text:
\begin{document} \title [Faltings' local-global principle  of local cohomology]
{A new proof of Faltings' local-global principle  for the finiteness of local cohomology modules}
\author{Davood Asadollahi and Reza Naghipour$^*$}
\address{Department of Mathematics, University of Tabriz, Tabriz, Iran;
and School of Mathematics, Institute for studies in Theoretical
Phisics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran.}

\email{naghipour@ipm.ir} \email {naghipour@tabrizu.ac.ir}
\email{d\_asadollahi@tabrizu.ac.ir}

\thanks{ 2010 {\it Mathematics Subject Classification}: 13D45, 14B15, 13E05.\\
This research was in part supported by a grant from IPM. \\
$^*$Corresponding author: e-mail: {\it naghipour@ipm.ir} (Reza
Naghipour)}

\keywords{Associated primes, Faltings' local-global principle,  Local cohomology.}

\begin{abstract}
 Let $R$  denote a commutative Noetherian ring.  Brodmann et al. in \cite{BRS} defined and studied the concept of the local-global principle for annihilation
 of local cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and $\frak b$ of $R$.
It was shown in \cite{BRS} that this principle holds at levels 1,2, over $R$ and at all  levels whenever $\dim R\leq 4$. The goal of this paper is to show that,
if  the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite or $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the  local-global principle holds at all levels $r\in\mathbb{N}_0$, for all ideals $\fa , \fb$ of $R$ and each finitely generated $R$-module  $M$,  where $c_{\fa}^{\fb}(M)$ denotes the first non $\fb$-cofiniteness of local cohomology module
$H^i_{\fa}(M)$. As a consequence of this, we provide  a new and  short proof of the  Faltings' local-global principle for finiteness dimensions.
Also, several new results concerning the finiteness dimensions are given.
\end{abstract}
\maketitle

\section{Introduction}
Throughout this paper, let $R$ denote a commutative Noetherian ring
(with identity) and $\frak a$ an ideal of $R$. For an $R$-module $M$, the
$i^{\rm th}$ local cohomology module of $M$ with support in $V(\frak a)$
is defined as:
$$H^i_{\frak a}(M) = \underset{n\geq1} {\varinjlim}\,\, \Ext^i_R(R/\frak a^n, M).$$  We refer the reader to \cite{BS} or \cite{Gr1} for more
details about local cohomology.  An important theorem in local
cohomology is Faltings' Local-global Principle for the Finiteness
Dimension of local cohomology modules \cite[Satz 1]{Fa1}, which
states that for a positive integer $r$,  the $R_{\frak p}$-module $H^i_{\frak a R_{\frak p}}(M_{\frak p})$ is finitely generated
for all $i\leq r$ and for all ${\frak p}\in  \Spec R$ if and only if
the $R$-module $H^i_{\frak a}(M)$ is finitely generated for all $i\leq r$.

Another formulation of the Faltings' local-global principal,
particularly relevant for this paper, is in terms  of the finiteness dimension $f_{\frak a}(M)$ of $M$ relative
to $\frak a$, where
$$f_{\frak a}(M):=\inf\{i\in \Bbb{N}_0\,\,|\,\,H^i_{\frak a}(M)\,\,{\rm is}\,\,{\rm not}\,\,{\rm finitely}\,\,{\rm generated}
\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\dag)$$
 With the usual convention that the infimum of the empty set of integers is interpreted as $\infty$.
We can restate the Faltings' local-global principal $(\dag)$, in the following form
 $$f_{{\frak a}R_{\frak p}}(M_{\frak p})>r \,\,\,\,\, \text{ for all } {\frak p}\in \Spec(R) \Longleftrightarrow f_{\frak a}(M)>r. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\dag\dag)$$

Now, let $\frak b$ be a second ideal of $R$.  Recall that the $\frak b$-{\it finiteness dimension} of $M$ relative to $\frak a$ is defined by
\begin{eqnarray*}
f_{\frak a}^{\frak b}(M)&:=& \inf\{i\in \Bbb{N}_0\,\,|\,\,\frak b\not\subseteq {\rm Rad}(0:_{R}H^i_{\frak a}(M))\}\\
&=& \inf\{i\in \Bbb{N}_0\,\,|\,\,{\frak b}^nH^i_{\frak a}(M)\neq 0\,\,{\rm for}\,\,{\rm all}\,\,n\in \Bbb{N}\}.
\end{eqnarray*}

So it is rather natural to ask whether the Faltings' local-global principal, as stated in  $(\dag\dag)$, generalizes in the obvious way to the
invariants $f_{\frak a}^{\frak b}(M)$. In other words, is the statement
$$f_{{\frak a}R_{\frak p}}^{{\frak b}R_{\frak p}}(M_{\frak p})>r \,\,\,\,\, \text{ for all } {\frak p}\in{\Spec}(R) \Longleftrightarrow f_{\frak a}^{\frak b}(M)>r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\dag\dag\dag)$$
 true for each integer $r>0$? We shall say that  the local-global principle for the annihilation of local cohomology modules holds at level $r$ (over the ring $R$) if  $(\dag\dag\dag)$  is true (for the given $r$) for every choice of ideals ${\frak a}$, ${\frak b}$ of $R$ and every choice of finitely generated $R$-module $M$.

In fact, recently  Brodmann et al. in \cite{BRS} defined and studied the concept of the local-global principle for annihilation
 of local cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and $\frak b$ of $R$.
It is shown in \cite{BRS} that the local-global principle for the annihilation of local cohomology modules holds at levels 1,2, over an arbitrary commutative Noetherian ring $R$ and at all  levels whenever $\dim R\leq 4$.

More recently, for an non-negative integer $n$, Doustimehr and Naghipour in \cite{DN}, defined the $n$th $\fb$-finiteness dimension
of $M$ relative to $\fa$ (resp. the $n$th $\fb$-minimum $\fa$-adjusted depth of $M$) by
$$f_{\fa}^{\fb}(M)_n={\rm inf}\{i\in\mathbb{N}_0|\,\dim\Supp\fb^tH^i_{\fa}(M)\geq n\,\, \text{for all}\,\, t\in\mathbb{N}_0 \}.$$
(resp. $\lambda_{\fa}^{\fb}(M)_n={\rm inf}\{\lambda_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\,\p\in\Spec(R),\dim R/\p\geq n\}),$
and using the theory of G-dimension, gave a nice generalization of Faltings' Annihilator theorem.

Now, for a non-negative integer $n$, we define the upper $n$th $\fb$-finiteness dimension of $M$ relative to $\fa$ by
$$f_{\fa}^{\fb}(M)^n={\rm inf}\{f_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\, \p\in\Spec(R),\dim(R/\p)\geq n\}.$$

Note that $f_{\fa}^{\fb}(M)^n$ is either a positive integer or $\infty$, and
$$f_{\fa}^{\fa}(M)^n=f_{\fa}^n(M):=\inf\{f_{\fa R_{\frak p}}(M_{\frak p})\,\,|\,\,{\frak p}\in \Supp(M/\fa M)\,\,{\rm and}\,\,\dim R/{\frak p}\geq n\},$$
is the $n$th finiteness dimension of $M$ relative to $\fa$ (cf. \cite{BNS1}).

The purpose of the present paper is to show that, if the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite or $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the  Faltings' local-global principle holds at all levels $r\in\mathbb{N}_0$, where $c_{\fa}^{\fb}(M)$ denotes the first non $\fb$-cofiniteness of local cohomology module $H^i_{\fa}(M)$.  As a consequence of this, we provide a short proof of the  Faltings' local-global principle for finiteness dimensions. More precisely,
as a main result in the second section, we shall show that

\begin{thm}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$. Let  $M$ be a finitely generated $R$-module such that the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite.  Then
$$f_{\fa}^{\fb}(M)={\rm inf}\{f_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\,\,\p\in\Spec(R)\}.$$
In particular,
$$f_{\fa}(M)={\rm inf}\{f_{\fa R_{\p}}(M_{\p})|\,\,\,\p\in\Spec(R)\}.$$
\end{thm}
The result in Theorem 1.1 is proved in Theorem 2.3 and Corollary 2.4. One of our tools for proving Theorem 1.1 is the following, which plays a key role in this paper.

\begin{prop}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$. Let  $M$ be a finitely generated $R$-module and $n$  a non-negative integer such that
 $(\Ass_R H_{\fa}^{f_{\fa}^{\fb}(M)_n}(M))_{\geq n}$ is finite. Then
$$f_{\fa}^{\fb}(M)_n=f_{\fa}^{\fb}(M)^n.$$
Here, for any subset $T$ of $\Spec(R)$ and a non-negative integer $n$, we set $$(T)_{\geq n}=\{ \p\in T | \dim R/\p\geq n\}.$$
\end{prop}

For a  finitely generated $R$-module $M$ and for ideals $\fa,\fb$  of $R$ with $\fb\subseteq\fa$,  the $\fb$-{\it cofiniteness dimension $c_{\fa}^{\fb}(M)$ of $M$ relative to} $\fa$ is define  by
$c_{\fa}^{\fb}(M)={\rm inf}\{i\in\mathbb{N}_0|\,H^i_{\fa}(M)\,\text{is not}\, \fb\text{-cofinite}\}.$
The notion of $\fb$-cofiniteness dimension $c_{\fa}^{\fb}(M)$ of $M$ relative to $\fa$ is introduced  and studied in \cite{BNS2}.

In Section 3, it is shown that  if   $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the  local-global principle holds at all levels $r\in\mathbb{N}_0$, for all ideals $\fa , \fb$ of $R$ and each finitely generated $R$-module  $M$. Moreover,  we obtain some new results about the finiteness dimensions of local cohomology modules. In this section among other things, we derive
the following consequence of Theorem 1.1, which shows that the Faltings' local-global principle holds at all levels.

\begin{thm}
Let $\fb\subseteq\fa$ be ideals of $R$  and let  $M$ be a finitely generated $R$-module such that $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$.  Then the
Faltings' local-global principle for the finiteness of local cohomology modules  holds at all levels.
\end{thm}

 Throughout this paper, $R$ will always be a commutative Noetherian
ring with non-zero identity and $\frak a$ will be an ideal of $R$.  Recall
that an $R$-module $M$ is called $\frak a$-{\it cofinite} if $\Supp M\subseteq V(\frak a)$ and ${\rm Ext}^i_R(R/\frak a, M)$ is finitely
generated for all $i\geq 0$. The concept of $\frak a$-cofinite modules were introduced by Hartshorne \cite{Ha}. Also, if  $n$ is a non-negative integer,
then  $M$ is said to be in dimension $< n$, if there is a finitely generated submodule $N$ of $M$ such that $\dim \Supp  M/ N <n$ (cf. \cite[Definition 2.1]{AN}); and moreover if $T$ is a subset of $\Spec(R)$, then we define $$(T)_{\geq n}=\{ \p\in T | \dim R/\p\geq n\}.$$
 Finally, for any ideal $\frak{b}$ of $R$, the {\it radical} of $\frak{b}$, denoted by $\Rad(\frak{b})$, is defined to
be the set $\{x\in R \,: \, x^n \in \frak{b}$ for some $n \in \mathbb{N}\}$. For any unexplained notation and terminology we refer
the reader to \cite{BS} and \cite{Mat}.

\section{ Faltings' local-global principle and finiteness}

In this section we establish that the  Faltings' local-global principle for the finiteness of local cohomology modules holds at all levels over an arbitrary commutative Noetherian ring $R$,  whenever the set  $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite. As a consequence,  we will provide  a new and  short proof of the  Faltings' local-global principle for finiteness dimensions. We begin with the following lemma which is needed in the proof of Proposition 2.2.
\begin{lem}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$, and let $M$ be a finitely generated $R$-module. Then, for every non-negative integer $n$,
$$f_{\fa}^{\fb}(M)_n\leq f_{\fa}^{\fb}(M)^n\leq \lambda_{\fa}^{\fb}(M)_n.$$
\end{lem}
\proof
Let $t:=f_{\fa}^{\fb}(M)^n$ and suppose that $t<f_{\fa}^{\fb}(M)_n$. Then, in view of the definition, there exists an integer $m$ such that $\dim\Supp (\fb^mH^{t}_{\fa}(M))<n$. Hence, for every prime ideal $\p$ of $R$ with $\dim R/\p\geq n$, we have
 $$(\fb^mH^{t}_{\fa}(M))_{\p}=0,$$
and so   $(\fb R_{\p})^mH^{t}_{\fa R_{\p}}(M_{\p})=0$, which is a contradiction.

Therefore $f_{\fa}^{\fb}(M)_n\leq f_{\fa}^{\fb}(M)^n$. The inequality $f_{\fa}^{\fb}(M)^n\leq \lambda_{\fa}^{\fb}(M)_n$, follows easily from the definition and  \cite[Theorem 9.3.5]{BS}. \qed\\

The following proposition  plays a key role in the proof of the main result of this section.
\begin{prop}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$. Let  $M$ be a finitely generated $R$-module and  let $n$ be a non-negative integer such that
 $(\Ass_R H_{\fa}^{f_{\fa}^{\fb}(M)_n}(M))_{\geq n}$ is finite. Then
$$f_{\fa}^{\fb}(M)_n=f_{\fa}^{\fb}(M)^n.$$
\end{prop}
\proof
Let $t:=f_{\fa}^{\fb}(M)_n$ and suppose that
$$(\Ass_R H_{\fa}^t(M))_{\geq n}=\{\p_1,\dots,\p_r\}.$$
In view of Lemma 2.1, it is enough for us to show that $f_{\fa}^{\fb}(M)^n\leqslant t$. Suppose the contrary that $t<f_{\fa}^{\fb}(M)^n$, and look for a contradiction.
To this end, in view of definition  $t<f_{\fa R_{\p_i}}^{\fb R_{\p_i}}(M_{\p_i})$, for all $1\leq i\leq r$. Hence, for all $1\leq i\leq r$, there exists an integer $n_i$ such that
$$(\fb R_{\p_i})^{n_i}H^{t}_{\fa R_{\p_i}}(M_{\p_i})=0.$$
Set $n:={\rm max}\{n_1,\ldots,n_r\}$.  Then, for all $1\leq i\leq r$, we have
$$(\fb^nH^{t}_{\fa}(M))_{\p_i}=0.$$
Now, it is easy to see that $\dim\Supp(\fb^nH^{t}_{\fa}(M))<n$, which is a contradiction.\qed\\

We are now ready to state and prove the main theorem of this section, which shows
that the Faltings' local-global principle for the finiteness of local cohomology modules is
valid at all levels over any Noetherian ring $R$,  whenever the set  $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite.
\begin{thm}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$. Let  $M$ be a finitely generated $R$-module such that the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite. Then
$$f_{\fa}^{\fb}(M)={\rm inf}\{f_{\fa R_{\p}}^{\fb R_{\p}}(M_{\p})|\,\,\,\p\in\Spec(R)\}.$$
\end{thm}
\proof
Put $n=0$ in the Proposition 2.2, and use the fact that $f_{\fa}^{\fb}(M)=f_{\fa}^{\fb}(M)_0$.\qed \\

An immediate  consequence of Theorem 2.3, we give a short proof of the  Faltings' local-global principle for finiteness dimensions.

\begin{cor}{\rm (\cite[\bf 9.6.2 Local-global Principle for Finiteness Dimensions]{BS})}
Let $M$ be a finitely generated $R$-module. Then for  any ideal $\fa$ of $R$,
$$f_{\fa}(M)={\rm inf}\{f_{\fa R_{\p}}(M_{\p})|\,\,\,\p\in\Spec(R)\}.$$
\end{cor}
\proof
Put $\fa=\fb$ in Theorem 2.3 and use \cite[Corollary 2.3 ]{BL}.\qed\\

\section{Faltings' local-global principle and cofiniteness}

The purpose of this section  is to show that  if  $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the  local-global principle holds at all levels $r\in\mathbb{N}_0$, for all ideals $\fa , \fb$ of $R$ and each finitely generated $R$-module  $M$. Moreover,  we obtain some new results about the finiteness dimensions of local cohomology modules. In order to do this, let us recall that for a  finitely generated $R$-module $M$ and for ideals $\fa,\fb$  of $R$ such that $\fb\subseteq\fa$,  the $\fb$-cofiniteness dimension $c_{\fa}^{\fb}(M)$ of $M$ relative to $\fa$ is defined  by
$c_{\fa}^{\fb}(M)={\rm inf}\{i\in\mathbb{N}_0|\,H^i_{\fa}(M)\,\text{is not}\, \fb\text{-cofinite}\}.$
The notion of $\fb$-cofiniteness dimension $c_{\fa}^{\fb}(M)$ of $M$ relative to $\fa$ is introduced  and studied in \cite{BNS2}.
\begin{prop}
Let $\fa$ and $\fb$ be ideals of $R$ such that $\fb\subseteq\fa$  and let  $M$ be a finitely generated $R$-module. Then
$$f_{\fa}(M)={\rm min}\{f_{\fa}^{\fb}(M),c_{\fa}^{\fb}(M)\}.$$
\end{prop}
\proof
It is clear that $f_{\fa}(M)\leq f_{\fa}^{\fb}(M)$. Now, if $t:=c_{\fa}^{\fb}(M)<f_{\fa}(M)$, then $H^t_{\fa}(M)$ is a finitely generated $R$-module. Since $$\Supp H^t_{\fa}(M)\subseteq V(\fa)\subseteq V(\fb),$$
it follows that the $R$-module $H^t_{\fa}(M)$ is $\fb$-cofinite, which is a contradiction. Whence $f_{\fa}(M)\leq c_{\fa}^{\fb}(M)$ and so
$$f_{\fa}(M)\leq{\rm min}\{f_{\fa}^{\fb}(M),c_{\fa}^{\fb}(M)\}.$$
Finally, suppose that
$$r:=f_{\fa}(M)<{\rm min}\{f_{\fa}^{\fb}(M),c_{\fa}^{\fb}(M)\},$$ and look for a contradiction.
To do this, as $r<f_{\fa}^{\fb}(M)$, there exists an integer $n$ such that $\fb^nH^r_{\fa}(M)=0$, and thus
$$H^r_{\fa}(M)\cong {\rm Hom}_R(R/\fb^n,H^r_{\fa}(M)).$$
Now, since $r<c_{\fa}^{\fb}(M)$,  it follows that the $R$-module $H^r_{\fa}(M)$ is $\fb$-cofinite, and so it yields from the above isomorphism that
$H^r_{\fa}(M)$ is finitely generated, which is a contradiction. \qed\\

We are now ready to state and prove the main theorem of this section, which shows
that the Faltings' local-global principle for the finiteness of local cohomology modules is
valid at all levels over any Noetherian ring $R$,  whenever $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$.

\begin{thm}
Let $\fb\subseteq\fa$ be ideals of $R$  and let  $M$ be a finitely generated $R$-module such that $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$.  Then the
Faltings' local-global principle for the finiteness of local cohomology modules  holds at all levels.
\end{thm}
\proof
Since $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, it follows from Proposition 3.1 that $f_{\fa}^{\fb}(M)\leq c_{\fa}^{\fb}(M)$, and so
$f_{\fa}(M)= f_{\fa}^{\fb}(M)$.  Now, the assertion follows from Theorem 2.3 and Brodmann-Lashgari's result \cite[Corollary 2.3 ]{BL}.  \qed\\

Before bringing the next result, we give a couple of lemmas that will be used in the proof of Theorem 3.5.

\begin{lem}
Let  $M$ be an  $R$-module and let $s$  be a non-negative integer. Then

 ${\rm(i)}$ If $M$ is in dimension $<s$, then $M_{\frak p}$ is a finitely generated $R_{\frak p}$-module for all $\frak p \in \Spec (R)$ with $\dim R/\frak p\geq s$.

${\rm(ii)}$  If $M$ is in dimension $<s$, then the set $(\Ass_R M)_{\geq s}$ is finite.
\end{lem}
\proof
The part $\rm(i)$ follows easily from the definition. In order to prove $\rm(ii)$,  since $M$ is in dimension $< s$, it follows from the definition that there is a finitely generated submodule $M'$ of $M$ such that $\dim \Supp M/M'<s$. Now, from the exact sequence $$0 \longrightarrow M' \longrightarrow M\longrightarrow M/M' \longrightarrow 0$$
we obtain
$$(\Ass_R M)_{\geq s}\subseteq (\Ass_RM')_{\geq s}\cup (\Ass_R M/M')_{\geq s}.$$
As $\dim \Supp M/M'<s$, it follows that $(\Ass_R M/M')_{\geq s}=\emptyset$,  and so  the set $(\Ass_R M)_{\geq s}$ is finite.\qed \\

Before bringing the next lemma, let us recall that a full subcategory $\mathcal{S}$ of the category of
$R$-modules is called a {\it Serre subcategory}, when it is closed under taking submodules, quotients
and extensions.
\begin{lem}
For any non-negative integer $n$, the class of in dimension $<n$ modules over a  Noetherian ring $R$ consists  a  Serre subcategory of
the category of $R$-modules.
\end{lem}

\proof See \cite[Corollary 2.3]{MNS}.  \qed \\

\begin{thm}
Let $(R,{\frak m})$ be a complete local ring and $M$ a finitely generated $R$-module. Let  $\fb\subseteq\fa$ be ideals of $R$  and let $n$  be a non-negative
integer such  that $f_{\fa}^{\fb}(M)_n=f_{\fa}^{\fb}(M)^n$.
Then $f_{\fa}^{\fb}(M)_n\geq {\rm depth}(\fb,M)$ if and only if $f_{\fa}^n(M)\geq {\rm depth}(\fb,M)$.
\end{thm}

\proof
Let ${\rm depth}(\fb,M)=s$.  Since $\fb\subseteq\fa$, it yields that  $f_{\fa}(M)\leq f_{\fa}^{\fb}(M)$ and so
$$f_{\fa}^n(M)\leq f_{\fa}^{\fb}(M)^n=f_{\fa}^{\fb}(M)_n.$$
Recall that $f_{\fa}^n(M):=\inf\{f_{\fa R_{\frak p}}(M_{\frak p})\,\,|\,\,{\frak p}\in \Supp(M/\fa M)\,\,{\rm and}\,\,\dim R/{\frak p}\geq n\}.$

Now, suppose that $f_{\fa}^{\fb}(M)_n\geq s$ and we show that $f_{\fa}^n(M)\geq s$.  We use induction on $s$. For $s=0$ there is nothing to show. So assume that $s>0$ and the result has been proved for $s-1$.

Since ${\rm depth}(\fb,M)>0$, it follows that $\fb$ contains an element $x$ which is a non-zero divisor on $M$.
Moreover, as $f_{\fa}^{\fb}(M)_n\geq s$, there exists an integer $t$ such that, for all $i<s$,  $\dim\Supp(\fb^t H^{i}_{\fa}(M))<n$.
Hence from the exact sequence
$$0\longrightarrow M \stackrel{x^t}\longrightarrow M\longrightarrow M/x^tM\longrightarrow 0,$$
and \cite[Lemma  2.9]{DN}, there exists an integer $l$ such that, for all $i<s-1$, $$\dim\Supp(\fb^lH^{i}_{\fa}(M/x^tM))<n,$$  and thus $f_{\fa}^{\fb}(M/x^tM)_n\geq s-1$.
Therefore, in view of the inductive hypothesis $$f_{\fa}^n(M/x^tM)\geq s-1,$$  and so by virtue of \cite[Theorem 2.5]{AN}, $H^{s-2}_{\fa}(M/x^tM)$ is in dimension $<n$.
Therefore, it follows from the exact sequence and Lemma 3.4
$$0\longrightarrow H^{s-2}_{\fa}(M)/x^tH^{s-2}_{\fa}(M) \longrightarrow H^{s-2}_{\fa}(M/x^tM)\longrightarrow(0:_{H^{s-1}_{\fa}(M)}x^t)\longrightarrow 0,$$
that $(0:_{H^{s-1}_{\fa}(M)}x^t)$ is in dimension $<n$. Hence in view of Lemma 3.3, the $R_{\frak p}$-module
$(0:_{(H^{s-1}_{\fa}(M))_{\p}}x^t/1)$ is finitely generated, for all $\frak p \in \Spec(R)$ with  $\dim R/\p\geq n$.  Now, since $\dim\Supp(\fb^t H^{s-1}_{\fa}(M))<n$, we have $(\fb^t H^{s-1}_{\fa}(M))_{\p}=0$, and hence
 $x^t/1 (H^{s-1}_{\fa}(M))_{\p}=0.$
 Consequently,
 $$(H^{s-1}_{\fa}(M))_{\p}=(0:_{(H^{s-1}_{\fa}(M))_{\p}}x^t/1),$$
is a finitely generated $R_{\frak p}$-module. Therefore, by virtue of Lemma 3.3,   the $R$-module  $ H^{s-1}_{\fa}(M)$ is also in dimension $<n$.
 Consequently,  we deduce from \cite[Theorem 2.5]{AN} that $f_{\fa}^n(M)\geq s$, as required.\qed\\

 \begin{cor}
Let $(R,{\frak m})$ be a complete local ring and $M$ a finitely generated $R$-module. Let  $\fb\subseteq\fa$ be ideals of $R$  and let $n$  be a non-negative
integer such  that ${\rm depth}(\fb,M)\leq f_{\fa}^{\fb}(M)_n$ and  the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite.
Then $${\rm depth}(\fb,M)\leq \min\{f_{\fa}^{\fb}(M)_n,  f_{\fa}^n(M)\}.$$
\end{cor}
\proof
The result follows from Proposition 2.2 and Theorem 3.5. \qed\\

 \begin{thm}
Let $(R,{\frak m})$ be a complete local ring and $M$ a finitely generated $R$-module. Let  $\fa$ be an ideal of $R$  and  let $s,t$ be   non-negative integers.
Then the following statements are equivalent.

$\rm(i)$  $H^{i}_{\fa}(M)$ is in dimension $<s$, for all $i<t$.

$\rm(ii)$ There exists an integer $n$ such that $\dim\Supp(\fa^nH^{i}_{\fa}(M))<s$, for all $i<t$.
\end{thm}
\proof
In order to show $\rm(i)\Longrightarrow \rm(ii)$, let $i$ be an integer such that $i<t$.  Then, since $H^{i}_{\fa}(M)$ is in dimension $<s$, in view of Lemma 3.3, the set
$$(\Ass_R H^{i}_{\fa}(M))_{\geq s}$$
is finite. Let
$$(\Ass_R H^{i}_{\fa}(M))_{\geq s}=\{\p_1,\dots,\p_n\}.$$
Thus, for all $j$ with $1\leq j\leq r$, the $R_{\p_j}$-module $(H^{i}_{\fa}(M))_{\p_j}$ is  finitely generated
and thus there exists $n_j\in\mathbb{N}$ such that $(\fa^{n_j}H^{i}_{\fa}(M))_{\p_j}=0$.

Put $n:=\max\{n_1,\ldots,n_r\}$. Then,  for all $j$ with $1\leq j\leq r$, we have $(\fa^{n}H^{i}_{\fa}(M))_{\p_j}=0$.
Now, we show that $\dim\Supp(\fa^{n}H^{i}_{\fa}(M))<s$.
For this, let $\q\in\Ass_R(\fa^{n}H^{i}_{\fa}(M))$.  Then, if $\dim R/\q\geq s$, then there exists $j$ such that $\q=\p_j$ and this is a contradiction. Thus $\dim R/\q< s$,  as desired.

Now, we show $\rm(ii)\Longrightarrow \rm(i)$.  To do this, for all $i<t$ and for all prime ideals $\p$  with $\dim R/\p\geq s$,
we have $(\fa^{n}H^{i}_{\fa}(M))_{\p}=0$.
 Thus, for all $i<t$ and for all prime ideals $\p$ with $\dim R/\p\geq s$, the $R_{\p}$-module $ H^{i}_{\fa}(M))_{\p}$ is  finitely generated. Hence $f_{\fa}^s(M)\geq t$. Now, since $$f_{\fa}^s(M)={\rm inf} \{0\leq i\in\inte|\,H^{i}_{\fa}(M)\text{ is not in dimension}<s\},$$
(cf.  \cite[Theorem 2.5]{AN}), it follows that $H^{i}_{\fa}(M)$ is in dimension $<s$ for all $i<t$, as required.\qed\\

\begin{cor}
Let $(R,{\frak m})$ be a complete local ring,  $M$ a finitely generated $R$-module and $\fa$  an ideal of $R$. Then
for any non-negative integer $n$,  $$f_{\fa}^n(M)=f_{\fa}^{\fa}(M)_n.$$
\end{cor}

 \proof
 The assertion follows from Theorem 3.7 and  the definitions of  $f_{\fa}^n(M)$ and $f_{\fa}^{\fa}(M)_n.$  \qed\\

\begin{thm}
Let $R$ be a Noetherian ring,  $M$ a finitely generated $R$-module and let  $\fb\subseteq\fa$ be  ideals of $R$. Then, for any non-negative integers
$i$ and $ s$,  the following statements are equivalent.

$\rm (i)$  There exists an integer $n$ such that $\dim\Supp(\fb^nH^{i}_{\fa}(M))<s$.

$\rm (ii)$  There exists an integer $m$ such that $\fb^mH^{i}_{\fa}(M)$ is in dimension $<s$.
\end{thm}

\proof
The implication $\rm (i) \Longrightarrow \rm (ii)$ is clear. In order to show  $\rm (ii) \Longrightarrow \rm (i)$,
since $\fb^mH^{i}_{\fa}(M)$ is in dimension $<s$,  it follows from Lemma 3.3 that the set
$$(\Ass_R \fb^mH^{i}_{\fa}(M))_{\geq s}$$
is finite.  Let
$$(\Ass_R \fb^mH^{i}_{\fa}(M))_{\geq s}=\{\p_1,\dots,\p_r\}.$$
Then, for all $j$ with $1\leq j\leq r$, the $R_{\p_j}$-module $(\fb^mH^{i}_{\fa}(M))_{\p_j}$ is finitely generated. Hence,  there exists $n_j\in\mathbb{N}$ such that $(\fa^{n_j}(\fb^mH^{i}_{\fa}(M))_{\p_j}=0$. Since $\fb\subseteq\fa$, it follows that $(\fb^{m+n_j}H^{i}_{\fa}(M))_{\p_j}=0$.

Set $n:=\max\{m+n_1,\ldots,m+n_r\}$. Then,  for all $j$ with $1\leq j\leq r$, we deduce that $(\fb^{n}H^{i}_{\fa}(M))_{\p_j}=0$.

We  show that $\dim\Supp(\fb^{n}H^{i}_{\fa}(M))<s$.  To do this, let $\p\in\Ass_R(\fb^{n}H^{i}_{\fa}(M))$. Then $\p\in \Ass_R(\fb^{m}H^{i}_{\fa}(M))$.
Now, if $\dim R/\p\geq s$, then there exists $j$ such that $\p=\p_j$ and this is a contradiction. Thus $\dim R/\p< s$,  as required.\qed\\

\begin{cor}
Let $R$ be a Noetherian ring and $M$ a finitely generated $R$-module.
Let $\fa$ and $\fb$ be two  ideals of $R$ such that $\fb\subseteq\fa$. Then, for any non-negative integer
$n$,
$$f_{\fa}^{\fb}(M)_n={\rm inf} \{0\leq i\in\inte|\,\fb^m H^{i}_{\fa}(M)\text{ is not in dimension} <n\,\, \text{for all}\,\, m\}.$$
\end{cor}

\proof
The assertion follows from Theorem 3.9 and the definition of $f_{\fa}^{\fb}(M)_n$. \qed\\

\begin{center}
{\bf Acknowledgments}
\end{center}
The authors would like to thank Professors  Hossein Zakeri and Kamal Bahmanpour for reading of the first draft and valuable discussions.
Also, we  would like to thank from the Institute for Research in Fundamental Sciences (IPM) for the financial support.

config: arxiv


TITLE: Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?
QUESTION [7 upvotes]: Most proofs of Galois theorem stating that "an equation is solvable in radicals if and only if its Galois group is solvable," show the left to right direction by induction on the height of the radical extension. Starting with a cyclotomic field, adjoining "enough" roots of the unity (e.g. adjoining the product of all the ) and continuing with successive simple radical extensions, which are all Galois and Abelian over the previous one (because we ensured that there were enough roots of unity to start with). The key element in proving that all these extensions are solvable over the base field is then to define a solvable extension as an extension which normal closure has solvable Galois group (equivalently such that there exist an extension which Galois group is solvable) (def (a)), this makes "being a solvable extension" transitive (it is distinguished) as proved in Lang Algebra. Doing this makes it possible to complete the proof by induction, while "being Galois and solvable" (def (b)) does not go through (since Galois extensions are not distinguished).
Many texts also state that the (radical) extensions arising in the above mentioned induction are not Galois in general. While this is indeed not the case in general (as with the traditional example $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ since it's missing $i\sqrt[4]{2}$), one might wonder if this is true or not for radical extensions where the base field contains enough roots of unity.
More precisely, the question is to determine whether the following statement is true.

Let $\omega_n$ be a primitive $n^\textrm{th}$ root of the unity.
Let $x = (x_1, \ldots, x_h)$ be sequence of elements in the algebraic closure of $\mathbb{Q}$, such that $x_{i+1}^{m_{i+1}} \in \mathbb{Q}(x_1, \ldots, x_{i})$ for some minimal $m_i > 1$ for all $i < h$.
Is there a natural number $N$, such that in the radical extension series
$$
 \mathbb{Q}(\omega_{N}) = K \subset K(x_1) \subset \ldots \subset K(x_1, \ldots, x_n)
$$
each $K(x_1, \ldots, x_{i+1})/K$ is Galois ?

Definition
Let's call $N_x$ the smallest of such natural numbers if it exists, and set $N_x = \infty$ if there is no such natural number.
The above question amounts to «Is $N_x$ always finite?»
In a typical proof of Galois theorem above, since the primitive root of the unity that is chosen is typically a divisor of $d_x = \prod_i m_i$ (one can also pick the lcm afaik).

If it were true that $N_x | d_x$ for all $x$ then the proof above could be carried out with def (b) instead of (a). (The answers seem to indicate this is not true though.)
However if $N_x$ is finite for all $x$, then for each $x$ one can use def (b) by starting the radical series with $\mathbb{Q}(\omega_{N_x})$ (instead of $\mathbb{Q}(\omega_{d_x})$ as in the proof above).

I conjectured the above statement is false, i.e. that there are $x$s such that $N_x > d_x$ and even such that $N_x = \infty$.
Hence I am searching for a (minimal?) counter example, i.e. a radical extension
$$
 \mathbb{Q}(\omega_N) = K \subset K(x_1) \subset \ldots \subset K(x_1, \ldots, x_h)
$$
such that each for all $N$ such that each $K(x_1, \ldots, x_{i+1})/K(x_1, \ldots, x_i)$ is Galois, $K(x_1, \ldots, x_n)/K$ is not Galois.
(Another way to state it is to start with $K$ being an extension of $\mathbb{Q}$ containing all roots of unit.)
I think the following is a radical extensions of total degree $6$ over $\mathbb{Q}(\omega)$, where $\omega = \frac{\sqrt{3}}{2} + \frac{i}{2}$ is a primitive 12th root of unity and where each intermediate extension is Galois over the previous one:
$$K = \mathbb{Q}(\omega) \subset F = K\left(\sqrt[3]{2}\right) \subset K\left(\sqrt[3]{2}, \sqrt{1+ \sqrt[3]{2}}\right) = E.$$
However, I would like to claim $E$ is not Galois over $K$, even though all 12th roots of unity are in $K$.
Indeed a conjugate of $\sqrt{1+ \sqrt[3]{2}}$ over $K$ is a square root $\alpha$ of $1 + j\sqrt[3]{2}$ -- their minimal polynomial is $X^6 - 3 X^4 + 3 X^2 - 3$ -- but $\alpha$ would not be in $E = \textrm{span}_F\left(1, \sqrt{1+ \sqrt[3]{2}}\right).$
I reduced the problem to checking that $\frac{1 + j\sqrt[3]{2}}{1+ \sqrt[3]{2}}$ is not a square in $F$ (unless I made a mistake) and some famous online computer algebra system tells that if I compare it to $(x+y\sqrt[3]{2}+z\sqrt[3]{2}^2)^2$ then $z$ needs to be expressed using a square root of an element of $F$ that does not look like it's in $F$, but I am still not 100% sure this is correct...
Does someone has indications on better ways to find a counter example, check this one or prove the theorem that I believe is wrong in the first place? (i.e. that all radical extension over a field with enough roots of unity are Galois.)

EDIT 25/01/2021
I made my question more precise by showing the statement for which I am searching for a counter example.
Unless I am mistaken, the answers and their comments below show that both

$x_1 ^ 2 = 3, \quad x_2 ^ 2= 1 + x_1$
$y_1 ^ 3 = 2, \quad y_2 ^ 2= 1 + y_1$

are indeed counter examples of $N | d$ since

$d_x = 4$ and $N_x = 8$
$d_y = 6$ and $N_y > 12$.

They also show that $x$ is not a counter example that $N_x < \infty$, and I think the finiteness status of $N_y$ is unknown.

REPLY [2 votes]: The polynomial $f:=(x^2-1)^3-2$ has Galois group $G \simeq S_4 \times C_2$ over $\mathbb{Q}$, so the commutator subgroup $[G,G] \simeq A_4$. Let $L$ be the splitting field (over $\mathbb{Q}$) of $f$. This contains the $12$-th roots of unity, i.e. contains your $K$. Since $[K:\mathbb{Q}]=4=[G:[G,G]]$, $L$ contains no other roots of unity and $\operatorname{Gal}(L/K)\simeq A_4$. Since $A_4$ has no normal subgroups of order $2$ the extension $E/K$ is not Galois.
Certifying that $G \simeq S_4 \times C_2$ could e.g. be done using (an exact version of a small part of) Algorithm 6.3.10 in Cohen's "A Course in Computational Algebraic Number Theory" or using more basic Galois resolvents, or...


text:
In this section, we will be concerned with the definition and
properties of the quaternion Fourier transform (QFT) of complex valued
signals. Before introducing the main definitions, we give some
prerequisites on quaternion valued signals.

\subsection{Preliminary remarks}
Quaternions were discovered by Sir W.R. Hamilton in 1843
\cite{Hamilton:1853}. Quaternions are 4D hypercomplex numbers that form a noncommutative
division ring denoted $\H$. A quaternion $q \in \H$
has a Cartesian form: $q=a+b\i+c\j+d\k$, with $a,b,c,d
\in {\mathbb R}$ and $\i,\j,\k$ roots of $-1$ satisfying
$\i^2=\j^2=\k^2=\i\j\k=-1$.
The {\em scalar part} of $q$ is $a$: ${\cal S}(q)=a$.
The {\em vector part} of $q$ is: ${\cal V}(q)=q-{\cal S}(q)$.
Quaternion multiplication is not commutative, so that in
general $qp\neq pq$ for $p,q \in \H$.
The conjugate of $q$ is $\qconjugate{q}=a-b\i-c\j-d\k$.
The norm of $q$ is $\norm{q}=|q|^2=(a^2+b^2+c^2+d^2)=q\qconjugate{q}$.
A quaternion with $a=0$ is called pure.
If $|q|=1$, then $q$ is called a {\em unit} quaternion.
The inverse of $q$ is $q^{-1}=\qconjugate{q}/\norm{q}$.
Pure unit quaternions are special quaternions,
among which are $\i$, $\j$ and $\k$.
Together with the identity of $\H$, they form a {\em quaternion basis}:
$(1,\i,\j,\k)$. In fact, given any two unit pure quaternions, $\mu$
and $\xi$, which are orthogonal to each other ({\em i.e.} ${\cal
  S}(\mu\xi)=0$), then $(1,\mu,\xi,\mu\xi)$ is a quaternion basis.

Quaternions can also be viewed as complex numbers with complex
components, {\em i.e.} one can write $q=z_1+\j z_2 $ in the
basis $(1,\i,\j,\k)$ with $z_1,z_2 \in \C^{\i}$, {\em i.e.} $z_{\alpha}=\Re(z_{\alpha})+\i
\Im(z_{\alpha})$ for $\alpha=1,2$\footnote{In the sequel, we note as $\C^{\mu}$
a complex number with root of $-1$ being $\mu$. Note that
these are degenerate quaternions.}. This is called the
Cayley-Dickson form. Now, it is possible to define a
generalized Cayley-Dickson form of a given quaternion using an
arbitrary quaternion basis $(1,\mu,\xi,\mu\xi)$. A quaternion $q$ can
then be written as $q=w_1+ \xi w_2 $ where $w_1,w_2 \in \C^{\mu}$.

Any quaternion $q$ also has a unique polar Cayley-Dickson
  form \cite{10.1007/s00006-008-0128-1} given by: $q=A \exp(B\j)$, where $A$ is the complex modulus of $q$
  and $B$ is its complex phase. Expressions of $A$ and $B$ can be
  found in \cite{10.1007/s00006-008-0128-1}. This notation will be of
  use in Section \ref{PhaseAttribsection}.

\subsection{1D Quaternion Fourier transform}
In this paper, we use a 1D version of the (right) QFT first
defined in discrete-time form in \cite{SangwineEll:1998}.
Thus, it is necessary to specify the axis (a pure unit
quaternion) of the transform. So, we will denote $\qfourier{\mu}{\,}$ a QFT with
transformation axis $\mu$. This will become clear from the definition
below. We now present the definition and some properties of the
transform used here.

\begin{definition}
\label{def:qft}
Given a complex valued signal $z(t)=z_r(t)+\xi z_i(t)$, its quaternion
Fourier transform with respect to axis $\mu$ is:
\begin{equation}
Z_{\mu}(\nu)=\qfourier{\mu}{z(t)}=\int\limits_{-\infty}^{+\infty}z(t)e^{-\mu 2 \pi \nu t} \dt
\label{Eqn:QFT2}
\end{equation}
and the inverse transform is:
\begin{equation}
z(t)=\iqfourier{\mu}{Z_{\mu}(\nu)}=\int\limits_{-\infty}^{+\infty}Z_{\mu}(\nu)e^{\mu 2 \pi \nu t} d\nu
\label{Eqn:QFT3}
\end{equation}
where in both cases,
the transform axis $\mu$ can be chosen to be any unit pure quaternion orthogonal to $\xi$
so that $\left(1,\xi,\mu,\xi\mu\right)$ is a quaternion basis.
\end{definition}

\begin{property}
\label{prop:symQFT}
Given a complex signal $z(t)=z_r(t)+\xi z_i(t)$ and its quaternion
Fourier transform denoted $Z_{\mu}(\nu)$, then the following properties hold:
\begin{itemize}
\item The even part of $z_r(t)$ is in          $\Re(Z_{\mu}(\nu))$.
\item The odd  part of $z_r(t)$ is in    $\Im_{\mu}(Z_{\mu}(\nu))$
\item The even part of $z_i(t)$ is in    $\Im_{\xi}(Z_{\mu}(\nu))$.
\item The odd  part of $z_i(t)$ is in $\Im_{\xi\mu}(Z_{\mu}(\nu))$
\end{itemize}
\end{property}
\begin{IEEEproof}
Expand \eqref{Eqn:QFT2} into real and imaginary parts with respect to $\xi$,
and expand the quaternion exponential into cosine and sine components:
\begin{align*}
Z_{\mu}(\nu)
&=\int\limits_{-\infty}^{+\infty}{
                                  \!\left[z_r(t)+\xi z_i(t)\right]
                                  }
                                   \left[\cos(2\pi\nu t) - \mu\sin(2\pi\nu t)\right]\dt\\
&=\phantom{+\xi}\!\int\limits_{-\infty}^{+\infty}\!z_r(t)\cos(2\pi\nu t)\dt
            -\mu\!\int\limits_{-\infty}^{+\infty}\!z_r(t)\sin(2\pi\nu t)\dt\\
&\phantom{=}+\xi\!\int\limits_{-\infty}^{+\infty}\!z_i(t)\cos(2\pi\nu t)\dt
         -\xi\mu\!\int\limits_{-\infty}^{+\infty}\!z_i(t)\sin(2\pi\nu t)\dt
\end{align*}
from which the stated properties are evident.
\end{IEEEproof}

These properties are central to the justification of the use of the QFT to
analyze a complex valued signal carrying complementary but different
information in its real and imaginary parts. Using the QFT, it is
possible to have the odd and even parts of the real and imaginary parts of
the signal in four different components in the transform
domain. This idea was also the initial motivation of B\"ulow,
Sommer and Felsberg when they developed the monogenic signal for images
\cite{Bulow:1999,BulowSommer:2001,FelsbergSommer:2001}. Note that the
use of hypercomplex Fourier transforms was
originally introduced in 2D Nuclear Magnetic Resonance image analysis \cite{Ernst:1987,Delsuc:1988}.

We now turn to the link between a complex signal and the quaternion
signal that can be uniquely associated to it.
\begin{property}
A one-sided QFT $X(\nu)$ with $X(\nu)=0$ ({\em i;e.}$\forall \nu < 0$)
corresponds to a quaternion valued signal in the time domain.
\end{property}
\begin{IEEEproof}
A complex signal with values in $\C^{\xi}$ has a $QFT_{\mu}$ with the symmetry
properties listed in property \ref{prop:symQFT}.
Now, cancelling its negative
frequencies is equivalent to adding a signal with a
QFT with the following
properties: $\Re$ part is odd, $\Im_{\mu}$ is odd, $\Im_{\xi}$ is even
and $\Im_{\xi\mu}$ is even. By linearity of the inverse QFT,
the former two parts of the QFT produce a complex signal in the time
domain with $\Re$ and $\Im_{\xi}$ parts, while the two latter produce a
quaternion signal in the time domain with $\Im_{\mu}$ and $\Im_{\xi\mu}$
parts. Together, and again by linearity, the resulting signal in the
time domain is a full 4D quaternion signal.
\end{IEEEproof}

\begin{property}
Given a complex signal $x(t)$, one can associate to it a unique
canonical pair corresponding to a modulus and phase. These
modulus and phase are uniquely defined through the hyperanalytic
signal, which is quaternion valued.
\end{property}
\begin{IEEEproof}
Cancelling the negative frequencies of the QFT leads to a quaternion
signal in the time domain. Then, any quaternion signal has a modulus
and phase defined using its CD polar form.
\end{IEEEproof}

\subsection{Convolution}
We consider the special case of convolution of a complex signal by a
real signal. Consider $g$ and $f$ such that: $g:{\mathbb
R}^+\rightarrow{\mathbb C}^{\xi}$ and $f:{\mathbb R}^+\rightarrow{\mathbb
R}$. Now, consider the $\QFT_{\mu}$ of their convolution:
\begin{equation}
\begin{aligned}
\qfourier{\mu}{g*f(t)} &= \int\limits_{-\infty}^{+\infty}
                     \int\limits_{-\infty}^{+\infty}
                     g(\tau)f(t-\tau)\dtau
                     e^{-2\mu \pi \nu t} \dt \\
                  &= \int\limits_{-\infty}^{+\infty}
                     \int\limits_{-\infty}^{+\infty}
                     g(\tau)e^{-\mu 2 \pi \nu (t'+\tau)}f(t')d\tau \dt'\\
                  &= \int\limits_{-\infty}^{+\infty}
                     g(\tau)e^{-2\mu \pi \nu \tau}\dtau
                     \int\limits_{-\infty}^{+\infty}
                     f(t')e^{-2\mu \pi \nu t'}\dt'\\
                  &= \qfourier{\mu}{g(t)}\qfourier{\mu}{f(t)}\\
                  &= \qfourier{\mu}{f(t)}\qfourier{\mu}{g(t)}
\end{aligned}
\label{Eqn:QFT3A}
\end{equation}
Thus, the definition used for the QFT here verifies the convolution
theorem in the considered case. This specific case will be of use in
our definition of the hyperanalytic signal.

\subsection{The quaternion Fourier transform of the Hilbert transform}

It is straightforward to verify that the quaternion Fourier transform
of a real signal $x(t)=\frac{1}{\pi t}$ is $-\mu\sign(\nu)$,
where $\mu$ is the axis of the transform.
Substituting $x(t)$ into \eqref{Eqn:QFT2}, we get:
\begin{equation*}
\qfourier{\mu}{\frac{1}{\pi t}}=\frac{1}{\pi}\int\limits_{-\infty}^{+\infty}\frac{e^{-\mu 2\pi\nu t}}{t}\,\dt
\end{equation*}
and this is clearly isomorphic to the classical complex case. The solution
in the classical case is $-i\sign(\nu)$, and hence in the quaternion case
must be as stated above.

It is also straightforward to see that, given an arbitrary real signal $y(t)$,
subject only to the constraint that its classical Hilbert transform
$\hilbert{y(t)}$ exists, then one can easily show that the classical Hilbert transform
of the signal may be obtained using a quaternion Fourier transform as
follows:
\begin{equation}
\label{eqn:hilbertbyqft}
\hilbert{y(t)} = \iqfourier{\mu}{-\mu\sign(\nu)Y_{\mu}(\nu)}
\end{equation}
where $Y_{\mu}(\nu)=\qfourier{\mu}{y(t)}$.
This result follows from the isomorphism between the quaternion and complex
Fourier transforms when operating on a real signal, and it may be seen to be
the result of a convolution between the signal $y(t)$ and the quaternion
Fourier transform of $x(t)=\frac{1}{\pi t}$.
Note that $\mu$ and $Y_{\mu}(\nu)$ commute as a consequence of $y(t)$ being
real.