $R$-module $M$, $\Gpd_R M$, is defined by declaring that $\Gpd_R M\leq n$ if and only if $M$ has a Gorenstein projective resolution of length $n$, that is, there is an exact sequence $0 \to G_n \to \cdots \to G_0 \to M \to 0$ with each $G_i$ Gorenstein projective. The definition of Gorenstein injective dimension, $\Gid_RM$, can be defined dually. We let $\Ggd R$ denote the global Gorenstein projective dimension of $R$, that is, $\Ggd R=\sup\{\Gpd_R M\ |\ M\ \text{is an}\ R\text{-module}\}$. The next result is proved by Bennis and Mahdou \thmcite[1.1]{BM10}, which is used frequently in the paper.
\end{ipg}

\begin{lem}\label{BMth1.1}
For any ring $R$, $\Ggd R=\sup\{\Gid_R M\ |\ M\ \text{is an}\ R\text{-module}\}$.
\end{lem}

\begin{ipg}\label{spli}
Let $\silp R$ denote the supremum of the injective lengths of projective $R$-modules, and $\spli R$ the supremum of the projective lengths of injective $R$-modules. Since an arbitrary direct sum of projective $R$-modules is projective, the invariant $\silp R$ is finite if and only if every projective $R$-module has finite injective dimension. It follows from Beligiannis and Reiten \thmcite[VII. 2.2]{BR07} that every injective $R$-module has finite projective dimension and $\silp R$ is finite if and only if both $\spli R$ and $\silp R$ are finite. Thus by Emmanouil \thmcite[4.1]{IEm12} one gets that the global Gorenstein projective dimension $\Ggd R$ is finite if and only if $R$ is left Gorenstein.
\end{ipg}

\begin{ipg}
Recall from \cite{BGH14} that an $R$-module $F$ is \emph{type FP$_\infty$} if $F$ has a degree-wise finitely generated projective resolution. An $R$-module $A$ is called \emph{absolutely clean} if $\Ext_{R}^{1}(F,A)=0$ for all $R$-modules $F$ of type FP$_\infty$, and an $\Rop$-module $L$ is called \emph{level} if $\Tor_1^{R}(L,F)=0$ for all $R$-modules $F$ of type FP$_\infty$.

Recall from \cite{BGH14} that an $R$-module $M$ is \emph{Gorenstein AC-projective} if there exists an exact sequence $\cdots\to P_1\to P_0\to P_{-1}\to\cdots$ of projective $R$-modules such that $M \cong \Coker(P_{1}\to P_{0})$, and it remains exact after applying the functor $\Hom_{R}(-,L)$ for each level $R$-module $L$.

Dually, an $R$-module $M$ is called \emph{Gorenstein AC-injective} if there exists an exact sequence $\cdots\to I_1\to I_0\to I_{-1}\to\cdots$ of injective $R$-modules such that $M \cong \Coker(I_{1}\to I_{0})$, and it remains exact after applying the functor $\Hom_R(A,-)$ for each absolutely clean $R$-module $A$.

Recall from \cite{BEI18} that an $R$-module $M$ is \emph{Gorenstein AC-flat} if there exists an exact sequence $\cdots\to F_1\to F_0\to F_{-1}\to \cdots$ of flat $R$-modules such that
$M\cong\Coker(F_1\to F_0)$, and it remains exact after applying the functor $A\otimes_R-$ for each absolutely clean $\Rop$-module $A$.

The symbol $\GPac$ (resp., $\GIac$, and $\GFac$) denotes the subcategory of Gorenstein AC-projective (resp., Gorenstein AC-injective, and Gorenstein AC-flat) $R$-modules. It is easy to see that $\GPac\subseteq{\sf GP}$ and $\GIac\subseteq{\sf GI}$.
\end{ipg}

From \thmcite[A.6]{BGH14}, one has the next lemma.

\begin{lem}\label{gacpf}
All Gorenstein AC-projective $R$-modules are Gorenstein AC-flat. That is, $\GPac\subseteq\GFac$.
\end{lem}

\begin{ipg}\label{global}
The Gorenstein AC-projective dimension of $R$-module $M$, $\Gacpd_R M$, is defined by declaring that $\Gacpd_R M\leq n$ if and only if $M$ has a Gorenstein AC-projective resolution of length $n$, that is, there is an exact sequence $0 \to G_n \to \cdots \to G_0 \to M \to 0$ with each $G_i$ Gorenstein AC-projective. The Gorenstein AC-injective and Gorenstein AC-flat dimensions are defined similarly, which are denoted $\Gacid_R M$ and $\Gacfd_R M$, respectively.

Let $\GPacgd R$ (resp., $\GIacgd R$, and $\GFacgd R$) denote the global Gorenstein AC-projective (resp., global Gorenstein AC-injective, and global Gorenstein AC-flat) dimension of $R$. For example,
$$\GPacgd R=\{\Gacpd_R M\ |\ M\ \text{is an}\ R\text{-module}\}.$$
By Lemma \ref{BMth1.1}, one has
\begin{equation}\label{2.7.1}
\Ggd R\leq\min\{\GPacgd R, \GIacgd R\}\:.
\end{equation}
\end{ipg}

\begin{ipg}
A pair $(\sf{X},\sf{Y})$ of subcategories of $R$-modules is called a \emph{cotorsion pair} if
$\sf{X}^\perp=\sf{Y}$ and $\sf{Y}={^\perp\sf{X}}$. Here ${\sf X}^\perp=\{ A~|~\Ext^1_{R}(X,A)=0\ \text{for all}\ X\in\sf{X}\}$, and similarly one can define ${^\perp\sf{X}}$.
A cotorsion pair $(\sf{X},\sf{Y})$ is said to be \emph{hereditary} if $\Ext^n_R(X,Y)=0$ for all $X\in \sf{X}$, $Y\in \sf{Y}$ and $n\geq 1$, or equivalently, if ${\sf Y}$ is injectively coresolving (that is, whenever $0 \to L' \to L \to L'' \to 0$ is exact with $L', L \in\sf{Y}$ then $L''$ is also in $\sf Y$). A cotorsion pair $(\sf{X},\sf{Y})$ is called \emph{complete} if for any $R$-module $A$, there exist exact
sequences $0\to Y\to X\to A\to 0$ and/or $0\to A\to Y'\to X'\to 0$ with $X,X'\in\sf{X}$ and $Y,Y'\in\sf{Y}$.
\end{ipg}

\section{Global Gorenstein AC-homological dimensions}\label{Gacgd}
\noindent
In this section we focus on the global Gorenstein AC-projective/injective dimension. We let $\silac R=\sup\{\id_RM~|~M\ \text{is an absolutely clean}\ R\text{-module}\}$.

\begin{lem}\label{m-abclean}
Let $R$ be a ring. Then there exists an equality
\begin{equation*}
\max\{\Ggd R, \silac R\}=\GIacgd R.
\end{equation*}
\end{lem}
\begin{prf*}
For the inequality ``$\geq$", we let $\max\{\Ggd R, \silac R\}=m<\infty$. Then all absolutely clean $R$-module have finite injective dimension.
This implies that $\GIac=\GInj$. Thus we have $\GIacgd R=\Ggd R\leq m$.

For the inequality ``$\leq$", we let $\GIacgd R= n<\infty$. It is easy to see that $\Ggd R\leq\GIacgd R=n$. Next we prove that $\silac R\leq n$. Let $A$ be an absolutely clean $R$-module. For each $R$-module $M$, one has $\GIacid M\leq n$. So there is an exact sequence $0 \to M \to G^0 \to \cdots \to G^n \to 0$ with each $G^i$ Gorenstein AC-injective. Thus $\Ext^{n+1}_R(A,M)\cong\Ext^{1}_R(A,G^n)=0$. This yields that $A$ has finite projective dimension at most $n$. So $A$ has finite injective dimension at most $n$ by \corcite[2.7]{BM10}, as $\Ggd R\leq\GIacgd R=n$; see (\ref{2.7.1}). Thus one has $\silac R\leq n$.
\end{prf*}

The next result is immediate by Lemma \ref{m-abclean}.

\begin{lem}\label{abclean}
Let $R$ be a ring with $\GIacgd R$ finite. Then all absolutely clean $R$-modules have finite injective dimension at most $\GIacgd R$. Hence, all Gorenstein injective $R$-modules are Gorenstein AC-injective.
\end{lem}

The following two results are proved dually, where we let
$$\spll R=\sup\{\pd_RM~|~M\ \text{is a level}\ R\text{-module} \}.$$

\begin{lem}\label{m-level}
Let $R$ be a ring. Then there exists an equality
\begin{equation*}
\max\{\Ggd R, \spll R\}=\GPacgd R.
\end{equation*}
\end{lem}

\begin{lem}\label{level}
Let $R$ be a ring with $\GPacgd R$ finite. Then all level $R$-modules have finite projective dimension at most $\GPacgd R$. Hence, all Gorenstein projective $R$-modules are Gorenstein AC-projective.
\end{lem}

\begin{thm}\label{GIacgd}
The following statements hold:
\begin{prt}
\item If\: $\GIacgd R<\infty$, then there is an inequality $\GIacgd R\leq\GPacgd R$.
\item If\: $\GIacgd\Rop<\infty$, then there is an inequality $\GPacgd R\leq\GIacgd R$.
\end{prt}
\end{thm}
\begin{prf*}
(a) By Lemma \ref{abclean} all Gorenstein injective $R$-modules are Gorenstein AC-injective, so one has $\GIacgd R=\Ggd R\leq \GPacgd R$.

(b) We may assume that $\GIacgd R$ is finite. By Lemma \ref{abclean} all absolutely clean $\Rop$-modules have finite injective dimension, as $\GIacgd\Rop<\infty$. Let $L$ be a level $R$-module. Then by \thmcite[2.12]{BGH14} $L^+$ is a absolutely clean $\Rop$-module, and so  $\fd_R L=\id_RL^{+}<\infty$. Thus $\pd_RL<\infty$ by \corcite[2.7]{BM10} as $\Ggd R<\infty$.
Hence, one has $\sf{GP}=\GPac$; see Lemma \ref{level}. Thus $\GPacgd R=\Ggd R\leq\GIacgd R$; see \ref{global}.
\end{prf*}

The next result is proved dually.

\begin{thm}\label{GPacgd}
The following statements hold:
\begin{prt}
\item If\: $\GPacgd R<\infty$, then there is an inequality $\GPacgd R\leq\GIacgd R$.
\item If\: $\GPacgd\Rop<\infty$, then there is an inequality $\GIacgd R\leq\GPacgd R$.
\end{prt}
\end{thm}

The next corollary advertised in the introduction is immediate by Theorems \ref{GIacgd} and \ref{GPacgd}.

\begin{cor}\label{gacgdim}
If $R$ is a commutative ring, then $\GIacgd R=\GPacgd R$.
\end{cor}

\begin{lem}\label{Gac p and f4}
Let $M$ be an $R$-module and $n$ a nonnegative integer. Then the following conditions are equivalent.
\begin{eqc}
 \item $\GFacpd M\leq n$.
 \item There is an exact sequence $0\to M \to F \to N \to 0$ of $R$-modules
with $\fd_R F\leq n$ and $N\in\GPac$.
\end{eqc}
\end{lem}
\begin{prf*}
\proofofimp{i}{ii} We prove the result by induction on $n$. The case where $n=0$ holds by Proposition \ref{Gac p and f}.
Now let $n>0$. Consider an exact sequence
$0\to K \to H \to M \to 0$ of $R$-modules
with $H$ flat. Then one has $\GFacpd K\leq n-1$, and so by induction,
there is an exact sequence
$0\to K\to H'\to G\to 0$ of $R$-modules
with $\fd_RH'\leq n-1$ and $G\in \GPac$.
Consider the following pushout diagram
$$\xymatrix@C=20pt@R=20pt{&0\ar[d] &0\ar[d]\\
  0 \ar[r] & K \ar[r]^{}\ar[d] & H\ar[r]^{}\ar[d] & M \ar[r]\ar@{=}[d] & 0\\
  0 \ar[r] & H' \ar[r]^{}\ar[d] & H'' \ar[r]^{}\ar[d] & M \ar[r] & 0.\\
  &G\ar@{=}[r]\ar[d]&G\ar[d]\\
  &0&0}$$
In the middle columnn, by Lemma \ref{gacpf} both $H$ and $G$ are in $\GFac$, so is $H''$.
Whence, by Proposition \ref{Gac p and f}, there is an exact sequence
$0\to H''\to L\to N\to 0$ of $R$-modules
with $L\in \mathsf{Flat}$ and $N\in \GPac$.
Now we obtain another pushout diagram
$$\xymatrix@C=20pt@R=20pt{&&0\ar[d] &0\ar[d]\\
  0 \ar[r] & H' \ar[r]^{}\ar@{=}[d] & H''\ar[r]^{}\ar[d] & M \ar[r]\ar[d] & 0\\
  0 \ar[r] & H' \ar[r]^{} & L \ar[r]^{}\ar[d] & F \ar[r]\ar[d] & 0.\\
  &&N\ar@{=}[r]\ar[d]&N\ar[d]\\
  &&0&0}$$
Since, in the middle row, $L\in \mathsf{Flat}$ and $\fd_RH'\leq n-1$, it follows that $\fd_RF\leq n$. So the condition \eqclbl{ii} holds by the rightmost non-zero column.

\proofofimp[]{ii}{i} Assume that there is an exact sequence $0\to M \to F \to N \to 0$ of $R$-modules
with $\fd_R F\leq n$ and $N\in\GPac$. Since $(\GPac,{\GPac}^{\perp})$ is a hereditary complete cotorsion pair by Gillespie \cite[fact. 10.2]{Gil16}, there is an exact sequence $0\to E \to L \to F \to 0$ of $R$-modules
with $L\in \GPac$ and $E\in{\GPac}^{\perp}$.
Consider the next pullback diagram
$$\xymatrix@C=20pt@R=20pt{&0\ar[d] &0\ar[d]\\
  & E \ar[d]^{}\ar@{=}[r]& E \ar[d]^{}\\
  0 \ar[r] & Q \ar[r]^{} \ar[d]^{}& L \ar[r]^{}\ar[d] & N \ar[r]\ar@{=}[d] & 0\\
 0 \ar[r] & M \ar[r]^{} \ar[d]^{}& F \ar[r]^{}\ar[d] & N \ar[r] & 0.\\
  &0&0}$$
By the middle column one gets $\GFacpd E\leq n-1$ since $\GFacpd F\leq \fd_RF\leq n$ and $L\in \GPac\subseteq\GFac$; see Lemma \ref{gacpf}.
By the middle row one gets that $Q$ is in $\GPac\subseteq\GFac$ since $N$ and $L$ are in $\GPac$.
Thus, by the first non-zero column one has $\GFacpd M\leq n$.
\end{prf*}

\begin{prp}\label{Gfacgf}
Let $R$ be a ring with $\GFacgd R<\infty$. Then all Gorenstein flat $R$-modules are Gorenstein AC-flat.
\end{prp}
\begin{prf*}
We assume that $\GFacgd R=n<\infty$. Let $M$ be a Gorenstein flat $R$-module. Then one has $\GFacpd M\leq n$, and so by Lemma \ref{Gac p and f4} there is an exact sequence $0\to M \to F \to N \to 0$ of $R$-modules with $\fd_R F\leq n$ and $N\in\GPac\subseteq\GFac$. Since $M$ and $N$ are Gorenstein flat, a recent result by \v{S}aroch and \v{S}\v{t}ov\'{\i}\v{c}ek \thmcite[3.11]{SS18} yields that $F$ is Gorenstein flat. Thus $F$ is flat, and hence $M$ is Gorenstein AC-flat by Proposition \ref{Gac p and f}.
\end{prf*}

\begin{ipg}\label{cot-id}
The cotorsion dimension of $R$-module $M$, ${\sf Cot}{\text-}\id_R M$, is defined by declaring that ${\sf Cot}{\text-}\id_R M\leq n$ if and only if $M$ has a cotorsion coresolution of length $n$, that is, there is an exact sequence $0 \to M \to C^0 \cdots \to C^{n} \to 0$ with each $C^i$ cotorsion. We let $\mathsf{Cot}{\text-}{\rm gldim}R=\sup\{{\sf Cot}{\text-}\id_R M\ |\ M\ \text{is an}\ R\text{-module}\}$.
\end{ipg}

The next result was proved by Mao and Ding in \thmcite[19.2.14]{MD06}.

\begin{lem}\label{fpd}
For each $R$-module $M$ there exists an inequality
\begin{equation*}
\pd_RM\leq\fd_RM+\mathsf{Cot}{\text-}{\rm gldim}R.
\end{equation*}
\end{lem}

The next result is used in the proofs of Corollaries \ref{main-pd and fd} and \ref{main-pd and fd-coherent 1}.

\begin{thm}\label{cotgldim}
Let $R$ be a ring. Then there exist inequalities
\begin{equation*}
\max\{\GFacgd R, \mathsf{Cot}{\text-}{\rm gldim}R\}\leq\GPacgd R\leq\GFacgd R+ \mathsf{Cot}{\text-}{\rm gldim}R.
\end{equation*}
In particular, $\GPacgd R$ is finite if and only if $\GFacgd R$ and $\mathsf{Cot}{\text-}{\rm gldim}R$ are finite.
\end{thm}
\begin{prf*}
For the first inequality one let $\GPacgd R=n<\infty$. We notice that all Gorenstein AC-projective modules are Gorenstein AC-flat. So one has $\GFacgd R$ $\leq n$. Let $F$ be an flat $R$-module. Since $\Ggd R\leq\GPacgd R=n$, one has $\pd_RF\leq n$ by \corcite[2.7]{BM10}. Thus \corcite[7.2.6]{MD06} yields $\mathsf{Cot}{\text-}{\rm gldim}R\leq n$.

For the second inequality we let $\GFacgd R=n<\infty$ and $\mathsf{Cot}{\text-}{\rm gldim}R=m<\infty$. Let $M$ be an $R$-module. Then $\GFacpd M\leq n$. By Lemma \ref{Gac p and f4}, one gets an exact sequence $0\to M \to F \to N \to 0$ of $R$-modules
with $\fd_RF\leq n$ and $N \in\GPac$. So $\pd_RH\leq n+m$; see Lemma \ref{fpd}. Similar to the proof of \proofofimp[]{ii}{i} in Lemma \ref{Gac p and f4} one gets $\GPacpd M\leq n+m$. Thus $\GPacgd R\leq n+m$.
\end{prf*}

Next we give some rings that have finite global Gorenstein AC-projective/injective dimension.

\begin{thm}\label{exa-pd}
For a ring $R$ with $\Ggd R<\infty$, the following statements hold:
\begin{prt}
\item If $R$ is left coherent, then $\GIacgd R<\infty$.
\item If $R$ is right coherent, then $\GPacgd R<\infty$.
\end{prt}
\end{thm}
\begin{prf*}
(a) By Lemma \ref{m-abclean}, it is suffices to show that $\silac R<\infty$.
Let $A$ be an absolutely clean $R$-module, and let $\Ggd R=n<\infty$.
Then \corcite[2.9]{BGH14} yields that $A$ is FP-injective since $R$ is left coherent.
Hence, there is a pure exact sequence $0\to A\to I\to C\to 0$ of $R$-modules with $I$ injective. By \corcite[2.7]{BM10}, one has $\fd_RI\leq n$. It follows that $\fd_RA\leq \fd_RI\leq n$,
and hence one has $\id_RA\leq n$ again by \corcite[2.7]{BM10}.
This gives that $\silac R\leq n<\infty$.

(b) By Lemma \ref{m-level}, it is suffices to show that $\spll R<\infty$.
Let $L$ be a level $R$-module, and let $\Ggd R=n<\infty$. Then \corcite[2.11]{BGH14} yields that $L$ is flat since $R$ is right coherent.
Hence one has $\pd_RL\leq n$ by \corcite[2.7]{BM10}. This gives that $\spll R\leq n<\infty$.
\end{prf*}

In the followin we give an example to show that the converses of the statements in Theorem \ref{exa-pd} are not true in general. Before that we give some facts.

\begin{ipg}\label{product}
Let $R=\prod_{i=1}^{n}R_i$ be a direct product of rings. If $M_i$ is an $R_i$-module for $i=1,2,\cdots,n$ then $M=M_1 \oplus M_2 \oplus \cdots \oplus M_n$ is an $R$-module. Conversely, if $M$ is an $R$-module then it is of the form $M=M_1 \oplus M_2 \oplus \cdots \oplus M_n$, where $M_i$ is an $R_i$-module for $i=1,2,\cdots,n$. It is easy to see that the following equalities hold
\begin{equation*}
\GPacpd M = \sup\{\GPacpd[R_i]M_i~|~i = 1, ..., n\}
\end{equation*}
and
\begin{equation*}
\GIacid M = \sup\{\GIacid[R_i]M_i~|~i = 1, ..., n\},
\end{equation*}
which are parallel to the well-known ones about projective and injective dimension, respectively.
So one gets that $R$ is of finite global Gorenstein AC-projective/injective dimension if and only if each $R_i$ is so; the same conclusion holds for global dimension.
On the other hand, it is known that $R$ is left/right coherent if and only if each $R_i$ is so.
\end{ipg}

\begin{exa} \label{exa-1}
Let $R=D+(x_1,x_2)K[x_1, x_2]$, where $D$ is a Dedekind domain and $K$ its
quotient field. According to Kirkman and Kuzmanovich \cite[Example in p.128]{KK88},
$R$ is a commutative non-coherent ring of finite global dimension.
On the other hand, there exists a commutative Iwanaga-Gorenstein ring $S$ of infinite global dimension; see Bennis \cite[p.857]{DB09}. So $S$ has finite global Gorenstein AC-projective dimension and finite global Gorenstein AC-injective dimension; see Theorem \ref{exa-pd}.
Hence, $R\times S$ has finite global Gorenstein AC-projective dimension and finite global Gorenstein AC-injective dimension.
However, $R\times S$ is neither of finite global dimension nor coherent.
\end{exa}

\section{Compactly generatedness of singularity categories}
\noindent
We now turn to study the compactly generatedness of singularity categories and stable categories with respect to Gorenstein AC-homological modules, and prove Theorem \ref{thm2} advertised in the introduction. We open this section with the following terminology.

\begin{ipg}\label{dim comp}
Let $(\sf{A}, \sf{B})$ be a cotorsion pair in $\M{R}$, and let $X$ be an $R$-complex. From Yang and Ding \cite{YD15}, the \emph{$\sf{A}$-projective dimension} of $X$, ${\sf A}{\text-}\pd_RX$, is defined as
\begin{equation*}
{\sf A}{\text-}\pd_RX= \inf\{\sup A~|~X \simeq A\ \mathrm{in}\ \D(R)~\text{with}~ A \in \sf{dgA}\}.
\end{equation*}
The \emph{$\sf{B}$-injective dimension} of $X$, ${\sf B}{\text-}\id_RX$, is defined as
\begin{equation*}
{\sf B}{\text-}\id_RX= \inf\{-\inf B~|~X \simeq B\ \mathrm{in}\ \D(R)~\text{with}~ B \in \sf{dgB}\}.
\end{equation*}
Here $\sf dgA$ and $\sf dgB$ denote the subcategories of dg-${\sf A}$ complexes and dg-$\sf B$ complexes, respectively; see Gillespie \cite{Gi04}.

It is known that $({\sf Flat}, {\sf Cot})$ is a complete hereditary cotorsion pair, and by \cite{BGH14} and Proposition \ref{Gac p and f1} $(\GPac, \GPac^{\perp})$, $({^{\perp}\GIac},\GIac)$ and $(\GFac, (\GPac)^{\perp}\cap \sf{Cot})$ are complete hereditary cotorsion pairs. So for an $R$-complex $X$ we have the definitions of ${\GPac}{\text-}\pd_RX$, ${\GFac}{\text-}\pd_RX$, ${\GIac}{\text-}\id_RX$ and ${\sf Cot}{\text-}\id_RX$. We let $\catD{R}_{\widehat{\GPac}}$ (resp., $\catD{R}_{\widehat{\GFac}\cap\widehat{\sf{Cot}}}$, and $\catD{R}_{\widehat{\GIac}}$) denote the the triangulated subcategory of $\catD{R}$
consisting of all homology bounded complexes $X$ with ${\GPac}{\text-}\pd_RX<\infty$ (resp, ${\GFac}{\text-}\pd_RX<\infty$ and ${\sf Cot}{\text-}\id_RX<\infty$, and ${\GIac}{\text-}\id_RX<\infty$). It is easy to see that for an $R$-module $M$ (viewed as an $R$-complex concentrated in degree $0$), the definitions of ${\GPac}{\text-}\pd_RM$, ${\GFac}{\text-}\pd_RM$, ${\GIac}{\text-}\id_RM$ and ${\sf Cot}{\text-}\id_RM$ are the same as in \ref{global} and \ref{cot-id}.
\end{ipg}

\begin{lem}\label{frobe}
The subcategory $\GPac$ $($resp., $\GFac\cap\sf{Cot}$, and $\GIac$$)$
together with all short exact sequences in $\GPac$ $($resp., $\GFac\cap\sf{Cot}$, and $\GIac$$)$
forms a Frobenius category with projective-injective objects all projective $($resp., flat-cotorsion, and injective$)$ $R$-modules.
\end{lem}
\begin{prf*}
We give a straight proof for the case $\GIac$; see \ref{Frobenius exact category} for the other ones.

The subcategory $\GIac$, together with all short exact sequences in $\GIac$, forms an exact category, as $\GIac$ is closed under extensions by \lemcite[5.6]{BGH14}.

For $I\in {\sf Inj}$ and $G\in \GIac$, one gets that $\Ext_R^{1}(G,I)=0=\Ext_R^{1}(I,G)$, which yields that all injective $R$-modules are both projectives and injectives in $\GIac$. Conversely, let $M$ (resp., $N$) be a injective (resp., projective) object in $\GIac$. Then there exist split exact sequences $0\to M\to I\to M' \to 0$ and $0\to N'\to H\to N\to 0$ with $I,H\in {\sf Inj}$ and $M',N'\in \GIac$. So both $M$ and $N$ are in $\sf Inj$. Thus projectives and injectives in $\GPac$ are exactly injective $R$-modules.

Finally, for every $G\in\GIac$ there exist exact sequences $0\to G\to I'\to G'\to0$ and $0\to G''\to I''\to G\to0$ with $I', I''\in {\sf Inj}$ and $G', G''\in \GIac$, so the subcategory $\GIac$ has enough injectives and enough projectives.
\end{prf*}

\begin{ipg}
By Lemma \ref{frobe}, the stable category $\underline{\GPac}$ (resp., $\underline{\GFac\cap\sf{Cot}}$, and $\underline{\GIac}$) modulo projectives (resp., flat-cotorsions, and injectives) is a triangulated category.
\end{ipg}

\begin{thm}\label{equivalence}
The following conditions are equivalent.
 \begin{eqc}
 \item $\GPacgd R<\infty$.
 \item There is an equality $\catD{R}_{\widehat{\GPac}}=\catD{R}$.
 \item The natural functor $\mathrm{F}: \underline{\GPac} \to \catD{R}/\catK{\sf Prj}$ induced by the compositions
     \begin{equation*}
     \sf GPac\hookrightarrow \catD{R}_{\widehat{\GPac}}\to\catD{R}_{\widehat{\GPac}}/\catK{\sf Prj}\hookrightarrow\catD{R}/\catK{\sf Prj}
     \end{equation*}
     is a triangulated equivalence.
\end{eqc}
\end{thm}
\begin{prf*}
\proofofimp{i}{ii} Fix $P\in\catD{R}$. It suffices to show that ${\GPac}{\text-}\pd_RP<\infty$. Without loss of generality, we may assume that $P$ is bounded as follows:
\begin{equation*}
P=0\to P_k \to P_{k-1} \to \cdots \to P_{1}\to P_{0}\to 0\:.
\end{equation*}
Consider the exact sequence $0 \to P_0 \to P \to \Thb{1}{P} \to 0$ of $R$-complexes. Since $P_0$ and $\Thb{1}{P}$ have finite Gorenstein AC-projective dimension by \eqclbl{i} and induction on $k$, respectively, so does $P$.

\proofofimp{ii}{i} Each $R$-module $M$, viewed as an $R$-complex concentrated in degree $0$, is in $\catD{R}$. Thus $M\in\catD{R}_{\widehat{\GPac}}$,
and so $\GPacpd{M}<\infty$; see \ref{dim comp}. Note that $(\GPac, \GPac^{\perp})$ is a complete hereditary cotorsion pair. It is a standard way to see that for any family $(M_i)_{i\in\Lambda}$ of $R$-modules there is an equality
$$\GPacpd{(\oplus_{i\in\Lambda} M_i)}=\sup\{\GPacpd{M_i}\ |\ i\in\Lambda\}.$$
Thus it is easy to verify that the condition \eqclbl{i} holds.

\proofofimp{ii}{iii} From a result by Di, Liu, Yang and Zhang \corcite[5.9]{DLYZ18}, the induced natural functor $\mathrm{F}: \underline{\GPac} \to \catD{R}_{\widehat{\GPac}}/\catK{\sf Prj}$ is a triangulated equivalence, so the statement \eqclbl{iii} follows from \eqclbl{ii}.

\proofofimp{iii}{ii} It is clear that $\catD{R}_{\widehat{\GPac}}\subseteq\catD{R}$. Conversely, we let $X\in\catD{R}$ ($X$ is also an object of $\catD{R}/\catK{\sf Prj}$). By \eqclbl{iii} and \corcite[5.9]{DLYZ18}, the functor
\begin{equation*}
\catD{R}_{\widehat{\GPac}}/\catK{\sf Prj}\hookrightarrow\catD{R}/\catK{\sf Prj}
\end{equation*}
is a triangulated equivalence. We notice that each triangulated equivalence is dense. So $X$ is isomorphic to an $R$-complex in $\catD{R}_{\widehat{\GPac}}/\catK{\sf Prj}$. It follows that $X$ is in $\catD{R}_{\widehat{\GPac}}$.
\end{prf*}

\begin{thm}\label{equivalence-f}
The following conditions are equivalent.
 \begin{eqc}
\item $\GFacgd R<\infty$ and $\mathsf{Cot}{\text-}{\rm gldim}R<\infty$.
\item There is an equality $\catD{R}_{\widehat{\GFac}\cap\widehat{\sf{Cot}}}=\catD{R}$.
\item There is a triangulated equivalence
   \begin{equation*}
   \underline{\GFac\cap\sf{Cot}}\simeq \catD{R}/\catK{\sf FlatCot}.
   \end{equation*}
\item The natural functor $\mathrm{F}: \underline{\GFac\cap\sf{Cot}} \to \catD{R}/\catK{\sf FlatCot}$ induced by the compositions
     $\GFac\cap\sf{Cot}\hookrightarrow \catD{R}_{\widehat{\GFac}\cap\widehat{\sf{Cot}}}\to \catD{R}_{\widehat{\GFac}\cap\widehat{\sf{Cot}}}/\catK{\sf FlatCot}\hookrightarrow\catD{R}/\catK{\sf FlatCot}$
     is a triangulated equivalence.
\end{eqc}
\end{thm}
\begin{prf*}
Analogous to the proof of Theorem \ref{equivalence}, using Corollary \ref{Gac p and f3} instead of \corcite[5.9]{DLYZ18}.
\end{prf*}

\begin{cor}\label{main-pd and fd}
Let $R$ be a ring with $\GPacgd R$ finite. Then
\begin{equation*}
\catD{R}/\catK{\sf Inj}\simeq\catD{R}/\catK{\sf Prj}\simeq\underline{\GPac}\simeq \underline{\GFac\cap\sf{Cot}}\simeq \catD{R}/\catK{\sf FlatCot}
\end{equation*}
are compactly generated.
\end{cor}
\begin{prf*}
The first equivalence in the statement holds by \thmcite[6.9]{Beligiannis2000} since $\Ggd R$ is finite; see \ref{2.7.1}. The second equivalence follows from Theorem \ref{equivalence}, the third one holds by Corollary \ref{triequivalence}, and the last one follows from Theorems \ref{cotgldim} and \ref{equivalence-f}. By a careful
reading of the proof of Gillespie \thmcite[6.2]{Gil19}, one gets that $\underline{\GPac}$ is compactly generated.
\end{prf*}

It is from \cite[lem. 6.6 and thm. 6.7]{Be01} that if $R$ is a right coherent and left perfect or left Morita ring with $\Ggd R<\infty$ then $\underline{\sf GP}\simeq\underline{\sf GI}$ are compactly generated. The same conclusion holds if $R$ is Iwanaga-Gorenstein; see \thmcite[9.4]{Ho02} or \thmcite[4.1]{Chen11}. We have the next improved result.

\begin{cor}\label{main-pd and fd-coherent 1}
Let $R$ be a right coherent ring with $\Ggd R<\infty$. Then
\begin{equation*}
\catD{R}/\catK{\sf Inj}\simeq\catD{R}/\catK{\sf Prj}\simeq\underline{\sf GP}\simeq\underline{\sf GI}\simeq \underline{\sf GF\cap\sf{Cot}}\simeq \catD{R}/\catK{\sf FlatCot}
\end{equation*}
are compactly generated.
\end{cor}
\begin{prf*}
By Theorem \ref{exa-pd} one has $\GPacgd R$ finite. We notice that $\underline{\sf GP}\simeq\underline{\sf GI}$ by \thmcite[6.9]{Beligiannis2000} as $\Ggd R<\infty$. On the other hand, by Lemma \ref{level} one has $\GPac={\sf GP}$, and the equality $\GFac={\sf GF}$ holds by Proposition \ref{Gfacgf} and Theorem \ref{cotgldim}. So the desired result in the statement follows from Corollary \ref{main-pd and fd}.
\end{prf*}

Dual to the proof of Theorem \ref{equivalence}, we have the following result.

\begin{thm}\label{equivalence-dual}
The following conditions are equivalent.
 \begin{eqc}
 \item $\GIacgd R<\infty$.
  \item There is an equality $\catD{R}_{\widehat{\GIac}}=\catD{R}$
 \item The natural functor $\mathrm{F}: \underline{\GIac} \to \catD{R}/\catK{\sf Inj}$ induced by the compositions
     \begin{equation*}
     \sf GIac\hookrightarrow \catD{R}_{\widehat{\GIac}}\to\catD{R}_{\widehat{\GIac}}/\catK{\sf Inj}\hookrightarrow\catD{R}/\catK{\sf Inj}
     \end{equation*}
     is a triangulated equivalence.
\end{eqc}
\end{thm}

From Gillespie \dfncite[5.1]{Gi17}, a complex $I$ of injective $R$-modules is called \emph{AC-injective} if each chain map into $I$ from an acyclic complex with each cycle absolutely clean
is null homotopic.

\begin{prp}\label{kac-inj}
Let $R$ be a ring with $\GIacgd R$ finite. Then all complexes of injective $R$-modules are AC-injective.
\end{prp}
\begin{prf*}
We let $\sf{dw}\widetilde{\sf Inj}$ denote the subcategory of complexes of injective $R$-modules. Let $I\in\sf{dw}\widetilde{\sf Inj}$, and let $\alpha: X\to I$ be an homomorphisms of $R$-complexes with $X$ acyclic and each cycle $\Z_{i}(X)$ absolutely clean. Next we prove that $\alpha$ is null homotopic. Set $n=\GIacgd R<\infty$. Then by Lemma \ref{abclean} each cycle $\Z_{i}(X)$ has finite injective dimension $\leq n$, and hence has finite flat dimension $\leq n$ as $\Ggd R\leq n$. On the other hand, by \prpcite[7.2]{Gil16} the pair $(^{\perp}{\sf{dw}\widetilde{\sf Inj}},\sf{dw}\widetilde{\sf Inj})$ is an injective cotorsion pair in $\Ch(R)$. Then it follows from Gillespie \corcite[3.3]{Gil17} that $X$ is in $^{\perp}{\sf{dw}\widetilde{\sf Inj}}$, and so is $\sf{\Sigma}X$. Thus one has $\Ext^{1}_{\Ch(R)}({\sf\Sigma}X,I)=0$. This yields that the exact sequence $0 \to I \to \sf{Cone}\alpha \to {\sf\Sigma}X \to 0$ is split. So $\alpha$ is null homotopic; see Enochs, Jenda and Xu \lemcite[3.2]{EJX96}.
\end{prf*}

\begin{cor}\label{main-id}
Let $R$ be a ring with $\GIacgd R$ finite. Then
\begin{equation*}
\underline{\GIac}\simeq\catD{R}/\catK{\sf Inj}\simeq\catD{R}/\catK{\sf Prj}
\end{equation*}
are compactly generated.
\end{cor}
\begin{prf*}
The first equivalence in the statement holds by Theorem \ref{equivalence-dual}, and the second one follows from \thmcite[6.9]{Beligiannis2000} as $\Ggd R<\infty$. Next we prove that $\underline{\GIac}$ is compactly generated. Let ${\sf S}(\sf ACInj)$ denote the homotopy category of all acyclic AC-injective $R$-complexes, and let $\sf{K}_{\sf ac}(\sf Inj)$ (resp., $\sf{K}_{\sf tac}(\sf Inj)$) denote the homotopy category of acyclic (resp., totally acyclic) complexes of injective $R$-modules. Consider the following equivalences:
  \begin{equation*}
    \underline{\GIac}=\underline{\mathsf{GI}}\simeq \sf{K}_{\sf tac}(\sf Inj)=\sf{K}_{\sf ac}(\sf Inj)={\sf S}(\sf ACInj)\:.
  \end{equation*}
Here the first equality holds by Lemma \ref{abclean}. Since $\Ggd R$ is finite by \ref{2.7.1}, all $R$-module have finite Gorenstein injective dimension by Lemma \ref{BMth1.1}. It follows that every acyclic complex of injective $R$-modules has Gorenstein injective cycles and so it is totally acyclic. This yields that the second equality holds. The last equality follows from Lemma \ref{kac-inj}; while the equivalence holds by Krause \prpcite[7.2]{HKr05}. Finally, from \thmcite[5.8 and 4.6]{Gi17} that ${\sf S}(\sf ACInj)$ is compactly generated.
\end{prf*}

Let $R$ be a ring with $\GIacgd R$ finite. Then one has $\underline{\sf GP}\simeq\underline{\sf GI}$ by \thmcite[6.9]{Beligiannis2000} as $\Ggd R<\infty$; see \ref{global}. On the other hand, by Lemma \ref{abclean}, the equality $\GIac={\sf GI}$ holds. So the next result is immediate by Theorem \ref{exa-pd} and Corollary \ref{main-id}.

\begin{cor}\label{main-pd and fd-coherent 2}
Let $R$ be a left coherent ring with $\Ggd R<\infty$. Then
\begin{equation*}
\catD{R}/\catK{\sf Inj}\simeq\catD{R}/\catK{\sf Prj}\simeq\underline{\sf GP}\simeq\underline{\sf GI}
\end{equation*}
are compactly generated.
\end{cor}

We close this section with the following example; it shows that coherent rings of finite global Gorenstein dimension may not be Iwanaga-Gorenstein nor perfect nor Morita\footnote{See \cite{Be01} for the definition of Morita rings. It is known that a ring $R$ is left Morita if and only if $R$ is left Artinian and $\M{R}$ has a finitely generated injective cogenerator.}. Let $R=\prod_{i=1}^{n}R_i$ be a direct product of rings (see \ref{product}). It is easy to see that $R$ is Iwanaga-Gorenstein (resp., left perfect and left Morita) if and only if each $R_i$ is Iwanaga-Gorenstein (resp., left perfect and left Morita).

\begin{exa}\label{exa-2}
Let $R=\mathbb{Z}$ and $S=\left(
         \begin{array}{cc}
           \mathbb{Q} & \mathbb{R} \\
           0 & \mathbb{Q} \\
         \end{array}
       \right)$.
Then $R$ is a commutative Iwanaga-Gorenstein ring that is neither perfect nor Artin; hence $R$ is commutative coherent with $\Ggd R<\infty$.
According to Wang \exacite[3.4]{Wang17} $S$ is a commutative perfect coherent (non-noetherian) ring with $\Ggd S<\infty$. Then the direct product $R\times S$ is a commutative coherent ring with $\Ggd (R\times S)=\sup\{\Ggd R, \Ggd S\}<\infty$, which is neither Iwanaga-Gorenstein nor perfect nor Morita.
\end{exa}

\appendix
\section*{Appendix. Gorenstein AC-flat modules}
\stepcounter{section}
\noindent
In this section we give some properties of Gorenstein AC-flat modules. We notice that all Gorenstein AC-projective $R$-modules are Gorenstein AC-flat. Actually, by \thmcite[A.6]{BGH14}, an $R$-module $M$ is Gorenstein AC-projective if and only if there exists an exact sequence $\cdots\to P_1\to P_0\to P_{-1}\to \cdots$ of projective $R$-modules such that
$M\cong\mathrm{Coker}(P_1\to P_0)$, and it remains exact after applying the functor $A\otimes_R-$ for each absolutely clean $\Rop$-module $A$. The next two results are from Estrada, Iacob and P\'{e}rez \thmcite[2.12]{EIP18} and \exacite[2.17(2)]{EIP18}.

\begin{prp}\label{Gac p and f}
The following conditions are equivalent for an $R$-module $M$.
 \begin{eqc}
 \item $M$ is Gorenstein AC-flat.
\item There is a short exact sequence $0\to K\to L\to M\to0$ of $R$-modules
with $K\in\sf{Flat}$ and $L\in\GPac$, and it remains exact after applying the functor
$\Hom_R(-,C)$ for any (flat) cotorsion $R$-module $C$.
\item $\Ext^1_{R}(M,C)=0$ holds for all cotorsion $R$-modules $C\in (\GPac)^\perp$.
\item There is a short exact sequence $0\to M \to F \to N \to 0$ of $R$-modules
with $F\in\sf{Flat}$ and $N\in\GPac$.
 \end{eqc}
\end{prp}

\begin{prp}\label{Gac p and f1}
The pair $(\GFac, (\GPac)^{\perp}\cap \sf{Cot})$ is a
complete and hereditary cotorsion pair with the equality
$\GFac\cap(\GPac)^{\perp}=\sf{Flat}$.
\end{prp}

Recall that a triple $(\sf{Q},\sf{W},\sf{R})$ of classes of $R$-modules is Hovey triple if $\sf{W}$ is thick
and $(\sf{Q}\cap \sf{W}, \sf{R})$ and $(\sf{Q}, \sf{W}\cap \sf{R})$ are complete cotorsion pairs. If furthermore the above two cotorsion pairs are hereditary then the Hovey triple $(\sf{Q},\sf{W},\sf{R})$ is called hereditary. From Hovey \thmcite[2.2]{Ho02}, an abelian model structure on $\M{R}$ is equivalent to a Hovey triple. This fact is known as ``Hovey correspondence'' in the literature. We hence always denote an abelian model structure $\mathcal{M}$ as a Hovey triple $\mathcal{M}=(\sf{Q},\sf{W}, \sf{R})$.

For an abelian model structure $\mathcal{M}=(\sf{Q},\sf{W}, \sf{R})$, we denote by Ho$(\mathcal{M})$ the homotopy category of $\mathcal{M}$.
By Gillespie \cite[sec 4 and 5]{Gil162}, for any hereditary Hovey triple $\mathcal{M}=(\sf{Q},\sf{W},\sf{R})$, there
is a Frobenius exact category $\sf{Q}\cap \sf{R}$ whose projective-injective objects are precisely those in $\sf{Q}\cap \sf{R}\cap\sf{W}$. Furthermore, the stable category $\underline{\sf{Q}\cap \sf{R}}$ is triangulated equivalent to $\Ho(\mathcal{M})$. This triangulated equivalence is known as the fundamental theorem of model categories in the literature.

\begin{ipg}\label{Frobenius exact category}
By \cite{BGH14}, the triple $\mathcal{M}=(\GPac, \GPac^{\perp}, \M{R})$
is a hereditary Hovey triple. As an immediate consequence of Proposition \ref{Gac p and f1} one gets that the triple $\mathcal{M}'=(\GFac, \GPac^\perp, \sf{Cot})$ is a hereditary Hovey triple, which can also be found in \corcite[4.3]{EIP18}. Thus the category $\GPac$ (resp., $\GFac\cap\sf{Cot}$) is a Frobenius category with projective-injective objects all projective (resp., flat-cotorsion) $R$-modules. By the fundamental theorem of model categories, $\underline{\GPac}$ (resp., $\underline{\GFac\cap \sf{Cot}}$) is triangulated equivalent to ${\rm Ho}(\mathcal{M})$ (resp., ${\rm Ho}(\mathcal{M}')$).
\end{ipg}

It follows from Estrada and Gillespie \lemcite[5.4]{EG19} that if two hereditary Hovey triples $\mathcal{M} =(\sf{Q}, \sf{W}, \sf{R})$ and $\mathcal{M}' = (\sf{Q}', \sf{W}, \sf{R}')$ on $\M{R}$ have the same class $\sf{W}$ of trivial objects and if $\sf{Q} \subseteq \sf{Q'}$ (or equivalently, $\sf{R'} \subseteq \sf{R}$), then there is a triangulated equivalence ${\rm Ho}(\mathcal{M})\simeq{\rm Ho}(\mathcal{M}')$. Applying this fact to the hereditary Hovey triples $\mathcal{M}=(\GPac, \GPac^\perp, \M{R})$ and $\mathcal{M}'=(\GFac, \GPac^\perp, \sf{Cot})$ (in view of \ref{Frobenius exact category}) we get

\begin{cor}\label{triequivalence}
There exits a triangulated equivalence $\underline{\GPac}\simeq\underline{\GFac\cap \sf{Cot}}$.
\end{cor}

Note that the pairs $({\sf Flat},{\sf Cot})$ and $(\GFac, (\GPac)^{\perp}\cap \sf{Cot})$ are complete hereditary cotorsion pairs with $\GFac\cap(\GPac)^{\perp}\cap{\sf Cot}=\sf{Flat}\cap{\sf Cot}$; see Proposition \ref{Gac p and f1}. The following result is immediate by \thmcite[4.5]{DLYZ18}.

\begin{cor}\label{Gac p and f3}
There exists a triangle equivalence
\begin{equation*}
\underline{\GFac\cap \sf{Cot}}\simeq \catD{R}_{\widehat{\GFac}\cap\widehat{\sf{Cot}}}/\catK{\sf FlatCot}.
\end{equation*}
\end{cor}

\section*{Acknowledgments}
\noindent
We thank James Gillespie for conversations and comments on an early draft of this paper, and
we extend our gratitude to the referee for valuable comments that have improved the presentation at several points.

\bibliographystyle{amsplain-nodash}

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TITLE: topological class of a flat 2-bundle with nontrivial extension class
QUESTION [3 upvotes]: So a $2$-group presented by a crossed module
$H\overset{t}{\to} G$,
where $t$ has nontrivial kernel and cokernel, is given just by the data of that kernel and cokernel, the action of the cokernel on the kernel, and the class in the group cohomology $H^3(\text{coker } t, \text{ker } t)$ classifying the extension above. Therefore, a principal $2$-bundle with fiber the $2$-group above (or at least a fake-flat one (?)) should also be presented by this data.
So, how do you do it?
I think that the extension class can be interpreted as a $2$-bundle with fiber $\text{ker t}$ associated via the action I mentioned to some principal $2$-bundle with fiber $\text{coker }t$ over the classifying space $B\text{coker t}$. Then the classifying map of the residual $\text{coker }t$ bundle on my space can be used to pull back this 2-bundle. Is its topological class the same as the principal $2$-bundle with fiber $\text{ker }t$ from the reduction of structure group of my original bundle (assuming it's fake-flat)?
Thanks.

REPLY [2 votes]: Here is the general story (as in section 4.3 of Principal infinity-bundles -- General theory).
So consider
$$
  A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G
$$
a "central" extension of higher groups, hence a long homotopy fiber sequence of the form
$$
  \array{
    \mathbf{B}A &\stackrel{}{\to}& \mathbf{B}\hat G
    \\
    && \downarrow
    \\
    && \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 A
  }
  \,,
$$
where the map $\mathbf{c}$ is the cocycle that classifies the extension. For instance if $A = \mathbf{B} ker(t)$ then this is a higher group 3-cocycle on $G$ with coefficients in $ker(t)$.
Now by the pasting law for homotopy pullbacks, if you have a $\hat G$-principal infinity-bundle modulated by a map
$$
  X \stackrel{\hat g}{\to} \mathbf{B}\hat G
$$
with, hence, underlying $G$-principal $\infty$-bundle modulated by
$$
  g : X \stackrel{\hat g}{\to} \mathbf{B}\hat G \stackrel{}{\to} \mathbf{B}G
$$
(this may be an ordinary principal bundle if $G$ is a 0-truncated $\infty$-group, as in your case)
then we get the pasting diagram of homotopy pullbacks of the form
$$
  \array{
    \hat G &\to& \hat P &\to& \ast
    \\
    \downarrow && \downarrow && \downarrow
    \\
    G &\to& P &\to& \mathbf{B}A  &\to& \ast
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    \ast &\stackrel{x}{\to}& X &\stackrel{\hat g}{\to}& \mathbf{B}\hat G &\stackrel{}{\to}& \mathbf{B}G
  }
  \,.
$$
Here we read off

$\hat P \to X$ is the $\hat G$-principal $\infty$-bundle modulated by $\hat g$
$P \to X$ is the underlying $G$-principal $\infty$-bundle;
$\hat P \to P$ is an $A$-principal $\infty$-bundle on the total space of $P$ with the special property that restricted to the fibers it becomes the extension $\hat G \to G$ that we started with

This is the stage-wise decomposition of principal $\infty$-bundles which is induced from the extension of $\infty$-groups that we started with. This construction establishes an equivalence of $\infty$-categories between $\hat G$-principal $\infty$-bundles on $X$ and $A$-principal $\infty$-bundles on total spaces of $G$-principal $\infty$-bundles satisfying these compatibility conditions.
An important example of this is the case where $\hat G$ is the smooth string 2-group sitting in the extension
$$
  \mathbf{B} U(1) \to String \to Spin
  \,.
$$
and classfied by the smooth refinement of the first fractional Pontryagin class $\frac{1}{2}\mathbf{p}_1$. Here the above tells us that String-2 bundles are equivalently circle 2-bundles on total spaces of Spin-principal bundles which restrict to the canonical bundle gerbe on each fiber.
(I could say more, but I need to stop as the updating of the formula typesetting almost kills my little computer now. This used to be better with the previous MO version, where I could select "one shot math preview"...)


TITLE: Dirichlet theorem on primes premise
QUESTION [0 upvotes]: Dirichlet theorem on primes in arithmetic progressions: Suppose a and
  b are positive coprime integers, then $\exists$ infinitely many primes
  $p$ such that $p\equiv a\pmod b$.

Why is it necessary for a and b to be coprime?
My approach is to suppose $gcd(a,b)=d$ with $d>1$ such that $a'=\frac{a}{d}$ and $b'=\frac{b}{d}$.
If $p\equiv a\pmod b$ this implies $p-a=bq$ for $q\in \mathbb{Z}$ and $p=bq+a$
If we divide both sides by $d$, we get $\frac{p}{d}=b'q+a'$
$p$ is prime so $d$ does not divide $p$ and $b'q+a'$ is an integer which means $\frac{p}{d}\ne b'q+a'$
So if $a$ and $b$ are not coprime, then $p$ cannot be congruent $a$ modulo $b$.
Is this approach correct?

REPLY [1 votes]: No, it is not correct. Note that $2$ is prime and $2$ and $2$ are not coprime, but $2\equiv2\pmod 2$.
What happens when $a$ and $b$ are not coprime is that there is at most one prime number $p$ such that $p\equiv a\pmod b$.


TITLE: What is the infinity norm on a continuous function space?
QUESTION [4 upvotes]: As I understand it the norm $\|f\|_2$ on the set of continuous functions on $[0,1]$ is defined by $$\|f\|_2 = \sqrt{\int_0^1|f(t)|^2dt}$$ but what is the infinity norm $\|f\|_\infty$? Is it $$\int_0^1 \max|f(t)|dt$$ so in other words just $\max|f(t)|$?

REPLY [8 votes]: Up there I think you meant to write
$$ \|f\|_2 := \sqrt{\int_0^1|f(t)|^2dt} $$
The $\sup$-norm is usually defined as follows
$$ \|f\|_\infty := \sup_{x \in [0,1]} |f(x)|$$
Where $f: [0,1] \to \mathbb R$ is continuous, for example. So you get a normed space of functions $(C([0,1]), \|\cdot\|_\infty)$.


TITLE: What's the simplest way to compute the average distance to the center of points in the n-ball?
QUESTION [0 upvotes]: Define the standard $n$-ball as usual:
$$B_n(R) = \{x\in\mathbb{R}^n:\|x\|_2\leq R\}$$
How to compute the mean distance to the center of $B_r(R)$?
Specifically I am looking for simple, intuitive ways to approach this problem.

What I tried (not very simple nor very intuitive...)
Divide the $n$-ball into all the (n-1)-spheres inside it (i.e. $S^{n-1}(r) = \{x\in\mathbb{R}^n:\|x\|_2=r\}$ for $0\leq r \leq R$) and denote its sufrace area by $S_{n-1}(r)$
Then one computes
$$\frac{\int_0^R r S_{n-1}(r)dr}{V_n(R)}$$
where $V_n(R)$ is the volume of the $n$-ball.
Simplifying the above expression and using the formulas for the surface area and volume of an $n$-sphere
$$S_{n}(r) = \frac{2\pi^{\frac{n + 1}{2}}}{\Gamma(\frac{n+1}{2})}r^n$$
$$V_n(r)=\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}r^n$$
One arrives at
$$\frac{\int_0^R r \frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}r^{n-1}dr}{\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n}=\frac{2\Gamma(\frac{n}{2}+1)}{R^n\Gamma(\frac{n}{2})}\int_0^R  r^ndr=\frac{2\Gamma(\frac{n}{2}+1)}{\Gamma(\frac{n}{2})}\frac{R}{n+1}=\frac{n}{n+1}R$$

Is there a simpler approach?

REPLY [1 votes]: If $X$ represents the distance to the origin then
$$\mathbb{P}(X\le r)=\cases{0& if $r<0$\cr r^n/R^n& if $0\le r\le R$\cr
1& if $r\ge R$}$$
So, the density function of $X$ is
$$f_X(r)=n\frac {r^{n-1}}{R^n}\mathbb{I}_{[0,R]}(r)$$
Hence
$$\mathbb{E}(X)=\frac{n}{R^n}\int_0^R r^ndr=\frac{n}{n+1}R$$
Which is the expected distance to the origine.


TITLE: Compute characteristic classes of principal bundle over closed surfaces
QUESTION [4 upvotes]: Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$.
The following is my question:

Let $G$ be a semisimple connected (or even compact) Lie group, $\beta\colon\pi_1(\Sigma)\to G$ a group homomorphism, and $\Sigma$ a closed oriented surface. Consider the universal cover $\tilde{\Sigma}\to\Sigma$ which is a principal $\pi_1(\Sigma)$-bundle. Form the associated bundle $\tilde{\Sigma}\times_\beta G$ which is necessarily a principal $G$-bundle over $\Sigma$. Hence, it should have a characteristic class $c(\tilde{\Sigma}\times_\beta G)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$. Is there a way to compute this characteristic class in terms of $\beta$?

The difficulty here is that $\pi_1(\Sigma)$ is not connected, and hence I cannot use the functoriality of the characteristic class. Another one is that I am not sure how to compute the induced map $B\beta\colon B\pi_1(\Sigma)\to BG$ concretely.
I've asked this question on MSE and haven't got any answers yet.

REPLY [1 votes]: Note sure if you can call it "calculating" but there is a rather straightforward construction. Let us fix some generators $A_i, B_i$ of $\pi_1(\Sigma)$ (i.e. a canonical system of curves such that slicing $\Sigma$ along them gives the fundamental polygon).
Consider the central extension of $\pi_1$
$$ \mathbb{Z} \to \Gamma \to \pi_1(\Sigma),$$
where $\Gamma$ is generated by $A_i, B_i$ and the central element $J = \Pi_i [A_i, B_i]$ (with the commutator being $[A, B] = ABA^{-1}B^{-1}$).
Since this extension is universal, every group homomorphism $\beta: \pi_1(\Sigma) \to G$ lifts to a group homomorphism $\hat{\beta}: \Gamma \to \tilde{G}$ where $\tilde{G}$ is the universal covering group. Then $\hat{\beta}(J) \in \pi_1(G)$ is the characteristic class of the bundle coming from $\beta$.
I think more details about this construction can be found in Atiyah and Bott's classical work on the Yang-Mills equation over surfaces.


TITLE: why must $|A\vec v_1|^2$ is the highest value ?if the $\vec v_1$ is the vector which is corresponding to the biggest singular value of A
QUESTION [0 upvotes]: If A is a $3$ by $2$ matrix,if we do the singular value decomposition (SVD) to A,that is
$A=
    \begin{bmatrix}
    \vec u_1  & \vec u_2 &\vec u_3
    \end{bmatrix}
    \begin{bmatrix}
    s_1  & 0 \\
     0 & s_2 \\
      0 & 0 \\
    \end{bmatrix}
 \begin{bmatrix}
    \vec v_1  \\
    \vec v_2 \\
    \end{bmatrix}
$
$\vec u_1$ , $\vec u_2$ and  $\vec u_3$ are both a column vector,that is,$3$ by $1$ vector
$\vec v_1$ and  $\vec v_2$ are both a row vector,that is,$1$ by $2$ vector
$s_1$ and $s_2$ are both singular value,and $s_1>s_2$.
Now ,why must $|A\vec v_1|^2$ is the highest value ?and why must $|A\vec v_1|^2$ > $|A\vec v_2|^2$?,Is there any theory can prove it?
$|f|^2=f^Hf$,if the f is a $N$ by $1$ vector

REPLY [1 votes]: It is not entirely clear from your question, but it looks like you're trying to prove that $$
\max_{x \in \Bbb C^2, |x| = 1} |Ax| = s_1.
$$
To that end: we have $A = U \Sigma V^T$.  Note the following:

For $x \in \Bbb C^2$, $|Ax| = |U (\Sigma V^Tx)| = |\Sigma V^Tx|$.
For any vector $x$, $y = V^Tx$ is a vector of the same magnitude.

We can therefore state that
$$
\max_{x \in \Bbb C^2, |x| = 1} |Ax| =
\max_{x \in \Bbb C^2, |x| = 1} |\Sigma (V^Tx)| =
\max_{y \in \Bbb C^2, |y| = 1} |\Sigma y|.
$$
From here, it's easy to prove the desired result by noting that if $y = (y_1,y_2)^T$, we have
$$
|\Sigma y|^2 = s_1 |y_1|^2 + s_2 |y_2|^2 .
$$


TITLE: Show that $(\mathbb{Z}/7\mathbb{Z})^\ast$ is isomorphic to $\mathbb{Z}/6\mathbb{Z}$
QUESTION [1 upvotes]: I know how to prove one to one and onto and I define the function $\phi:\mathbb{Z}/6\mathbb{Z} \to (\mathbb{Z}/7\mathbb{Z})^\times$ by $\phi([n]) = [n+1]$, but i don't know how to prove $\phi(ab)= \phi(a)\phi(b)$.
Thanks

REPLY [2 votes]: Hint: $\phi([n]) = [n+1]$ does not define a group isomorphism: this would imply
$$
[3] = \phi([2]) = \phi([1] + [1]) = \phi([1]) \phi([1]) = [2][2] = [4] \quad \mbox{in $\mathbb{Z}/7\mathbb{Z}^\ast$}
$$
which is false. Instead, consider $\phi([n]) = [3^n]$.


text:
\begin{document}
\title[The fifth-order KPII equation]
{Global well-posedness for the fifth-order Kadomtsev-Petviashvili II equation in anisotropic Gevrey spaces}

\author[A. Boukarou, D. O. da Silva, K. Guerbati, Kh. Zennir]
{A. Boukarou, D. O. da Silva, K. Guerbati and Kh. Zennir}
\address{Aissa Boukarou \newline
	Laboratoire de Math\'ematiques et Sciences appliqu\'ees Universit\'e de Ghardaia, Algeria}
\email{boukarou.aissa@univ-ghardaia.dz}
\address{Daniel Oliveira da Silva \newline
	Department of Mathematics, Nazarbayev University, Nur-Sultan, Kazakhstan}
\email{daniel.dasilva@nu.edu.kz}
\address{Kaddour Guerbati \newline
	Laboratoire de Math\'ematiques et Sciences appliqu\'ees Universit\'e de Ghardaia, Algeria}
\email{guerbati\_k@yahoo.com}
\address{Khaled Zennir \newline
Department of Mathematics, College of
Sciences and Arts, Qassim University, Ar-Rass, Kingdom of Saudi
Arabia}
\email{k.zennir@qu.edu.sa}

\subjclass[2010]{35Q35, 35Q53}
\keywords{KPII equation, Gevrey space, radius of spatial analyticity}
\begin{abstract}
We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
 \newtheorem{thm}{Theorem}[section]
 \newtheorem{cor}[thm]{Corollary}
 \newtheorem{lem}[thm]{Lemma}
 \newtheorem{prop}[thm]{Proposition}
 \theoremstyle{definition}
 \newtheorem{defn}[thm]{Definition}
 \theoremstyle{remark}
 \newtheorem{rem}[thm]{Remark}
 \newtheorem*{ex}{Example}
 \numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
 The fifth-order Kadomtsev-Petviashvili II (KPII) equation is the partial differential equation
 \begin{equation}
 \left\{
 \begin{array}{ll}\label{p1}
 \partial_{t}u - \partial_{x}^{5}u + \partial^{-1}_{x} \partial^{2}_{y} u + u \partial_{x} u = 0, \\
 u(x,y,0) = f(x,y),
 \end{array}
 \right.
 \end{equation}
 where $u = u(x,y,t)$ and $(x,y,t) \in \mathbb{R}^{3}$.  This equation belongs to the class of KP equations, which are models for the propagation of long dispersive nonlinear waves which are essentially unidirectional and have weak transverse effects.  Due to the asymmetric nature of the equation with respect to the spatial derivatives, it is natural to consider the Cauchy problem for \eqref{p1} with initial data in the anisotropic Sobolev spaces $H^{s_1, s_2} (\mathbb{R}^{2})$, defined by the norm
 \[
 \| u \|_{H^{s_1, s_2}(\mathbb{R}^{2})} = \left( \int \langle \xi \rangle^{2 s_1} \langle \eta \rangle^{2 s_2} | \tilde{u}(\xi, \eta) |^{2} d\xi d\eta \right)^{1/2},
 \]
 where $\tilde{f}$ denotes the spatial Fourier transform
\[
\tilde{f}(\xi, \eta, t) = \int_{\mathbb{R}^{2}} f(x,y,t) e^{-i (x \xi + y \eta)}\ dx dy.
\]
For initial data in these spaces, Saut and Tzvetkov \cite{ST2000} proved that the problem \eqref{p1} is locally well-posed for initial data $f$ in $H^{0,0}(\mathbb{R}^{2}) = L^{2}(\mathbb{R}^{2})$.  This result was improved in \cite{ILM2006} by Isaza, L\'opez and Mej\'{i}a, who reduced the minimal regularity for initial data to $s_{1} > -5/4$ and $s_{2} \geq 0$.  They also showed that the problem is globally well-posed in $H^{s_{1},0}(\mathbb{R}^{2})$ with $s_{1} > -4/7$.

In the present work, we will consider the Cauchy problem for equation \eqref{p1} with initial data in an anisotropic Gevrey space $G^{\sigma_1, \sigma_2}(\mathbb{R}^{2})$, which we define as the completion of the Schwartz functions with respect to the norm
\[
\| f \|_{G^{\sigma_1, \sigma_2}(\mathbb{R}^{2})} = \left(\int_{\mathbb{R}^{2}} e^{2 \sigma_1 |\xi|} e^{2 \sigma_2 |\eta|} | \tilde{f}(\xi,\eta) |^{2} d\xi d\eta\right)^{1/2}.
\]
The primary reason for considering initial data in these spaces is because of the following theorem:
\begin{prop}[Paley-Wiener Theorem]\label{wiener}
Let $\sigma > 0$, and suppose $f \in L^{2}(\mathbb{R})$.  Then the following are equivalent:
\begin{enumerate}
\item The function $f$ is the restriction to the real line of a function $F$ which is holomorphic in the strip
$$ S_{\sigma} = \lbrace x+iy \in \mathbb{C}:\  | y | < \sigma \rbrace,$$ and satisfies
\[
\sup_{| y | < \sigma} \| F(x+iy) \|_{L^{2}}< \infty.
\]
\item $e^{\sigma |\xi|} \hat{f}(\xi) \in L^{2}_{\xi}(\mathbb{R})$.
\end{enumerate}
\end{prop}

A proof of this can be found in chapter 4, section 7 of \cite{K1976}.  The quantity $\sigma$ is known as the \emph{radius of analyticity}.  Thus, for functions in $G^{\sigma_1, \sigma_2}$, if we hold one variable fixed, the resulting function in the other variable will have a holomorphic extension satisfying the stated bounds.

In addition to the holomorphic extension property, Gevrey spaces satisfy the embeddings $G^{\sigma_1, \sigma_2} \hookrightarrow G^{\sigma'_1, \sigma_2'}$ for $\sigma_{i}' < \sigma_{i}$, which follow from the corresponding estimates
\begin{equation}\label{embed}
\| f \|_{G^{\sigma_1', \sigma_2'}} \lesssim \| f \|_{G^{\sigma_1, \sigma_2}}.
\end{equation}
With these facts in mind, we may state our main results for this paper.  The first result relates to the short-term persistence of analyticity of solutions.
\begin{thm} \label{the1.2}
Let  $\sigma_1 \geq 0$ and $\sigma_2 \geq 0$. Then for all initial data $f \in  G^{\sigma_1, \sigma_2}$, there exists $\delta = \delta(\parallel f\parallel_{G^{\sigma_1, \sigma_2}}) > 0$ and a unique solution $u$ of \eqref{p1} on the time interval $[0, \delta]$ such that $$u \in C\left([0, \delta]; G^{\sigma_1, \sigma_2} (\mathbb{R}^{2})\right).$$  Moreover the solution depends continuously on the data $f$.  In particular, the time of existence can be chosen to satisfy
\[
\delta = \frac{c_{0}}{(1 + \| f \|_{G^{\sigma_1, \sigma_2}})^{\gamma}},
\]
for some constants $c_{0}> 0$ and $\gamma > 1$.  Moreover, the solution $u$ satisfies
\[
\sup_{t \in [0,\delta]}\| u(t) \|_{G^{\sigma_1, \sigma_2}} \leq 2 \| f \|_{G^{\sigma_1, \sigma_2}}.
\]
\end{thm}

Our second main result concerns the evolution of the radius of analyticity for the $x$-direction.
\begin{thm} \label{the1.3}
Let  $\sigma_1 > 0$ and $\sigma_2 \geq 0$, and assume $f \in  G^{\sigma_1, \sigma_2}$.  Then the solution $u$ given by Theorem \ref{the1.2} extends globally in time, and for any $T > 0$, we have
$$
u \in C\left([0, T], G^{\sigma(T), 0}(\mathbb{R}^{2}) \right) \quad \text{with} \quad \sigma(T) = \min\left\lbrace \sigma_1, C T^{-1}\right\rbrace,$$
where $C > 0$ is a constant which does not depend on $T$.
\end{thm}
\noindent The method used here for proving lower bounds on the radius of analyticity was introduced in \cite{ST2015} in the study of the 1D Dirac-Klein-Gordon equations. It was applied to the modified Kawahara equation \cite{PD2019} and the non-periodic KdV equation in \cite{SD2017}, to the dispersion-generalized periodic KdV equation in \cite{HKS2017}, the Ostrovsky equation \cite{BZGG2020a}, and to the quartic generalized KdV equation on the line in \cite{ST2017}.

The rest of the paper is organized as follows: in section \ref{prel}, we introduce the various tools which will be used in the proofs of our main theorems.  We then prove Theorems \ref{the1.2} and \ref{the1.3} in sections \ref{local} and \ref{global}, respectively.

\section{Preliminaries}\label{prel}

In this section, we introduce the preliminary results necessary for our proofs.  We begin by fixing the notation to be used.  We begin by noting that, in addition to the spatial Fourier transform  stated above, we will also need the full spacetime Fourier transform, which we denote by
\[
\hat{f}(\xi, \eta, \tau) = \int_{\mathbb{R}^{3}} f(x,y,t) e^{-i (x \xi + y \eta + t \tau)}\ dx dy dt.
\]
In both cases, we will denote the corresponding inverse transform of a function $f = f(\xi, \eta)$ or $f = f(\xi, \eta, \tau)$ by $\mathfrak{F}^{-1}(f)$.

To simplify the notation, we introduce some operators which will be introduced later.  We first introduce the operator $A^{\sigma_1, \sigma_2}$, which we define as
\begin{equation}\label{1.1.3}
    A^{\sigma_1, \sigma_2}f = \mathfrak{F}^{-1}\left( e^{\sigma_1 |\xi|} e^{\sigma_2 |\eta|} \hat{f} \right).
\end{equation}
With this, we may then define another useful operator
\begin{equation}\label{nu}
    N(f) = \partial_{x}\left[(A^{\sigma_1, \sigma_2}f)^2 - A^{\sigma_1, \sigma_2}(f^2)\right].
\end{equation}
For $x \in \mathbb{R}^{n}$, we denote $\langle x \rangle = (1 + |x|^{2})^{1/2}$.  Finally, we write $a \lesssim b$ if there exists a constant $C > 0$ such that $a \leq Cb$, and $a \sim b$ if $a \lesssim b \lesssim a$.  If the constant $C$ depends on a quantity $q$, we denote this by $a \lesssim_{q} b$.

Since our proofs will rely heavily on the theory developed by Isaza, L\'{o}pez and Mej\'{i}a, let us state the function spaces they used explicitly, so that we can state their useful properties which we will exploit in our modifications of their spaces.  The main function spaces they used are the so-called Bourgain spaces, adapted to the KP II equation, whose norm is given by
\[
\| u \|_{X^{s_1, s_2, b, \varepsilon}} =
\left(\displaystyle \int_{\mathbb{R}^{3}} \lambda^{2}(s_1, s_2, b, \varepsilon) |\hat{u}(\xi, \eta, \tau)|^{2} d\xi d\eta d\tau \displaystyle\right) ^{\frac{1}{2}},
\]
where
\[
\lambda(s_1, s_2, b, \varepsilon) = \langle \xi \rangle^{s_1} \langle \eta \rangle^{s_2} \langle \tau - m(\xi, \eta) \rangle^{b} \left\langle \frac{\tau - m(\xi,\eta)}{1 + |\xi|^{5}}\right\rangle^{\varepsilon}.
\]
with $m(\xi, \eta)=\xi^{5}-\frac{\eta^{2}}{\xi}$.  As is well-known (see, for example, Corollary 2.10 of \cite{T2006}), these spaces satisfy the embedding $X^{s_1, s_2, b, \varepsilon} \hookrightarrow C \left(\mathbb{R}; H^{s_1, s_2} (\mathbb{R}^{2})\right)$.  Thus, solutions constructed in $X^{s_1, s_2, b, \varepsilon}$ belong to the natural solution space.

When considering local solutions, it is useful to consider localized versions of these spaces.  For a time interval $I$ and a Banach space $Z$, we define the localized space $Z(I)$ by the norm
\[
\| u \|_{Z(I)} = \inf \{ \| v \|_{Z}:\ v = u \textrm{ on } I \}.
\]
It is easy to see that estimates which hold for the original spaces also hold for the localized spaces.  For simplicity, we will omit the interval from the notation when there is no chance of confusion.

The final preliminary fact we must state is the following bilinear estimate, which is Lemma 1.1 of \cite{ILM2006}:
\begin{lem}\label{bilinear}
Let $s_1 > - 5/4$, $s_2 \geq 0$, $b > 1/2$, and define $s = \max \{0, -s_1\}$.  If $\varepsilon$ and $\beta$ satisfy the inequalities
\[
0 \leq \varepsilon \leq \min \left\{ \frac{2}{5}\left( \frac{5}{4} - s \right), \frac{3}{20} \right\}
\]
and
\[
\max \left\{ \frac{9}{20}, \frac{1}{2} - \frac{1}{2}\left( \frac{5}{4} - s \right) + \varepsilon \right\} \leq \beta < \frac{1}{2},
\]
then
\[
\| \partial_{x} (uv) \|_{X^{s_1, s_2, -\beta, \varepsilon}} \lesssim \| u \|_{X^{s_1, s_2, b, \varepsilon}} \| v \|_{X^{s_1, s_2, b, \varepsilon}}.
\]
\end{lem}

\section{Proof of Theorem \ref{the1.2}}\label{local}

We may now begin the proof of Theorem \ref{the1.2}.  Since this is a modification of the proof of Isaza, L\'{o}pez, and Mej\'{i}a in \cite{ILM2006}, we will merely outline the essential steps.  To begin, consider the linear problem
\begin{align*}
& \partial_{t} u - \partial_{x}^{5} u + \partial^{-1}_{x} \partial^{2}_{y} u = F, \\
& u(0) = f.
\end{align*}
By Duhamel's principle the solution can be written as
\begin{equation}\label{12.11}
u(t) = S(t)f - \dfrac{1}{2} \int_{0}^{t} S(t-t') F(t') dt',
\end{equation}
where $$\widetilde{S(t)f}(\xi,\eta) = e^{i t m(\xi,\eta)} \tilde{f}(\xi,\eta).$$
Let $\psi \in C^{\infty}_{0}(\mathbb{R})$ be supported in the interval $(-2,2)$ such that $0 \leq \psi(t) \leq 1$ and $\psi = 1$ on $[-1,1]$, and let $\phi_{\delta} \in C^{\infty}_{0}(\mathbb{R})$ be supported on $(-2 T^{-1/2}, 2 T^{-1/2})$ such that $0 \leq \phi_{\delta}(t) \leq 1$ and $\phi_{\delta} = 1$ on $[-T^{-1/2},T^{-1/2}]$.  We then observe that we may decompose the integral operator on the right-hand side of  equation \eqref{12.11} in the form
\begin{equation}\label{112}
-\dfrac{1}{2}\int_{0}^{t}S(t-t')f(t')dt'=I_{\phi_{\delta}}(f)+II_{\phi_{\delta}}(f)+III_{\phi _{\delta}}(f),
\end{equation}
where
$$I_{\phi_{\delta}}(f) = C \int e^{i(x\xi+y\eta)} e^{itm(\xi,\eta)} \dfrac{e^{itp(\xi, \eta, \tau)} - 1}{i p(\xi, \eta, \tau)} \phi_{\delta}(p(\xi, \eta, \tau)) \hat{f}(\xi,\eta,\tau)\ d\xi d\eta d\tau,  $$

$$II_{\phi_{\delta}}(f) = C \int e^{i(x\xi+y\eta)} \dfrac{e^{it\tau}}{ip(\xi, \eta, \tau)}(1-\phi_{\delta}(p(\xi, \eta, \tau))) \hat{f}(\xi,\eta,\tau)\ d\xi d\eta d\tau,  $$

$$III_{\phi_{\delta}}(f) = -C \int e^{i(x\xi+y\eta)} e^{itm(\xi,\eta)} \dfrac{(1-\phi_{\delta}(p(\xi, \eta, \tau))}{i p(\xi, \eta, \tau)} \hat{f}(\xi,\eta,\tau)\ d\xi d\eta d\tau.  $$
Here, $p(\xi, \eta, \tau) = \tau - m(\xi,\eta)$.  We then define a modification $G_{\delta}(f)$ of the integral operator in  (\ref{112}),
$$G_{\delta}(f)= \psi(t/\delta)I_{\phi_{\delta}}(f)+II_{\phi_{\delta}}(f)+\psi(t/\delta)III_{\pi_{\delta}}(f). $$
A simple computation will show that if $0 < \delta < 1$ and $t \in [-\delta, \delta]$, then
\[
G_{\delta}(f)(t) = -\dfrac{1}{2}\int_{0}^{t}S(t-t')f(t')dt'.
\]

Now define a sequence $\{ u_{n} \}_{n = 0}^{\infty}$ of functions which are solutions to the equations
\begin{align*}
    & \partial_{t} u_{0} - \partial_{x}^{5} u_{0} + \partial_{x}^{-1} \partial_{y}^{2} u_{0} = 0 & \qquad & \partial_{t} u_{n} - \partial_{x}^{5} u_{n} + \partial_{x}^{-1} \partial_{y}^{2} u_{n} = - u_{n-1}\partial_{x} u_{n-1} \\
    & u_{0}(x, y, 0) = f(x, y) & & u_{n}(x, y, 0) = f(x, y)
\end{align*}
By the discussion above, for $t \in (-\delta, \delta)$, we have the identity $u_{n}(x, y, t) = \Phi(u_{n-1}(x, y, t))$, where $$\Phi(u) = \psi(t) S(t) f + G_{\delta}\left( \partial_{x}(u^2) \right).$$  To this operator, we apply the following estimate, which follows from equations (1.2) and (1.4) of \cite{ILM2006} and the commutativity of Fourier multipliers:
\begin{lem}\label{lem2.2}
Let $\sigma_1 \geq 0$, $\sigma_2 \geq 0$, $\frac{1}{2} < b < 1$, $\beta \in (0, 1-b)$, and $0 < \delta < 1$.  Then
\[
\| \Phi(u) \|_{Y^{\sigma_1, \sigma_2, b}} \leq \| f \|_{G^{\sigma_1, \sigma_2}} + C \delta^{\gamma} \| \partial_{x} (u^2) \|_{Y^{\sigma_1, \sigma_2, -\beta}}
\]
for some $0 < \gamma < 1$.
\end{lem}
\noindent To this result, we apply the following lemma, which is a corollary of Lemma \ref{bilinear}:
\begin{lem}\label{goodest}
For $\sigma_1 \geq 0$, $\sigma_2 \geq 0$, and $b > 1/2$, we have
\[
\| \partial_{x}(uv) \|_{Y^{\sigma_1, \sigma_2, 9/20}} \lesssim \| u \|_{Y^{\sigma_1, \sigma_2, b}} \| v \|_{Y^{\sigma_1, \sigma_2, b}}.
\]
\end{lem}
\begin{proof}
From the triangle inequality, it is easy to see that
\begin{align*}
  & e^{2 (\sigma_1 |\xi| + \sigma_2 |\eta|)} \left| \widehat{uv}(\xi,\eta, \tau) \right|^{2} \\
  & \quad = e^{2 (\sigma_1 |\xi| + \sigma_2 |\eta|)} \left| \int \hat{u}(\xi - \xi_1, \eta - \eta_1, \tau - \tau_1) \hat{v}(\xi_1, \eta_1, \tau_1)\ d\xi d\eta d\tau \right|^{2} \\
  & \quad \leq \left| \int e^{\sigma_1 |\xi - \xi_1| + \sigma_2 |\eta - \eta_1|}\hat{u}(\xi - \xi_1, \eta - \eta_1, \tau - \tau_1) \right. \times \\
  & \qquad \qquad \left. \times e^{\sigma_1 |\xi_1| + \sigma_2 |\eta_1|} \hat{v}(\xi_1, \eta_1, \tau_1)\ d\xi d\eta d\tau \right|^{2} \\
  & \quad = \left| \widehat{A^{\sigma_1, \sigma_2}u A^{\sigma_1, \sigma_2}v} \right|^{2}.
\end{align*}
It follows that
\begin{align*}
\| \partial_{x}(uv) \|_{Y^{\sigma_1, \sigma_2, 9/20}} & \leq \| \partial_{x}(A^{\sigma_1, \sigma_2}u A^{\sigma_1, \sigma_2}v) \|_{X^{0,0,\frac{9}{20},0}} \\
& \lesssim \| A^{\sigma_1, \sigma_2}u \|_{X^{0, 0, b, 0}} \| A^{\sigma_1, \sigma_2}v \|_{X^{0, 0, b, 0}} \\
& = \| u \|_{Y^{\sigma_1, \sigma_2, b}} \| v \|_{Y^{\sigma_1, \sigma_2, b}}
\end{align*}
by Lemma \ref{bilinear}.
\end{proof}
With these results, it is a simple matter to show that
\[
\| u_{n} \|_{Y^{\sigma_1, \sigma_2, b}} \leq \| f \|_{G^{\sigma_1, \sigma_2}} + C \delta^{\gamma} \| u_{n-1} \|_{Y^{\sigma_1, \sigma_2, b}}^{2}.
\]
Using a simple proof by induction, one may show that
\[
\| u_{n} \|_{Y^{\sigma_1, \sigma_2, b}} \leq 2 \| f \|_{G^{\sigma_1, \sigma_2}}
\]
for all $n \in \mathbb{N} \cup \{ 0 \}$, if we choose $\delta$ such that
\begin{equation}\label{prop1}
\delta < \frac{1}{(C \| f \|_{G^{\sigma_1, \sigma_2}})^{1/\gamma}}.
\end{equation}
The final step is to show that the sequence converges.  Applying Lemmas \ref{lem2.2} and \ref{goodest} once again, a similar computation will show that
\begin{align*}
\| u_{n} - u_{n-1} \|_{Y^{\sigma_1, \sigma_2, b}} & = \| \Phi(u_{n}) - \Phi(u_{n-1}) \|_{Y^{\sigma_1, \sigma_2, b}} \\
& \leq C \delta^{\gamma} M_{n-1} \| u_{n-1} - u_{n-2} \|_{Y^{\sigma_1, \sigma_2, b}} \\
& \leq 4 C \delta^{\gamma} \| f \|_{G^{\sigma_1, \sigma_2}} \| u_{n-1} - u_{n-2} \|_{Y^{\sigma_1, \sigma_2, b}}.
\end{align*}
where
\[
M_{n-1} = \| u_{n-1} \|_{Y^{\sigma_1, \sigma_2, b}} + \| u_{n-2} \|_{Y^{\sigma_1, \sigma_2, b}}.
\]
Thus, the sequence will converge if $\delta$ satisfies
\begin{equation}\label{prop2}
\delta < \frac{1}{(4 C \| f \|_{G^{\sigma_1, \sigma_2}})^{1/\gamma}}.
\end{equation}
sufficiently small.  Thus, the sequence converges to a solution $u \in Y^{\sigma_1, \sigma_2, b} \subset C([0,\delta]; G^{\sigma_1, \sigma_2})$.

To show uniqueness, suppose $u$ and $v$ are solutions to \eqref{p1}, and let $w = u - v$.  Then $w$ satisfies the equation
\[
\partial_{t}w - \partial_{x}^{5}w + \partial_{x}^{-1} \partial_{y}^{2} w = v \partial_{x}v - u \partial_{x}u,
\]
with $w(x, y, 0) = 0$.  Multplying by $w$ and integrating in $x$ and $y$ yields
\[
\frac{1}{2} \frac{d}{dt} \| w(t) \|_{L^{2}_{x,y}} \leq \left( \| u_{x}(t) \|_{L^{\infty}_{x,y}} + \| v_{x}(t) \|_{L^{\infty}_{x,y}} \right) \| w(t) \|_{L^{2}_{x,y}}.
\]
By the embedding
\[
\| u_{x}(t) \|_{L^{\infty}_{x,y}} \lesssim \| u(t) \|_{G^{\sigma_1, \sigma_2}} \leq 2 \| f \|_{G^{\sigma_1, \sigma_2}},
\]
which also holds for $v$, it will follow from Gr\"{o}nwall's Inequality that $w = 0$.  This completes the proof of Theorem \ref{the1.2}.

As a concluding remark, we observe that for later convenience, we may choose the time of existence to be
\[
\delta = \frac{C}{(1 + \| f \|_{G^{\sigma_1, \sigma_2}})^{1/\gamma}}.
\]
For appropriate choice of $C$, this will satisfy inequalities \eqref{prop1} and \eqref{prop2}.

\section{Proof of Theorem \ref{the1.3}}\label{global}

In this section, we begin the proof of Theorem \ref{the1.3}.  The first step is to obtain estimates on the growth of the norm of the solutions.  For this, we will need the following approximate conservation law:
\begin{prop}\label{th03}
Let $\sigma_1 \geq 0$.  Then there is a $b \in (1/2, 1) $ and a $C > 0$, such that $u \in Y^{\sigma_1, 0, b}(I)$ is a solution to the Cauchy problem (\ref{p1}) on the time interval $[0, \delta]$, we have the estimate
\begin{equation}\label{3.3.1}
\sup_{ t\in[0, T]} \| u(t) \|^{2}_{G^{\sigma_1, 0}} \leq \| f \|^{2}_{G^{\sigma, 0}} + C \sigma \| u \|^{3}_{Y^{\sigma_1, 0, b}(I)}.
\end{equation}

\end{prop}
\noindent Before we may state the proof, let us first state some preliminary lemmas.  The first is an immediate consequence of Lemma 12 in \cite{SD2017} and the comments immediately following it:
\begin{lem}\label{200}
For $\sigma > 0$ and $\xi, \xi_{1} \in \mathbb{R}$, we have
\begin{align*}
e^{\sigma |\xi - \xi_1|} e^{\sigma |\xi_1|} - e^{\sigma |\xi|} \lesssim \sigma \frac{\langle \xi - \xi_1 \rangle \langle \xi_1 \rangle}{\langle \xi \rangle} e^{\sigma |\xi - \xi_1|} e^{\sigma |\xi_1|}.
\end{align*}
\end{lem}
\noindent This will be used to prove the following key estimate:
\begin{lem}\label{lem3.3}
Let $N(u)$ be as in equation \eqref{nu} for $\sigma_1 \geq 0$ and $\sigma_2 = 0$.  Then for $b$ and $\beta$ as in Lemma \ref{bilinear}, we have
\[
\| N(u) \|_{Y^{0, 0, -\beta}} \leq C \sigma \| u \|^{2}_{Y^{\sigma_1, 0, b}}.
\]
\end{lem}
\begin{proof}
We first observe that the inequality in Lemma \ref{bilinear}, for the case $\varepsilon = 0$, is equivalent to
\begin{align*}
& \left\| \xi \lambda(s_1, s_2, -\beta, 0) \int \frac{\hat{f}(\xi - \xi_1, \eta - \eta_1, \tau - \tau_1)}{\langle \xi - \xi_1 \rangle^{s_1} \langle \eta - \eta_1 \rangle^{s_2} \langle \phi(\xi - \xi_1, \eta - \eta_1, \tau - \tau_1) \rangle^{b}} \right. \times \\
& \times \left. \frac{\hat{g}(\xi_1, \eta_1, \tau_1)}{\langle \xi_1 \rangle^{s_1} \langle \eta_1 \rangle^{s_2} \langle \phi(\xi_1, \eta_1, \tau_1) \rangle^{b}} \ d\xi_1 d\eta_1 d\tau_1 \right\|_{L^{2}_{\xi,\eta}} \lesssim \| f \|_{L^{2}_{x,y}} \| g \|_{L^{2}_{x,y}}
\end{align*}
where we denote $\phi(\tau, \xi, \eta) = \langle \tau - m(\xi,\eta)\rangle$.
With this in mind, we observe that the left side of the inequality in Lemma \ref{lem3.3} can be estimated by Lemma \ref{200} as
\begin{align*}
\| N(u) \|_{Y^{\sigma_1, 0, -\beta}} & \lesssim \sigma \left\| \frac{\xi \langle \xi \rangle^{-1}}{\langle \phi(\tau, \xi, \eta)\rangle^{\beta}} \int \frac{e^{\sigma_1 |\xi - \xi_1|} \hat{u}(\xi - \xi_1, \eta - \eta_1, \tau - \tau_1)}{\langle \xi - \xi_1 \rangle^{-1}} \right. \times \\
& \qquad \qquad \times \left. \frac{e^{\sigma_1 |\xi_1|} \hat{u}(\xi_1, \eta_1, \tau_1)}{\langle \xi_1 \rangle^{-1}}  \ d\xi_1 d\eta_1 \right\|_{L^{2}_{\xi,\eta}}.
\end{align*}
If we apply Lemma \ref{bilinear} with $s_1 = -1$, $s_2 = 0$, it will follow from the comments above that
\[
\| N(u) \|_{Y^{\sigma_1, 0, -\beta}} \lesssim \| u \|_{Y^{\sigma_1, 0, b}}^{2}.
\]
\end{proof}

\subsection{Proof of Proposition \ref{th03}}
Begin by applying the operator $A^{\sigma_1, 0}$ to equation \eqref{p1}.  If we let $U = A^{\sigma_1, 0} u$, then equation \eqref{p1} becomes
\[
U_{t} - U_{xxxxx} + \partial_{x}^{-1} U_{yy} + U U_{x} = N(u),
\]
where $N(u)$ is as defined in Lemma \ref{lem3.3}.  Multiplying this by $U$ and integrating with respect to the spatial variables, we obtain
\[
\int U U_{t} - U U_{xxxxx} + U \partial^{-1}_{x} U_{yy} + U^2 U_{x}\ dxdy = \int U N(u)\ dxdy.
\]
If we apply integration by parts, we may rewrite the left-hand side as
\[
\frac{d}{dt} \int \frac{1}{2} U^{2}\ dxdy + \int U_{xx} U_{xxx} dx dy - \int U_{y} \partial_{x}^{-1} U_{y}\ dxdy + \int U^2 U_{x}\ dxdy,
\]
which can then be rewritten as
\begin{align*}
& \frac{1}{2} \frac{d}{dt} \int U^{2}\ dxdy + \frac{1}{2} \int \partial_{x}(U_{xx}^{2}) dx dy - \frac{1}{2} \int \partial_{x} [( \partial_{x}^{-1} U_{y})^2] \ dxdy \\
& \qquad \qquad + \frac{1}{3} \int \partial_{x} (U^3) \ dxdy.
\end{align*}
For $U$ and its derivatives vanishing at infinity, we thus obtain the formal identity
\[
\frac{d}{dt} \int U^{2}(x,y,t)\ dxdy = 2\int U(x,y,t) N(u)(x,y,t)\ dx dy.
\]
Integrating with respect to time yields
\begin{align*}
\int U^{2}(x,y,t)\ dxdy & = \int U^{2}(x,y,0)\ dxdy \\
& \qquad + 2\int_{0}^{t} \int U(x,y,t') \partial_{x} N(u)(x,y,t')\ dx dy dt'.
\end{align*}
Applying Cauchy-Schwarz and the definition of $U$, we obtain
\[
\| u(t) \|_{G^{\sigma_1, 0}}^{2} \leq \| f \|_{G^{\sigma_1, 0}}^{2} + \| u \|_{Y^{\sigma_1, 0, b}} \| N(u) \|_{Y^{0, 0, -\beta}(I)},
\]
where $\beta$ is as defined previously.  If we now apply Proposition \ref{th03} and the fact that $\beta < 1/2 < b$, we can further estimate this by
\begin{equation}\label{almost}
\| u(t) \|_{G^{\sigma_1, 0}}^{2} \leq \| f \|_{G^{\sigma_1, 0}}^{2} + C\sigma \| u \|_{Y^{\sigma_1, 0, b}}^{3},
\end{equation}
as desired.

\subsection{Proof of Theorem \ref{the1.3}}

With the tools established in the previous section, we may begin the proof of Theorem \ref{the1.3}.  By the embedding in equation \eqref{embed}, it suffices to consider the case $\sigma_2 = 0$.  To begin, let us first suppose that $T^{*}$ is the supremum of the set of times $T$ for which
\[
 u \in C([0,T]; G^{\sigma_1,0}).
\]
If $T^{*} = \infty$, there is nothing to prove, so let us assume that $T^{*} < \infty$.  In this case, it suffices to prove that
\begin{equation}\label{3.3.6}
u \in C\left([0, T], G^{\sigma(T), 0} \right)
\end{equation}
for some $\sigma(T) > 0$ and all $T > T^{*}$.  To show that this is the case, we will use Theorem \ref{the1.2} and Proposition \ref{th03} to construct a solution which exists over subintervals of width $\delta$, using the parameter $\sigma$ to control the growth of the norm of the solution.  Thus, the desired result will follow from the following proposition:

\begin{prop}\label{finalprop}
Let $T > 0$ and $\delta > 0$ be numbers such that $n \delta \leq T < (n + 1) \delta$.  Then the solution $u$ to the Cauchy problem \eqref{p1} satisfies
\begin{equation}\label{ineq1}
\sup_{t \in [0, n \delta]} \| u(t) \|_{G^{\sigma(T), 0}}^{2} \leq \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C \sigma(T) n \| f \|_{G^{\sigma_1, 0}}^{3}
\end{equation}
and
\begin{equation}\label{ineq2}
\sup_{t \in [0, n \delta]} \| u(t) \|_{G^{\sigma(T), 0}}^{2} \leq 4 \| u(t) \|_{G^{\sigma_1, 0}}^{2}
\end{equation}
if
\[
\sigma(T) \leq \sigma_1 \quad \textrm{and} \quad \sigma(T) \leq \frac{C (1 + \| f \|_{G^{\sigma_1, 0}})^{1/\gamma - 1}}{T}
\]
for some constant $C > 0$.
\end{prop}
\begin{proof}
By induction on $n$.  The base case $n = 1$ follows from equation \eqref{almost}, Theorem \ref{the1.2}, and the embedding $G^{\sigma_1, \sigma_2} \hookrightarrow G^{\sigma_1', \sigma_2'}$ when $\sigma_1' \leq \sigma_1$ and $\sigma_2' \leq \sigma_2$.  Suppose, then, that the result holds for $n \leq k$.  The inductive hypothesis then tells us that
\[
\sup_{t \in [0, k \delta]} \| u(t) \|_{G^{\sigma(T), 0}} \leq \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C \sigma(T) k \| f \|_{G^{\sigma_1, 0}}^{3}
\]
and
\[
\sup_{t \in [0, k \delta]} \| u(t) \|_{G^{\sigma(T), 0}}^{2} \leq 4 \| f \|_{G^{\sigma_1, 0}}^{2}
\]

If we apply the inductive hypothesis on the interval $[k\delta, (k+1) \delta]$, then
\[
\sup_{t \in [k \delta, (k+1) \delta]} \| u(t) \|_{G^{\sigma(T), 0}} \leq \| u(k\delta) \|_{G^{\sigma(T), 0}}^{2} + 2^3 C \sigma(T) \| u(k\delta) \|_{G^{\sigma_1, 0}}^{3}
\]
and
\[
\sup_{t \in [k \delta, (k + 1) \delta]} \| u(t) \|_{G^{\sigma(T), 0}}^{2} \leq 4 \| u(k\delta) \|_{G^{\sigma_1, 0}}^{2}.
\]
If we apply equations \eqref{ineq1} and \eqref{ineq2} to these, we get
\[
\| u(k \delta) \|_{G^{\sigma(T), 0}} \leq \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C k \sigma(T) \| u(k\delta) \|_{G^{\sigma_1, 0}}^{3}
\]
and
\[
\| u(k \delta) \|_{G^{\sigma_1, 0}}^2 \leq 4 \| f \|_{G^{\sigma_1, 0}}^2.
\]
Combining these together, we obtain
\begin{align*}
\sup \| u(t) \|_{G^{\sigma(T),0}} & \leq \left( \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C k \sigma(T) \| f \|_{G^{\sigma_1, 0}}^{3}\right) + 2^3 C \sigma(T) \| f \|_{G^{\sigma_1, 0}}^{3} \\
& = \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C (k+1) \sigma(T) \| f \|_{G^{\sigma_1, 0}}^{3}.
\end{align*}
It follows that
\[
\sup_{t \in [0, (k+1) \delta]} \| u(t) \|_{G^{\sigma(T),0}} \leq \| f \|_{G^{\sigma(T), 0}}^{2} + 2^3 C (k+1) \sigma(T) \| f \|_{G^{\sigma_1, 0}}^{3}.
\]
To complete the proof, we need to show that
\begin{equation}\label{complete}
\sup_{t \in [0, (k+1) \delta]} \| u(t) \|_{G^{\sigma(t),0}}^2 \leq 4 \| f \|_{G^{\sigma_1, 0}}^2.
\end{equation}
Next, we observe that the assumption $n \delta \leq T < (n+1)\delta$ implies that
\[
n \leq \frac{T}{\delta} < n + 1 \leq \frac{T}{\delta} + 1 \leq \frac{2T}{\delta}.
\]
Since $k + 1 \leq n + 1$, we have
\begin{align*}
2^3 C (k+1) \sigma(T) \| f \|_{G^{\sigma_1, 0}} & \leq 2^4 C \frac{2T}{\delta} \sigma(T) \| f \|_{G^{\sigma_1, 0}} \\
& \leq \frac{C \| f \|_{G^{\sigma_1, 0}}}{(1 + \| f \|_{G^{\sigma_1, 0}})^{1/\gamma}} T \sigma(T).
\end{align*}
Recalling that $1/\gamma > 1$, we thus have
\[
2^3 C (k+1) \sigma(T) \| f \|_{G^{\sigma_1, 0}} \leq \frac{CT}{(1 + \| f \|_{G^{\sigma_1, 0}})^{1/\gamma - 1}} \sigma(T).
\]
Thus, for equation \eqref{complete} to hold, it suffices to have
\[
\sigma(T) \leq \frac{C (1 + \| f \|_{G^{\sigma_1, 0}})^{1/\gamma - 1}}{T}.
\]
Since we have shown that the result holds for $n = 1$, and we have shown that the result for $n = k$ implies it for $n = k+1$, then the result holds for all $n$.  This completes the proof of Proposition \ref{finalprop}, from which Theorem \ref{the1.3} follows as an immediate corollary.

\end{proof}

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config: arxiv


TITLE: Show $A$ is "real-equivalent" to its transpose
QUESTION [4 upvotes]: The problem statement:
Let $A$ be a real $9\times 9$ matrix with transpose $B$. Prove that the matrices are real equivalent in the following sense: There exists a real invertible $9\times 9$ matrix $H$ such that $AH=HB$.
Unfortunately, I don't have much of an attempt at a solution.  My first thought was to notice that $A$ and $B$ have the same Jordan normal form so they're similar. But the Jordan form may have complex entries as the eigenvalues may be complex. Also confusing is why the matrix is $9\times 9$.  How could this matter?  Nine is a perfect square and it's odd, but so are a lot of other numbers.
Any help is welcomed.

REPLY [7 votes]: The key is to prove the following:

Theorem. Let $A$ and $B$ be matrices with real coefficients. If $A$ and $B$ are similar over $\mathbb{C}$, then they are similar over $\mathbb{R}$.

This problem appears, among other places, in Berkeley Problems in Mathematics, by de Souza and Nuno Silva (Problem 7.7.10 in the second edition; the problem has been used many times in the Prelim exam at Berkeley). The book offers two answers, though I think that one of the solutions (using the rational canonical form) is a bit sloppy.
Here is the other approach. Assume $A = UBU^{-1}$, with $U$ a complex matrix. Write $U=K+iL$, where $K$ and $L$ are real matrices, and assume that $L\neq 0$. Then
$$A(K+iL) = AU = UB = (K+iL)B$$
and so the real parts and imaginary parts of these matrices are equal. That is, $AK = KB$ and $AL = LB$ (the problem is we don't know that $K$, by itself, is invertible).
Now, for any complex number $z$ we have $A(K+zL) = (K+zL)B$. Let $p(z) = \det(K+zL)$. Prove that $p$ is not identically zero, and take a real number that is not a root.
Once you have this theorem, your observation that $A$ and $B$ are similar over $\mathbb{C}$ implies the result you want.
(For what it is worth, the other argument given by the book is that the Rational canonical forms of $A$ and $B$ have to be the same, since they are the same over $\mathbb{C}$, and so $A$ and $B$ are similar over $\mathbb{R}$; the reason this is sloppy is that the rational canonical form is usually defined, for uniqueness purposes, in terms of the irreducible factors of the minimal/characteristic polynomials, and the rational blocks are companion matrices of powers of these irreducible factors. But the irreducible factors over $\mathbb{R}$ are not necessarily the same as the irreducible factors over $\mathbb{C}$, so that the "uniqueness" of the rational canonical form does not directly apply; an argument needs to be made that the equality over $\mathbb{C}$ implies an equality when the irreducible factors are taken over $\mathbb{R}$ instead, and this argument is missing in the book).


TITLE: Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?
QUESTION [1 upvotes]: Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded?
If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ convergence? or at least, are the $L^p$ norms of the $f_j$ bounded?

REPLY [2 votes]: In the case $q<p^*$ the Rellich–Kondrachov theorem applies, if your domain $\Omega$ has the required properties.
If $q=p^*$, the $L^q$ norm is still controlled by the $W^{1,p}$ norm; and since a weakly convergent sequence is norm-bounded, $L^q$ norm is bounded. Moreover, you will have weak convergence in $L^q$, since linear maps preserve weak convergence. But it will not be strong in general. For example, pick $1<p<n$ and consider the sequence $f_k(x) = k^{\frac{n}{p}-1} (1-k|x|)^+$. This is a bounded sequence in $W^{1,1}(B^n)$, where $B^n$ is the $n$-dimensional unit ball. It converges weakly in $W^{1,p}$, specifically to zero. (Indeed, it converges to $0$ in the sense of distributions; on the other hand $W^{1,p}$ is reflexive, so a weakly convergent subsequence can be extracted; the weak limit has to be the same $0$, so the whole sequence converges weakly to $0$.)  On the other hand, the $L^q$ norm for $q=p^*=np/(n-p)$ stays the same for all $k$.


TITLE: Number of solutions of a Diophantine equation
QUESTION [3 upvotes]: I'm currently studying about the solutions of the $x_1\pm x_2\pm\cdots\pm x_m=0$, where the $x_i$'s are positive integers. They started with a trigonometric identity
\begin{equation}
\cos(x_1) \cos(x_2)\cdots\cos(x_m)=\dfrac{1}{2^{m-1}}\sum \cos(x_1\pm x_2\pm\cdots\pm x_m)
\end{equation}
where the sum is taken over all possible choices of signs $+$ and $-$. Later, they arrived to the conclusion that the number of solutions of the equation $x_1\pm x_2\pm\cdots\pm x_m=0$ (all the possible choices of signs) is given by the integral
\begin{equation}
\dfrac{2^{m-1}}{2\pi}\int_{-\pi}^{\pi}\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt)dt
\end{equation}
which of course can be simplified further (by using the fact that the integrant is an even function) as
\begin{equation}
\dfrac{2^{m-1}}{\pi}\int_{0}^{\pi}\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt)dt
\end{equation}
How they did arrived to that integral representation? I can't figure out this. Any help given will be appreciated.

REPLY [3 votes]: For every $t$, you have the given trigonometric identity
$$\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt)=\dfrac{1}{2^{m-1}}\sum \cos(x_1t\pm x_2t\pm\cdots\pm x_mt)$$
where, as you said, the sum is taken over all possible choices of signs $+$ and $−$.
Now integrate this between $-\pi$ and $\pi$ : you get
$$\int_{-\pi}^{\pi}\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt) dt=\dfrac{1}{2^{m-1}}\sum \int_{-\pi}^{\pi}\cos((x_1\pm x_2\pm\cdots\pm x_m)t) dt$$
But the integral $$\int_{-\pi}^{\pi}\cos((x_1\pm x_2\pm\cdots\pm x_m)t) dt$$
is equal to $2\pi$ if $x_1\pm x_2\pm\cdots\pm x_m = 0$, and is equal to $0$ otherwise (because the $x_i$'s are integers). So if $N$ denotes the number of solutions of the equation, you get directly that
$$\int_{-\pi}^{\pi}\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt) dt=\dfrac{1}{2^{m-1}} 2\pi N$$
so
$$N = \frac{2^{m-1}}{2\pi}\int_{-\pi}^{\pi}\cos(x_1t) \cos(x_2t)\cdots\cos(x_mt) dt$$
which is the given integral representation.


TITLE: Does the inequality $a^{m-1}(m-a)<m-1$, where $0<a<1$, hold for $m=2,3,\cdots$ ?
QUESTION [0 upvotes]: Does the inequality $a^{m-1}(m-a)<m-1$, where $0<a<1$, hold for $m=2,3,\cdots$ ?
I think it holds (from the numerical results). However, I cannot prove it. Does anyone have some hints?

REPLY [1 votes]: Let $m= 2,3,..$ be fixed.
$f(x)_m := (m-1) - x^{m-1}(m-x),$  $0\le x\le 1.$
We would like to show that for
$m=2,3,..$ and $x \in (0,1)$  :
$ f_m(x) >0.$
At $x=0$, and $x=1:$
$f_m(0) = m-1>0.$
$f_m(1) = 0.$
Now :
$f_m'(x)=$
$-(m-1)x^{m-2}(m-x) +x^m <0.$
Since: $x^{m-2} >x^m$ for $x\in (0,1)$ and $(m-1)(m-x) > 1$, $ m=2,3,..$,
we have :
$(m-1)(m-x) x^{m-2}> x^m$, I.e.
$ f_m'(x) <0 $.
$\rightarrow:$
$ f_m(x)$ is strictly decreasing in $(0,1).$
$\rightarrow:$
$f_m(x) >0$ for $x\in (0,1)$, $m=2,3,..$


TITLE: Question about sums with a negative limit for the index
QUESTION [0 upvotes]: To me, it looks like we have $\;\sum_{i = 1}^{0} x_i = 0\;$ and $\;\sum_{i = 1}^{1} x_i = x_1\;$. What happens if I write the following?
$$\;\sum_{i = 1}^{-123} x_i\;$$ Would this be defined?

REPLY [2 votes]: There are different conventions:

the sum is $0$;
the expression is undefined;
the expression is interpreted as $\displaystyle\sum_{i=-123}^1x_i$.

In my experience the first is the most common, and the last is quite uncommon.


TITLE: Advice for computing the Jacobson radical of rings
QUESTION [3 upvotes]: Could you please assist me the way how to compute radical of the rings given below. Is there some simple techniques how I could use easily?
a) $ \mathbb{Z} / n \mathbb{Z} $ of modulo $n$ congruence class
b) Polynomial ring $F[X]$ over field $F$
c)The ring of $n \times n $ upper triangular matrix over a field $F$

REPLY [5 votes]: Well, some general tips:

If your ring is commutative and finite, then the Jacobson radical is just going to be the set of nilpotent elements.
If your ring is just commutative, then it will contain the nilpotent elements, but may possibly be bigger.
The Jacobson radical contains every nilpotent ideal. (meaning $I^n=\{0\}$.
If $R/I$ is a ring that you know has radical zero, then $J(R)\subseteq I$.

So my hints on your questions are these:

This depends directly on $n$. The first hint above is the easiest thing to apply.
There is a theorem that tells you the Jacobson radical of $F[X]$ equals the nilradical, as well. I'm not sure there's an easy way to compute nilpotent polynomials in general, but of course when $F$ is a field $F[x]$ is a domain so...
The strictly upper triangular matrices are a nilpotent ideal, so you know it is contained in $J(R)$. And the quotient by this ideal is a product of fields, which has Jacobson radical zero. So the Jacobson radical of the triangular matrices is the ideal of strictly upper triangular matrices.


TITLE: Applying the complex interpolation method to $L^p$ spaces
QUESTION [2 upvotes]: I am currently reading about the abstract complex interpolation method, which is a generalization of the Riesz-Thorin theorem. I have been trying to see how to apply it to $L^p$ spaces.
My question is: If you apply the abstract complex interpolation method with exponent $\theta$ to the couple $(L^{p_0},L^{p_1})$, do you get the space $L^p$ ($1/p =(1-\theta)/p_0 + \theta/p_1$)
To keep the question self-contained, I will describe the essential problem I am facing:
Let $S$ be the strip $\{z \in \mathbb{C}, 0\leq Re(z)\leq 1\}$. Then, the class of functions $\mathcal{F}(p_0,p_1)$ consists of all functions $f:S\rightarrow L^{p_0}+L^{p_1}$ with the property that

$f$ is continuous and bounded on $S$, and weakly analytic in the interior.
$f(j+it) \in L^{p_j}, j=0,1$.

Let
$$ \frac{1}{p(\theta)}=\frac{1-\theta}{p_0} + \frac{\theta}{p_1} $$
For $0\leq \theta \leq 1$, does it follow that
$f(\theta) \in L^{p(\theta)}$?

REPLY [2 votes]: See page $91$ in this article by Bill Johnson.


TITLE: How does spot size of a laser depend on wavelength?
QUESTION [0 upvotes]: Does the spot size of a laser depend on wavelength? We have a laser source that is fed into a spectrometer to set the wavelength we want. No other optics are used after the spectrometer. I was going to do the "knife-edge" experiment at a bunch of wavelengths (400-1100nm) to see if the spot size changes, but I figured I would ask here first.

REPLY [1 votes]: How does spot size of a laser depend on wavelength?

Largely, it doesn't. The spot size of the laser can be set independently on the wavelength.
That said, there is a nontrivial relationship that links the wavelength $\lambda$, the focal waist $w_0$, and the Rayleigh range
$$
z_\mathrm{R} = \frac{\pi w_0^2}{\lambda},
$$
where the latter is the distance from the focal plane at which the area of the beam is doubled. This relation encodes the diffraction of the beam from the focal spot: at a fixed wavelength, the smaller the spot size, the faster the beam diverges.

In practice, of course, the details of how things change will depend on the details of the configuration. In particular, for the configuration you report,

We have a laser source that is fed into a spectrometer to set the wavelength we want. No other optics are used after the spectrometer.

there isn't enough information to determine the behaviour of the system.


TITLE: Definition of Strong Convergence in $L^p$
QUESTION [2 upvotes]: Is strong convergence in $L^p$, ie $f_i \overset{strongly}\longrightarrow$, just $||f_i-f||_{L^p} \rightarrow 0$. If so why dont we just call it convergence?

REPLY [4 votes]: Yes your definition is correct. Usually people do simply say convergence, but if you want to really emphasise that you do not mean another form of convergence e.g. weak convergence or almost everywhere convergence, you can use the word strong.