# A note on some top local cohomology modules

Majid Eghbali  Islamic Azad University, branch Soofian, Soofian, Iran.
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathfrak{a}$ be an ideal of a $d$-dimensional commutative Noetherian
ring $R$. In this paper we give some information on some last non-zero local
cohomology modules known as top local cohomology modules in particular,
$H^{d-1}_{\mathfrak{a}}(R)$.

###### Key words and phrases:

Top local cohomology, Attached primes, One dimensional ideal

###### 2000 Mathematics Subject Classification:

13D45, 13C14.

##  1\. Introduction

Throughout this note, we assume $R$ is a commutative Noetherian ring and
$\mathfrak{a}$ is an ideal of $R$. Let $M$ be an $R$-module. For an integer
$i\in\mathbb{Z}$ let $H^{i}_{\mathfrak{a}}(M)$ denote the $i$-th local
cohomology module of $M$ with respect to $\mathfrak{a}$ as introduced by
Grothendieck (cf. [Gr] and [Br-Sh]). The first non-vanishing cohomological
degree of the local cohomology modules $H^{i}_{\mathfrak{a}}(R)$ is well
understood. It is the maximal length of a regular sequence on $R$ consisting
of elements of $\mathfrak{a}$. The last non-vanishing amount of local
cohomology modules, instead, is more mysterious. It is known as the
cohomological dimension of $\mathfrak{a}$ in $R$, denoted by
$\operatorname{cd}(\mathfrak{a},R)$. It is a well-known fact that

$\begin{array}[]{ll}\
\operatorname{grade}(\mathfrak{a},R)\leq\operatorname{cd}(\mathfrak{a},R)\leq\dim
R.\end{array}$

Let $c:=\operatorname{cd}(\mathfrak{a},R)$ then, $\mathfrak{a}$ can not be
generated by the elements fewer than $c$. Hartshorne-Lichtenbaum vanishing
Theorem gives a characterization for vanishing of $H^{d}_{\mathfrak{a}}(R)$
i.e. for the case $\operatorname{cd}(\mathfrak{a},R)\leq\dim R-1$. In [E-Sch,
Theorem 3.3] the author and Schenzel have expressed the isomorphism
$H^{d}_{\mathfrak{a}}(R)\cong H^{d}_{\mathfrak{m}\widehat{R}}(\widehat{R}/J)$
for a certain ideal $J$ of $\widehat{R}$. It indicates the properties of
$H^{d}_{\mathfrak{a}}(R)$ via $H^{d}_{\mathfrak{m}\widehat{R}}(\widehat{R})$,
which is more well-known.

Ogus [O, Corollary 2.11] in equicharacteristic $0$ and Peskine and Szpiro
[P-S, 5.5] in equicharacteristic $p>0$ generalized the vanishing Theorem of
Hartshorne [Ha] for the cohomological dimension of the complement of a
subvariety of projective space. Huneke and Lyubeznik [Hu-L, Theorem 2.9] gave
a new characteristic-free proof of it. They proved the following Theorem:

######  Theorem 1.1.

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$ containing a
field and let $\mathfrak{a}$ be an ideal of $R$. Then the followings are
equivalent:

  1. (1)

$H^{i}_{\mathfrak{a}}(R)=0$ for $i=d-1,d$.

  2. (2)

$\dim R/\mathfrak{a}\geq 2\ $ and $\
\operatorname{Spec}(R/\mathfrak{a})\setminus\\{\mathfrak{m}\\}$ is formally
geometrically connected.

In general the case $\operatorname{cd}(\mathfrak{a},R)\leq\dim R-2$ is more
complicated than the first one. On the other hand, $H^{d-1}_{\mathfrak{a}}(R)$
is more mysterious and its properties are not so much known. It provides
motivation to do research on this subject. The aim of the present paper is to
obtain more information about the structure of $H^{d-1}_{\mathfrak{a}}(R)$ to
shed light on the obscure aspects of this issue. The layout of it is as
follows. In Section $2$, we examine
$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$ the set of attached prime
ideals of $H^{d-1}_{\mathfrak{a}}(R)$. As a technical tool we use the concept
of colocalization introduced by Richardson in [R]. As the main result in
Section $2$ we get the following result (cf. Theorem 2.2 ):

######  Theorem 1.2.

Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d$. Let
$\mathfrak{a}$ be an ideal of $R$. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is
representable and $H^{d}_{\mathfrak{a}}(R)=0$. Then

  1. (1)

$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\cup\operatorname{Assh}(R)$.

  2. (2)

$\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.

In Section $3$, we focus on $H^{d-1}_{\mathfrak{a}}(R)$ with a one-dimensional
ideal $\mathfrak{a}$. Among the other results in Theorem 3.1 we show the
relation of $H^{d-1}_{\mathfrak{a}}(R)$ with some known modules of the form
$H^{i}_{\mathfrak{m}}(R)$ as a main result. In Section $4$ we consider
$H^{d-1}_{\mathfrak{a}}(R)$ with an arbitrary ideal $\mathfrak{a}$. We define
a certain ideal $J$ to get an epimorphism
$H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{m}}(R/J)$ (cf. 4.1).

Part of this work is the author’s doctoral dissertation at the MLU Halle-
Wittenberg. My thanks are due to my adviser, Professor Peter Schenzel who draw
my attention to the importance of $H^{d-1}_{\mathfrak{a}}(R)$, helpful
conversations during the preparation of this note and careful reading of the
first draft of my paper.

##  2\. Attached primes of $H^{d-1}_{\mathfrak{a}}(R)$

Throughout this section, let $(R,\mathfrak{m})$ be a local ring and $S$ a
multiplicative closed subset of $R$. Recently, A. S. Richardson [R] has
proposed the definition of colocalization of an $R$-module $M$ relative to $S$
as the $S^{-1}R$-module $S_{-1}M=D_{S^{-1}R}(S^{-1}D_{R}(M))$, where $D(-)$
stands for the Matlis duality functor. See also [E] for more information.

In the light of [R, Theorem 2.2], representable modules are preserved under
colocalization and in case $M$ is a representable module, we have

$\begin{array}[]{ll}\
\operatorname{Att}S_{-1}M=\\{S^{-1}\mathfrak{p}:\mathfrak{p}\in\operatorname{Att}M\text{
and }S\cap\mathfrak{p}=\emptyset\\},\ \ \ \ \ \ \ (\ast)\end{array}$

where the notation $\operatorname{Att}M$ is used to denote the set of attached
prime ideals of $M$. This section is based on the use of this property. We
examine the set of attached prime ideals of last non-vanishing value of local
cohomology.

######  Proposition 2.1.

Let $(R,\mathfrak{m})$ be a local ring, $\mathfrak{a}$ be an ideal of $R$ and
$\mathfrak{p}\in\operatorname{Spec}R$. Let $c$ be an integer such that
$H^{i}_{\mathfrak{a}}(R)=0$ for every $i>c$. Assume that
$H^{c}_{\mathfrak{a}}(R)$ is representable. Then, we have the followings

  1. (1)

$\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{c}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{q}R_{\mathfrak{p}}:\dim
R/\mathfrak{q}\geq c,\mathfrak{q}\subseteq\mathfrak{p}\text{ and
}\mathfrak{q}\in\operatorname{Spec}R\\}$.

  2. (2)

In case $R$ is complete, we get the following equality

$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{\dim
R}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\dim R/\mathfrak{q}=\dim
R\text{ ,
}\mathfrak{q}\subseteq\mathfrak{p},\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}\text{
and }\mathfrak{q}\in\operatorname{Spec}R\\}.\end{array}$

Proof.

  1. (1)

By virtue of $(\ast)$ we have

$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{c}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\mathfrak{q}\in\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R))\text{
and }\mathfrak{q}\subseteq\mathfrak{p}\\}.\end{array}$

Since $H^{c}_{\mathfrak{a}}(R/\mathfrak{q})$ is representable then,

$\begin{array}[]{ll}\
\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R/\mathfrak{q}))=\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R))\cap\operatorname{Supp}(R/\mathfrak{q})\end{array}$

(cf. [A-M, Lemma 2.11]), which contains $\mathfrak{q}$. Hence,
$\operatorname{Att}_{R}(H^{c}_{\mathfrak{a}}(R/\mathfrak{q}))\neq\emptyset$ it
implies that $H^{c}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$ and consequently
$\dim R/\mathfrak{q}\geq c$.

  2. (2)

Put $d:=\dim R$. If $H^{d}_{\mathfrak{a}}(R)=0$ we are done. Then we assume
that $H^{d}_{\mathfrak{a}}(R)\neq 0$.

$"\subseteq"$: Let
$\mathfrak{q}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d}_{\mathfrak{a}}(R))$.
As we have seen in part one, $H^{d}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$ so
$\dim R/\mathfrak{q}=d$ and by Hartshorne-Lichtenbaum vanishing Theorem
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$.

$"\supseteq"$: again using $(\ast)$ we have

$\begin{array}[]{ll}\
\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d}_{\mathfrak{a}}(R))=\\{\mathfrak{q}R_{\mathfrak{p}}:\mathfrak{q}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R),\mathfrak{q}\subseteq\mathfrak{p}\\},\end{array}$

so it is enough to show that
$\mathfrak{q}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$.

As $\dim R/\mathfrak{q}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$, so Independence
Theorem implies that $H^{d}_{\mathfrak{a}}(R/\mathfrak{q})\neq 0$. Hence
easily one can see that

$\begin{array}[]{ll}\
\emptyset\neq\operatorname{Att}_{R}(H^{d}_{\mathfrak{a}}(R/\mathfrak{q}))=\operatorname{Att}_{R}(H^{d}_{\mathfrak{a}}(R))\cap\operatorname{Supp}_{R}(R/\mathfrak{q}).\
\ \ \ \ \ (\ast\ast)\end{array}$

In the contrary assume that
$\mathfrak{q}\notin\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$. Then by
virtue of $(\ast\ast)$ there exists a prime ideal
$\mathfrak{q}_{0}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$ such that
$\mathfrak{q}_{0}\supset\mathfrak{q}$ and so $\dim R/\mathfrak{q}_{0}<d$. On
the other $\mathfrak{q}_{0}\in\operatorname{Att}_{R}H^{d}_{\mathfrak{a}}(R)$
if and only if
$\mathfrak{q}_{0}R_{\mathfrak{q}_{0}}\in\operatorname{Att}_{R_{\mathfrak{q}_{0}}}(^{\mathfrak{q}_{0}}H^{d}_{\mathfrak{a}}(R))$.
By virtue of part one $\dim R/\mathfrak{q}_{0}=d$ which is a contradiction.
Now the proof is complete.

$\ \ \ \ \ \ \ \Box$

After this preparation we can state the main result of this section as
follows:

######  Theorem 2.2.

Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d$. Let
$\mathfrak{a}$ be an ideal of $R$. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is
representable and $H^{d}_{\mathfrak{a}}(R)=0$. Then

  1. (1)

$\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))\subseteq\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\cup\operatorname{Assh}(R)$.

  2. (2)

$\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1,\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.

Proof.

  1. (1)

Let $\mathfrak{p}\in\operatorname{Att}_{R}H^{d-1}_{\mathfrak{a}}(R)$, then
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d-1}_{\mathfrak{a}}(R))$.
Hence by Theorem 2.1 $\dim R/\mathfrak{p}\geq d-1$.

When $\dim R/\mathfrak{p}=d$ it follows that
$\mathfrak{p}\in\operatorname{Assh}(R)$. In the case $\dim
R/\mathfrak{p}=d-1$, as
$\mathfrak{p}\in\operatorname{Att}_{R}H^{d-1}_{\mathfrak{a}}(R)$ and
$H^{d-1}_{\mathfrak{a}}(-)$ is a right exact functor so one can deduce that
$H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})\neq 0$. By Hartshorne-Lichtenbaum
vanishing Theorem there exists a prime ideal
$\mathfrak{q}\supseteq\mathfrak{p}$ of $R$ of dimension $d-1$ with
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q})=\mathfrak{m}$. So, we must have
$\mathfrak{q}=\mathfrak{p}$.

  2. (2)

Let $\dim R/\mathfrak{p}=d-1$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}$, then Theorem
2.1(2) implies that
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Att}_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p}))$.
By $(\ast)$ we deduce that
$\mathfrak{p}\in\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p}))$.
On the other hand the epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R/\mathfrak{p})\rightarrow 0\end{array}$

implies that
$\mathfrak{p}\in\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R))$.

$\ \ \ \ \ \ \ \Box$

It is noteworthy to say that in the situation of Theorem 2.2 if $\mathfrak{a}$
is an ideal of dimension one, the inclusion at (1) will be an equality, see
[Hel, Theorem 8.2.3].

##  3\. One dimensional ideal

Of particular interest are the so-called top local cohomology modules, that
is, $H^{t}_{\mathfrak{a}}(R)\neq 0$ where $H^{i}_{\mathfrak{a}}(R)=0$ for all
$i>t$. In the light of Theorem 1.1, vanishing of the local cohomology modules
$H^{i}_{\mathfrak{a}}(R)$, for $i=\dim R,\dim R-1$ paving the ground for
connectedness results. $H^{\dim R}_{\mathfrak{a}}(R)$ has been studied in
[E-Sch] and some of its properties have been realized. As
$H^{d-1}_{\mathfrak{a}}(R)$ is mostly not Artinian so it rarely could be
possible to control $H^{d-1}_{\mathfrak{a}}(R)$ via
$H^{d-1}_{\mathfrak{m}}(R)$, however in the next Theorem we try to examine the
$H^{d-1}_{\mathfrak{a}}(R)$ via some known modules of the form
$H^{i}_{\mathfrak{m}}(R)$ for a one-dimensional ideal $\mathfrak{a}$ .

######  Theorem 3.1.

Let $\mathfrak{a}$ be a one dimensional ideal of a $d$-dimensional local ring
$(R,\mathfrak{m})$. Suppose $\operatorname{Rad}(\mathfrak{a})$ is not a prime
ideal. Then

  * (1)

Let $\widehat{R}$ the $\mathfrak{m}$-adic completion of $R$ be an integral
domain, then there exists an epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$

Moreover, in case $R$ is complete and unmixed, we have
$\operatorname{Ann}_{R}H^{d-1}_{\mathfrak{a}}(R)=0$.

  * (2)

If $R$ is a Gorenstein ring and $\widehat{R}$ is a domain then, there exist a
prime ideal $\mathfrak{p}$ and an element $y\in R\setminus\mathfrak{p}$ to
have the following exact sequence

$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow
E_{R_{\mathfrak{p}}}(k(\mathfrak{p}))\rightarrow 0.\end{array}$

  * (3)

If $H^{d}_{\mathfrak{a}}(R)=0$ then, for some element $x\in R$ with
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ we get the following
epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/xR)\rightarrow 0.\end{array}$

If $\mathfrak{a}=\mathfrak{p}$ is a prime ideal and $R$ is Cohen-Macaulay ring
then, there exists a short exact sequence

$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{p}}(R)\rightarrow
H^{d-1}_{\mathfrak{p}}(R)_{y}\rightarrow H^{d}_{\mathfrak{m}}(R)\rightarrow
0,\end{array}$

for an element $y\in\mathfrak{m}\setminus\mathfrak{p}$. Moreover
$H^{0}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=0$ and
$H^{1}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=H^{d}_{\mathfrak{m}}(R)$. In
particular, $H^{d-1}_{\mathfrak{p}}(R)$ is not Artinian.

Proof.

  * (1)

Put

$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\
\ \ \ n\geq 2\end{array}$

where $\mathfrak{p}_{i}$’s are distinct minimal prime ideals of $\mathfrak{a}$
for $i=1,...,n$. Therefore, $\mathfrak{p}_{i}$’s are not
$\mathfrak{m}$-primary. Choose $\mathfrak{p}_{t}$ for some integer
$t\in\\{1,...,n\\}$ such that
$\operatorname{Rad}(\mathfrak{p}_{t}+\cap^{n}_{j=1,j\neq
t}\mathfrak{p}_{j})=\mathfrak{m}$.

Put $\mathfrak{a}_{1}:=\mathfrak{p}_{t},\mathfrak{a}_{2}:=\cap^{n}_{j=1,j\neq
t}\mathfrak{p}_{j}$, by Mayer-Vietoris sequence we have the following exact
sequence

$\begin{array}[]{ll}\
H^{d-1}_{\mathfrak{a}_{1}\cap\mathfrak{a}_{2}}(R)\rightarrow
H^{d}_{\mathfrak{a}_{1}+\mathfrak{a}_{2}}(R)\rightarrow
H^{d}_{\mathfrak{a}_{1}}(R)\oplus H^{d}_{\mathfrak{a}_{2}}(R).\end{array}$

Hartshorne-Lichtenbaum vanishing Theorem implies that
$H^{d}_{\mathfrak{a}_{1}}(R)=0=H^{d}_{\mathfrak{a}_{2}}(R)$ then, we get the
following epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$

In the light of the above epimorphism and by the assumptions, it follows that
$\operatorname{Ann}_{R}H^{d-1}_{\mathfrak{a}}(R)=0$. To this end note that
$\operatorname{Ann}_{R}H^{d}_{\mathfrak{m}}(R)=0$ (see for example [E-Sch,
Theorem 4.2(i)]).

  * (2)

In this part we follow the lines of an argument given in [Hu-K, Lemma 4.7].
Let

$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\end{array}$

where $\mathfrak{p}_{i}$’s are distinct minimal prime ideals of $\mathfrak{a}$
for $i=1,...,n$. Choose $y\in\cap^{n}_{i=1,i\neq
j}\mathfrak{p}_{i}\setminus\mathfrak{p}_{j}$ for some integer
$j\in\\{1,...,n\\}$.

Mayer-Vietoris sequence implies the following long exact

$\begin{array}[]{ll}\ (H^{d-2}_{\mathfrak{a}}(R))_{y}\rightarrow
H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow(H^{d-1}_{\mathfrak{a}}(R))_{y}\rightarrow
H^{d}_{\mathfrak{a}+yR}(R).\end{array}$

Note that

$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a}+yR)=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{j}^{\wedge}\cap...\cap\mathfrak{p}_{n},\end{array}$

by $\wedge$ we mean the omitting the $j$th component and
$\operatorname{Rad}(\mathfrak{a}R_{y})={\mathfrak{p}}_{j}R_{y}$ which is
maximal ideal of $R_{y}$. Hence,

$\begin{array}[]{ll}\
(H^{d-2}_{\mathfrak{a}}(R))_{y}=H^{d-2}_{\mathfrak{a}R_{y}}(R_{y})\cong
H^{d-2}_{\mathfrak{p}_{j}R_{y}}(R_{y})=0.\end{array}$

To this end note that $\dim
R_{y}=\operatorname{ht}\mathfrak{p}_{j}R_{y}=\operatorname{ht}\mathfrak{p}_{j}=d-1$
and $R_{y}$ is a Gorenstein ring.

As $\widehat{R}$ is a domain and
$\operatorname{Rad}(\mathfrak{a}+yR)\neq\mathfrak{m}$ then,
$H^{d}_{\mathfrak{a}+yR}(R)=0$ using Hartshorne-Lichtenbaum vanishing Theorem.
Finally we have the following isomorphism

$\begin{array}[]{ll}\
(H^{d-1}_{\mathfrak{a}}(R))_{y}=(H^{d-1}_{\mathfrak{a}}(R))_{y}\otimes_{R_{y}}R_{\mathfrak{p}_{j}}\cong
E_{R_{\mathfrak{p}_{j}}}(k(\mathfrak{p}_{j})).\end{array}$

It is worthy to note that $R_{\mathfrak{p}_{j}}\cong R_{y}$ (cf. [At-Mac,
Exercise 8, pp.44]), $R_{\mathfrak{p}_{j}}$ is a Gorenstein local ring and
$\operatorname{Rad}(\mathfrak{a}R_{\mathfrak{p}_{j}})=\operatorname{Rad}(\mathfrak{a})R_{\mathfrak{p}_{j}}=\mathfrak{p}_{j}R_{\mathfrak{p}_{j}}$.
Hence, we get the desired short exact sequence

$\begin{array}[]{ll}\ 0\rightarrow H^{d-1}_{\mathfrak{a}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow
E_{R_{\mathfrak{p}_{j}}}(k(\mathfrak{p}_{j}))\rightarrow 0.\end{array}$

  * (3)

Put

$\begin{array}[]{ll}\
\operatorname{Rad}(\mathfrak{a})=\mathfrak{p}_{1}\cap...\cap\mathfrak{p}_{n},\end{array}$

Choose $x\in\mathfrak{p}_{t}$ for some integer $t\in\\{1,...,n\\}$ with
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ to get the following short
exact sequence

$\begin{array}[]{ll}\ 0\rightarrow R/(0:_{R}x)\stackrel{{\scriptstyle
x}}{{\rightarrow}}R\rightarrow R/xR\rightarrow 0.\end{array}$

It yields the long exact sequence

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R/(0:_{R}x))\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{a}}(R/xR)\rightarrow
H^{d}_{\mathfrak{a}}(R/(0:_{R}x)),\end{array}$

where $H^{d}_{\mathfrak{a}}(R/(0:_{R}x))$ is zero by assumption. So, by
Independence Theorem we get the epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/xR)\rightarrow 0.\end{array}$

For the last part, Choose $y\in\mathfrak{m}\setminus\mathfrak{p}$. So,
$\operatorname{Rad}(\mathfrak{p}+yR)=\mathfrak{m}$. Then, by Mayer-Vietoris
sequence we have

$\begin{array}[]{ll}\ (H^{d-2}_{\mathfrak{p}}(R))_{y}\rightarrow
H^{d-1}_{\mathfrak{p}+yR}(R)\rightarrow
H^{d-1}_{\mathfrak{p}}(R)\rightarrow(H^{d-1}_{\mathfrak{p}}(R))_{y}.\end{array}$

As $R$ is Cohen-Macaulay then, it yields the next short exact sequence

$\begin{array}[]{ll}\ 0\rightarrow
H^{d-1}_{\mathfrak{p}}(R)\rightarrow(H^{d-1}_{\mathfrak{p}}(R))_{y}\rightarrow
H^{d}_{\mathfrak{m}}(R)\rightarrow 0.\end{array}$

By virtue of [Br-Sh, Corollary 2.2.18] $\
H^{0}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=0$ and
$H^{1}_{yR}(H^{d-1}_{\mathfrak{p}}(R))=H^{d}_{\mathfrak{m}}(R)$. Note that in
case $H^{d-1}_{\mathfrak{p}}(R)$ is Artinian, it follows that
$H^{d}_{\mathfrak{m}}(R)=0$ which is a contradiction. $\ \ \ \ \ \ \ \Box$

######  Remark 3.2.

  * (1)

Keep the notations and hypotheses in Theorem 3.1(3), with some additional
assumptions we can remove the assumption $H^{d}_{\mathfrak{a}}(R)=0$. Let $x$
be in some minimal prime ideals of $\mathfrak{p}_{t}$ such that
$\operatorname{Rad}(\mathfrak{a}+x)=\mathfrak{m}$ then, we show that
$H^{d}_{\mathfrak{a}}(R/(0:_{R}x))=0$. Note that if $x\in\mathfrak{p}$ for all
$\mathfrak{p}\in\min R$ then, $\operatorname{Rad}(\mathfrak{a})=\mathfrak{m}$
which is a contradiction.

Let $(0)=\mathfrak{q}_{1}\cap...\cap\mathfrak{q}_{n}$ be a minimal primary
decomposition and $(x)=\mathfrak{q}_{1}\cap...\cap\mathfrak{q}_{r}$, $r<n$
such that $\dim R/\mathfrak{q}_{i}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q}_{i})=\mathfrak{m}$ for all
$i\in\\{1,...,r\\}$ but this is not true for all $\mathfrak{q}_{i}$ where
$r<i\leq n$.

If $H^{d}_{\mathfrak{a}}(R/(0:_{R}x))\neq 0$ then, there exists a prime ideal
$\mathfrak{P}$ containing $(0:_{R}x)$ such that $\dim R/\mathfrak{P}=d$ and
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{P})=\mathfrak{m}$ by Hartshorne-
Lichtenbaum vanishing Theorem. Since $\mathfrak{P}\supseteq(0:_{R}x)$ so
$\mathfrak{P}\supseteq\mathfrak{q}_{i}$ for some $i\geq r+1$. Therefore $\dim
R/\mathfrak{q}_{i}=d$ and $\mathfrak{P}=\operatorname{Rad}(\mathfrak{q}_{i})$.
This implies that
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{q}_{i})=\operatorname{Rad}(\mathfrak{a}+\mathfrak{P})=\mathfrak{m}$,
contracting the fact that $i\geq r+1$.

  * (2)

Let $\mathfrak{a}$ be a one dimensional ideal of a $d$-dimensional local ring
$(R,\mathfrak{m})$. Suppose $\operatorname{Rad}(\mathfrak{a})$ is not a prime
ideal. If for some element $x$ in a minimal prime ideal of $R$,
$\operatorname{Rad}(\mathfrak{a}+xR)=\mathfrak{m}$ then, by what we have seen
in (1) and Theorem 3.1(3) we get

$\begin{array}[]{ll}\
\operatorname{Assh}(R/xR)\subseteq\operatorname{Att}_{R}(H^{d-1}_{\mathfrak{a}}(R)).\end{array}$

$\ \ \ \ \ \ \ \Box$

It follows by Hartshorne’s result, (cf. [Ha, p. 417]) that
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},N)=0$
for a finitely generated $R$-module $N$ of dimension $s$. It is natural to
investigate the vanishing case for
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},N)$.
To be more precise we have the following result:

######  Proposition 3.3.

Let $(R,\mathfrak{m})$ be a local ring and $M$ a finite $R$-module of
dimension $s$ and $\mathfrak{a}$ an ideal of dimension one. Then,

  * (1)

There exists an epimorphism
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))$.

  * (2)

If $H^{s-1}_{\mathfrak{a}}(M)=0$ then, it implies that
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)=0$.

Proof. As $\dim R/\mathfrak{a}=1$ then, there exists an element
$x\in\mathfrak{m}\setminus\mathfrak{a}$. By virtue of [Sch, Corollary 1.4]
there is the following exact sequence

$\begin{array}[]{ll}\ 0\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))\rightarrow
H^{s}_{(\mathfrak{a},xR)}(M)\rightarrow
H^{0}_{xR}(H^{s}_{\mathfrak{a}}(M))\rightarrow 0.\end{array}$

As $H^{s}_{\mathfrak{a}}(M)$ is an Artinian module so every element of
$H^{s}_{\mathfrak{a}}(M)$ is annihilated by some power of $\mathfrak{m}$ and
also by some power of $x$, i.e.
$H^{0}_{xR}(H^{s}_{\mathfrak{a}}(M))=H^{s}_{\mathfrak{a}}(M)$.

On the other hand there is the long exact sequence

$\begin{array}[]{ll}\
{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{s}_{\mathfrak{m}}(M)\rightarrow
H^{s}_{\mathfrak{a}}(M)\rightarrow{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M),\end{array}$

where
${\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)=0$
as it has been mentioned above.

Now combine the above results to get the following commutative diagram with
exact rows

$\begin{array}[]{cccccccc}H^{s-1}_{\mathfrak{a}}(M)&\rightarrow&{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)&\rightarrow
H^{s}_{\mathfrak{m}}(M)&\rightarrow&H^{s}_{\mathfrak{a}}(M)&\rightarrow&0\\\
&&&\ \ \ \ \ \ \ \downarrow=&&\ \ \ \ \downarrow=&\\\
0&\rightarrow&H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))&\rightarrow
H^{s}_{\mathfrak{m}}(M)&\rightarrow&H^{s}_{\mathfrak{a}}(M)&\rightarrow&0\end{array}$

By snake lemma we get the epimorphism
$\phi:{\operatornamewithlimits{\varinjlim}}_{n}\operatorname{Ext}^{s-1}_{R}(\mathfrak{m}^{n}/\mathfrak{a}^{n},M)\rightarrow
H^{1}_{xR}(H^{s-1}_{\mathfrak{a}}(M))$. It is clear that $\phi$ is isomorphism
when $H^{s-1}_{\mathfrak{a}}(M)=0$, this proves the claim.

$\ \ \ \ \ \ \ \Box$

##  4\. A general case

In this Section we try to examine the structure of $H^{d-1}_{\mathfrak{a}}(R)$
for an ideal $\mathfrak{a}$ which is not in general of dimension one. For this
reason we consider the following case:

Let $(R,\mathfrak{m})$ be a local ring. Assume that

$\begin{array}[]{ll}\
\Sigma:=\\{\mathfrak{p}\in\operatorname{Spec}\widehat{R}:\dim\widehat{R}/\mathfrak{p}=d-1\text{
and
}\operatorname{Rad}(\mathfrak{a}\widehat{R}+\mathfrak{p})=\mathfrak{m}\widehat{R}\\}\end{array}$

is non-empty. Here $\widehat{R}$ is the $\mathfrak{m}$-adic completion of $R$.
When we have an ideal $\mathfrak{b}$ of $R$ of dimension $d-1$ such that
$H^{d-1}_{\mathfrak{a}}(R/{\mathfrak{b}})\neq 0$ then, by Hartshorne-
Lichtenbaum vanishing Theorem the above assumption will be verified.

######  Proposition 4.1.

Let $(R,\mathfrak{m})$ be a complete local ring of dimension $d>0$. Let
$\mathfrak{a}$ be an ideal of $R$ such that $H^{d}_{\mathfrak{a}}(R)=0$.
Suppose that $\Sigma:=\\{\mathfrak{p}\in\operatorname{Spec}R:\dim
R/\mathfrak{p}=d-1\text{ and
}\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})=\mathfrak{m}\\}\neq\emptyset$.
Then

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{m}}(R/J)\rightarrow 0\end{array}$

where $J=(\cap_{\mathfrak{p}_{i}\in\Sigma}\mathfrak{p}_{i})$ is an ideal of
$R$. ($J=R$, if for every $\mathfrak{p}\in\operatorname{Spec}R$ $\dim
R/\mathfrak{p}\neq d-1$ or
$\operatorname{Rad}(\mathfrak{a}+\mathfrak{p})\neq\mathfrak{m}$).

Proof. Consider the short exact sequence

$\begin{array}[]{ll}\ 0\rightarrow J\rightarrow R\rightarrow R/J\rightarrow
0\end{array}$

which implies the following long exact sequence

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(J)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{a}}(R/J)\rightarrow
H^{d}_{\mathfrak{a}}(J).\end{array}$

As the last module in the above exact sequence is zero so, we get the
following epimorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R/J)\rightarrow 0.\end{array}$

By virtue of Independence Theorem we get the claim:

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)&\rightarrow
H^{d-1}_{\mathfrak{a}}(R/J)\\\ &\cong H^{d-1}_{\mathfrak{a}+J/J}(R/J)\\\
&\cong H^{d-1}_{\mathfrak{m}}(R/J).\end{array}$

To this end note that $\operatorname{Rad}(\mathfrak{a}+J)=\mathfrak{m}$ $\ \ \
\ \ \ \ \Box$

Let $M$ denote a $s$-dimensional, finitely generated $R$-module. Here
$(R,\mathfrak{m})$ is a local ring. The functor $\widehat{\cdot}$ denotes the
completion functor.

For an $R$-module $M$ let $0=\cap_{i=1}^{n}Q_{i}(M)$ denote a minimal primary
decomposition of the zero submodule of $M.$ That is $M/Q_{i}(M),i=1,\ldots,n,$
is a $\mathfrak{p}_{i}$-primary $R$-module. Clearly
$\operatorname{Ass}_{R}M=\\{\mathfrak{p}_{1},\ldots,\mathfrak{p}_{n}\\}.$ It
is defined in [E-Sch] that $Q_{\mathfrak{a}}(M)=\cap_{\mathfrak{p}_{i}\in
U}Q_{i}(M)$, where $U=\\{\mathfrak{p}\in\operatorname{Ass}_{R}M|\dim
R/\mathfrak{p}=s\text{ and }\dim R/\mathfrak{a}+\mathfrak{p}=0\\}$ and
$P_{\mathfrak{a}}(M)$ as the intersection of all the primary components of
$\operatorname{Ann}_{R}M$ such that $\dim R/\mathfrak{p}=\dim M$ and $\dim
R/\mathfrak{a}+\mathfrak{p}=0$. Clearly $P_{\mathfrak{a}}(M)$ is the pre-image
of $Q_{\mathfrak{a}R/\operatorname{Ann}_{R}M}(R/\operatorname{Ann}_{R}M)$ in
$R$. It has been shown in [E-Sch, Corollary 3.5] that when $(R,\mathfrak{m})$
is a local ring there is a natural isomorphism

$H^{s}_{\mathfrak{a}}(M)\simeq
H^{s}_{\mathfrak{m}\widehat{R}}(\widehat{M})/P_{\mathfrak{a}}(\widehat{M})H^{s}_{\mathfrak{m}\widehat{R}}(\widehat{M}),$

where $P_{\mathfrak{a}}(\widehat{M})\subset\widehat{R}$ is the ideal.

######  Proposition 4.2.

Let $(R,\mathfrak{m})$ be a local ring of dimension $d>0$. Let $\mathfrak{a}$
be an ideal of $R$ such that $H^{d}_{\mathfrak{a}}(R)=0$. Let
$J=\cap_{\mathfrak{p}_{i}\in\Sigma}\mathfrak{p}_{i}$ be as above such that
$\Sigma$ is finite. Assume that $H^{d-1}_{\mathfrak{a}}(R)$ is Artinian. Then,

$\begin{array}[]{ll}\
H^{d-1}_{\mathfrak{a}}(R)/JH^{d-1}_{\mathfrak{a}}(R)\cong
H^{d-1}_{\mathfrak{m}\widehat{R}}(S)/KH^{d-1}_{\mathfrak{m}\widehat{R}}(S),\end{array}$

where $K:=P_{\mathfrak{a}}(\widehat{R}/J\widehat{R})\subset
S:=\widehat{R}/J\widehat{R}$ is the ideal.

Proof. First note that as $H^{d-1}_{\mathfrak{a}}(R)$ is an Artinian
$R$-module so, it admits a unique $\widehat{R}$-module structure. Then,
without loss of generality we may assume that $R$ is complete. It is clear
that $\dim R/J=d-1$ so, by assumption and Grothendieck’s non-vanishing Theorem
one can see that $H^{d-1}_{\mathfrak{a}}(R/J)\neq 0$. Hence, by the above
explanations there exists a natural isomorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R/J)\cong
H^{d-1}_{\mathfrak{m}}(R/(P_{\mathfrak{a}}(R/J),J)).\end{array}$

Now the claim is clear as $H^{d}_{\mathfrak{a}}(R)=0$. $\ \ \ \ \ \ \ \Box$

######  Remark 4.3.

Assume that $0=\cap^{n}_{i=1}q_{i}$ is a minimal primary decomposition of the
zero ideal of a ring $R$ with dimension $d\geq 3$, where at least one of
$q_{i}$ for $1\leq i\leq n$ is of dimension one. Let $\mathfrak{a}$ be an
ideal of $R$ such that $\operatorname{Spec}R\setminus V(\mathfrak{a})$ is
disconnected. Then, there are ideals $\mathfrak{b}$ and $\mathfrak{c}$ such
that $\operatorname{Rad}(\mathfrak{b})\neq
0\neq\operatorname{Rad}(\mathfrak{c})$,
$\operatorname{Rad}(\mathfrak{b}\cap\mathfrak{c})=\operatorname{Rad}(0)$ and
$\operatorname{Rad}(\mathfrak{b}+\mathfrak{c})=\operatorname{Rad}(\mathfrak{a})$.
If we choose one of them to be one-dimensional component of $0$, say
$\mathfrak{b}$ then, Mayer-Vietoris sequence implies the following long exact
sequence

$\begin{array}[]{ll}\ H^{d-2}_{\mathfrak{b}\cap\mathfrak{c}}(R)\rightarrow
H^{d-1}_{\mathfrak{a}}(R)\rightarrow H^{d-1}_{\mathfrak{b}}(R)\oplus
H^{d-1}_{\mathfrak{c}}(R)\rightarrow
H^{d-1}_{\mathfrak{b}\cap\mathfrak{c}}(R),\end{array}$

where
$H^{d-2}_{\mathfrak{b}\cap\mathfrak{c}}(R)=0=H^{d-1}_{\mathfrak{b}\cap\mathfrak{c}}(R)$.
Hence, we get the following isomorphism

$\begin{array}[]{ll}\ H^{d-1}_{\mathfrak{a}}(R)\cong
H^{d-1}_{\mathfrak{b}}(R)\oplus H^{d-1}_{\mathfrak{c}}(R).\end{array}$

Moreover, in case $R$ is a Cohen-Macaulay ring $H^{d-1}_{\mathfrak{a}}(R)$ is
not finitely generated because, $H^{d-1}_{\mathfrak{b}}(R)$ is not finitely
generated.

######  Problem 4.4.

Let $(R,\mathfrak{m})$ be a local ring and $\mathfrak{a}$ be an ideal of $R$.
By virtue of [E, Theorem 1.1] in the case $0ptR\geq 2$, $\dim R=d$, we have
the following equivalent statements of formal local cohomology modules:

$\begin{array}[]{ll}\
{\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})=0={\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\end{array}$

if and only if

$\begin{array}[]{ll}\
{\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\text{\
and \
}{\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})\end{array}$

are Artinian.

When $R$ is a Gorenstein ring the above statements are equivalent to
$H^{d}_{\mathfrak{a}}(R)=0=H^{d-1}_{\mathfrak{a}}(R)$ for $d\geq 2$. Now it
implies to find more precise conditions for Artinianness of
${\operatornamewithlimits{\varprojlim}}_{n}H^{0}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$
and
${\operatornamewithlimits{\varprojlim}}_{n}H^{1}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$.

It should be noted that if
$\operatorname{Coass}({\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(R/\mathfrak{a}^{n}))\not\subset
V(\mathfrak{a})$ then,
${\operatornamewithlimits{\varprojlim}}_{n}H^{i}_{\mathfrak{m}}(R/\mathfrak{a}^{n})$
is not Artinian (cf. [E, Theorem 1.4 and Corollary 3.5]).

For more information on formal local cohomology modules see [Sch2].

## References

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  * [Br-Sh] M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Univ. Press, 60, Cambridge, (1998).
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  * [E-Sch] M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Comm. Algebra, 40: 4295-4305, 2012.
  * [Gr] A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lect. Notes in Math., 20, Springer, 1966.
  * [Ha] R. Hartshorne, Cohomological dimension of algebraic varieties, Annals of Math. 88 (1968), 403-450.
  * [Hel] M. Hellus, Local Cohomology and Matlis Duality, Habilitationsschrift, Leipzig, (2006).
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  * [Hu-L] C. Huneke and G. Lyubeznik, On the vanishing of local cohomology modules, Invent. Math. 102, 73-93(1990).
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  * [Sch2] P. Schenzel, On formal local cohomology and connectedness, J. Algebra, 315(2), (2007), 894-923.


# Py3DFreeHandUS: a library for voxel-array reconstruction using
Ultrasonography and attitude sensors

Davide Monari∗†, Francesco Cenni†, Erwin Aertbeliën†, Kaat Desloovere† *
Corresponding author<EMAIL_ADDRESS>KULeuvenCopyright © 2014
Davide Monari et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original
author and source are credited. http://creativecommons.org/licenses/by/3.0/

###### Abstract

In medical imaging, there is a growing interest to provide real-time images
with good quality for large anatomical structures. To cope with this issue, we
developed a library that allows to replace, for some specific clinical
applications, more robust systems such as Computer Tomography (CT) and
Magnetic Resonance Imaging (MRI). Our python library _Py3DFreeHandUS_ is a
package for processing data acquired simultaneously by ultra-sonographic
systems (US) and marker-based optoelectronic systems. In particular, US data
enables to visualize subcutaneous body structures, whereas the optoelectronic
system is able to collect the 3D position in space for reflective objects,
that are called markers. By combining these two measurement devices, it is
possible to reconstruct the real 3D morphology of body structures such as
muscles, for relevant clinical implications. In the present research work, the
different steps which allow to obtain a relevant 3D data set as well as the
procedures for calibrating the systems and for determining the quality of the
reconstruction.

###### Index Terms:

medical imaging, free-hand ultrasonography, optoelectronic systems,
compounding

##  1 Introduction

In medical imaging, 3D data sets are an essential tool to explore anatomical
volumes and to extract clinical features, which can describe a particular
condition of the patient. These data are usually recorded by CT or MRI for
identifying hard or soft tissue, respectively, and provide a high image
quality together with a large field of view. On the other hand, these systems
are _very_ expensive (espacially MRI ones) and time consuming both for
operators and patients. Plus, radioations from CT are an issue. Therefore, for
some clinical applications, it could be interesting to replace these systems
with others that can allow to provide 3D data sets quickly, although without
the same high image quality. Ultrasonography (US) devices are systems largely
used to collect medical images. For example, it is very common to examine
pregnant women. This system, compared to other medical imaging systems, has
several advantages: real-time images, portability, no ionizing radiation.
However, one of the major drawbacks in US is the limited field of view and the
lack of spatial information among different images acquired. Therefore a
technique called 3D Freehand Ultrasound (3DUS) was originally proposed in the
90s [Rankin93], [Prager99] with the aim of reconstructing large 3D anatomical
parts. The idea is to combine US images and the corresponding position and
orientation (POS) of the US transducer; by simultaneously scanning a series of
2D images and recording spatial information it is possible to perform the
relevant reconstruction and then the visualization of the entire volume
acquired. The aim of the present work is to customize the 3DUS implementation
by pushing on vectorization in NumPy / SciPy along with memory waste
avoidance, for speeding up the processing phase as much as possible. These
aspects are essential in this context, since for commodity hardware: i) memory
resources are relatively limited and 3D volumes involved here can quickly
reach large dimensions, ii) computation time can become unrealistic if very
large for- or while- loops are used in Python. In addition the few existing
applications for applying this technique have at least one of the following
disadvantages: i) not open-source; ii) only supporting data streams from a
limited number of US/POS sensors; iii) they are written in low-level languages
such as C++, making rapid development and prototyping more difficult. We
developed a pure Python library called Py3DFreeHandUS that solves all the
above issues.

##  2 Requirements

Py3DFreeHandUS was developed in Python 2.7 (Python 3 not yet supported), and
uses the following libraries:

  * •

NumPy

  * •

SciPy (0.11.0+)

  * •

matplotlib

  * •

SymPy

  * •

pydicom

  * •

b-tk (Biomechanical ToolKit) [Barre14]

  * •

VTK

  * •

OpenCV (2.4.9+)

  * •

Cython + gcc (optional, we are "cythonizing" bottlenecks but leaving pure
Python implementation available)

We used the Python distribution _Python(x,y)_ for development and testing,
since it already includes all libraries but b-tk.

##  3 Description of the package

The present package is able to process synchronized data by US and POS, being
as input DICOM and C3D files, respectively. The operations flowchart is
composed by: US probe _temporal_ and _spatial calibration_ and _3D voxel array
reconstruction_.

###  3.1 US probe temporal calibration

The aim of the temporal calibration is to estimate the time delay between US
and the POS devices. This procedure is foundamental whenever it is not
possible to hardware-trigger US and POS devices, so data needs to be time-
shifted later. Time delay resolution cannot be lower than the inverse of the
lower frequency (normally US). Briefly, we moved vertically up and down the US
probe (rigidly connected to the POS sensor) in a water-filled tank and
generated two curves: the first one being the vertical coordinate of the the
POS sensor, the second one being the vertical coordinate (in the US image) of
the center of the line representing the edge between water and tank. These two
sine-like signal were demeaned, normalized to be inside the range [-1,+1] and
cross-correlated with the function matplotlib.pyplot.xcorr. The time of the
first peak for the cross-correlation estimates the time delay.

###  3.2 US probe spatial calibration

The probe spatial calibration is an essential procedure for image
reconstruction which allows to determine the _pose_ (position and orientation)
of the US images with respect to the POS device. The corresponding results
take the form of six parameters, three for position and three for orientation.
The quality of this step mainly influences the reconstruction quality of the
anatomical shape. To perform the probe calibration we used two different
steps. First we applied an established procedure already published in the
literature [Prager98] and later we tuned the results by using an image
compounding algorithm [Wein08]. The established procedure was proposed by
Prager et al. [Prager98] and improved by Hsu [Hsu06], with the idea of
scanning the floor of a water tank by covering all the degrees of freedom (see
Figure 1); this scanning modality produces clear and consistent edge lines
(between water and tank bottom) in the US images (B-scans). All the pixels
lying on the visible line in the B-scan should satisfy equations that come
from the different spatial transformations, which leave to solve 11
identifiable parameters. Each B-scan can be used to write 2 equations. The
overdetermined set of equations is solved using the Levenberg-Marquardt
algorithm. We found that it is essential to move the US transducers following
the sequence of movements suggested in [Prager98], in order to have reasonable
results. The equation that a pixel with image coordinates $(u,v)$ must satisfy
(see [Prager98] for details) is as follows:

  *

$\begin{pmatrix}0\\\ 0\\\ 0\\\ 1\end{pmatrix}=\ ^{C}T_{T}\ ^{T}T_{R}\
^{R}T_{P}\begin{pmatrix}s_{x}u\\\ s_{y}v\\\ 0\\\ 1\end{pmatrix}$,

  *

where $s_{x}$ and $s_{y}$ are conversion factors from _pixel_ to _mm_.

Figure 1: The aim of the US probe spatial calibration is to find the roto-
translation matrix ${}^{R}T_{P}$ from the image reference frame (P) to the
transducer reference frame (R). The other two roto-translation matrices
${}^{T}T_{R}$ and ${}^{C}T_{T}$ (respectively, from transducer to
optoelectronic system and from optoelectronic system to calibration phantom)
are known for every time frame of the calibration acquisition.

This is the code snippet for the equation creation:

  *

from sympy import Matrix, Symbol, varfrom sympy import cos as c, sin as s#
Ppsx = Symbol(’sx’)sy = Symbol(’sy’)u = Symbol(’u’)v = Symbol(’v’)Pp =
Matrix(([sx * u],\ [sy * v],\ [0],\ [1]\))# rTprTp, syms =
creatCalibMatrix()[x1, y1, z1, alpha1, beta1, gamma1] = syms# tTrtTr =
MatrixOfMatrixSymbol(’tTr’, 4, 4)tTr[3, 0:4] = np.array([0,0,0,1])# cTtx2 =
Symbol(’x2’)y2 = Symbol(’y2’)z2 = Symbol(’z2’)alpha2 = Symbol(’alpha2’)beta2 =
Symbol(’beta2’)gamma2 = Symbol(’gamma2’)cTt = Matrix(([c(alpha2)*c(beta2), ...
[s(alpha2)*c(beta2), ... [-s(beta2), c(beta2)*s(gamma2), ... [0, 0, 0, 1]\)) #
see [Prager98] for full expressions# Calculate full equationsPc = cTt * tTr *
rTp * PpPc = Pc[0:3,:]# Calculate full Jacobiansx = Matrix([sx, sy, x1, y1,
z1, alpha1, beta1,gamma1, x2, y2, z2, alpha2, beta2, gamma2])J =
Pc.jacobian(x)The equations system was solved by using the function
scipy.optimize.root with method=’lm’.

To validate the solution, the calibration part in this package allows to
visualize the corresponding covariance matrix; this can be exploited to
understand if some variable is not well constrained. In addition, since in
each B-scan it is necessary to have the position for at least two pixels that
belong to the edge line, we developed an automatic tool for extracting the
corresponding lines in each image, based on the Hough transform:

  *

import cv2# Threshold imagemaxVal = np.iinfo(I.dtype).maxth, bw =
cv2.threshold(I,np.round(thI*maxVal), maxVal,cv2.THRESH_BINARY)# Detect
edgesedges = cv2.Canny(bw,thCan1,thCan2, apertureSize=kerSizeCan)# Dilate
edgeskernel = np.ones(kerSizeDil,I.dtype)dilate = cv2.dilate(edges, kernel,
iterations=1)# Find longest linelines =
cv2.HoughLinesP(dilate,1,np.pi/180,thHou, minLineLength,maxLineGap)maxL = 0if
lines == None: a, b = np.nan, np.nanelse: for x1,y1,x2,y2 in lines[0]: L =
np.linalg.norm((x1-x2,y1-y2)) if L > maxL: maxL = L a = float(y1 - y2) / (x1 -
x2) b = y1 - a * x1# a, b being line parameters: y = a * x + b

  *

Since we experienced unsatisfactory calibration results (in terms of later
reconstruction compounding) at this stage, we passed those through an image
compounding algorithm which allows to achieve a good tuning. This is an image
based method which uses as input 2 perpendicular sweeps, at approximately 90
degrees, for the same 3D volume [Wein08]. Briefly, a similarity measure
(Normalized Cross Correlation, NCC) between the two sweeps was applied to
maximize this measure with the final aim to find the calibration parameters
relative to the best overlapping between the images. The initial values of
this iterative method are the results of the equations-based approach. A
calibration quality assessment was also implemented in terms of precision and
accuracy of the calibration parameters obtained. Precision gives an indication
of the dispersion of measures around their mean, whereas the accuracy gives an
indication of the difference between the mean of the measures and the real
value [Hsu06]. For example, this measure can be the known position of a point
in space (_Point accuracy_) or the known dimension of an object (_Distance
accuracy_).

###  3.3 3D voxel array reconstruction

The 3D reconstruction is performed by positioning the 2D US scans in the 3D
space by using the corresponding pose. The first step is to import the images
(DICOM file, standard format for medical imaging) and the synchronized
kinematics files (C3D format) containing pose data. A 3D voxel array is then
initialized. The 3D voxel array (a parallelepipedon) should be the smallest
one containing the sequence of all the repositioned scans, as seen in Figure
2, in order to avoid RAM waste. To face this issue, in the present package two
options are presented: reorienting manually the global reference frame in
order to be approximately aligned with the scan direction during the
acquisition; on the other hand, by using the Principal Component Analysis
(PCA), it is also possible to find the scan direction and thereby realigning
the voxel array according to this direction.

Figure 2: V’ is the smallest 3D voxel array parallelepipedon able to contain
all the US images. Others can be created, such as V, but they are bigger,
occupy more memory and contain more empty voxels.

The grey values of the original pixels in the 2D slices are then copied in the
new corresponding 3D position. This procedure is performed by using an
algorithm called Pixel Nearest Neighbor (PNN) which runs through each pixel in
every image and fills the nearest voxel with the value of that pixel; in case
of multiple contributions to the same voxel, the values are averaged. Below
the code to perform this is shown. Each 2D scan is positioned in the 3D volume
in a vectorized way.

  *

# x, y, z: arrays for 3D coordinates of# the pixels in image I# idxV: unique
ID for each voxel of the# 3D voxel array# V: 1D array containing grey values
for the# 3D voxel-array# contV: 1D array containing current number of#
contributions for voxels# I: 2D array containing US slice grey valuesidxV =
xyz2idx(x, y, z, xl, yl, zl).astype(np.int32)V[idxV] = (contV[idxV] * V[idxV])
/ (contV[idxV] + 1) + I.ravel() / (contV[idxV] + 1) # iterative avg

  *

Only 2 outer loops exist, one for the DICOM file number and one for the scan
number. After all the scans are correctly positioned in the 3D space, gaps can
occur in the voxel array when the voxel size is small compared to the distance
between the acquired images (e.g. scanning velocity significantly different
from 0). Therefore interpolation methods are applied for filling these empty
voxels. For optimizing this process, a robust method was also used, i.e.
convex hull (see Figure 3), for restricting the gap filling operation only to
the voxels contained between 2 consecutive slices:

Figure 3: Considering 2 US images consecutive in time, the convex hull is the
smaller object able to contain them. An easier shape can be created, such as
the parallelepipedon, but this is always bigger in volume.

The quick-and-dirty way, known as VNN (Voxel Nearest Neighbour), consists of
filling a gap by using the closest voxel having an assigned grey value. We
also implemented another (average cube) solution which consist of the
following steps:

  * •

Create a cube with side 3 voxels, centered around the gap;

  * •

Search the minimum percentage of non-gaps inside the cube (100% = number of
voxels in the cube);

  * •

If that percentage is found, a non-gap voxels average (weighted by the
Euclidean distances) is performed into the cube;

  * •

If that percentage is not found, the cube size in incremented by 2 voxels
(e.g. 5);

  * •

If cube size is lesser or equal than a maximum size, start again from point 2.
Otherwise, stop and don’t fill the gap.

The entire voxel array can be subdivided in N parallelepipedal blocks, and the
gap filling is performed on each one at a time, to spare some of the RAM. The
bigger the number of blocks, the bigger the number of iterations to go, but
the smaller the block size, the RAM used and the time spent per iteration.
Finally, both the voxel array scans silhouette (previously created with the
wrapping convex hulls) and the grey scale data voxel array are exported to VTI
files, after being converted to vtk.vtkImageData. These can be opened with
software like MeVisLab or Paraview for visualization and further processing.

###  3.4 Preliminary results

The calibration quality assessments were 1.9 mm and 3.9 mm for the distance
accuracy and reconstruction precision, respectively. The average data
processing time (calibration + reconstruction + gap filling) over 3 trials on
a human calf, shown in Figure 4, was 5.9 min, on a 16 GB RAM Intel i7 2.7 GHz
machine.

Figure 4: Three transversal and one longitudinal section of a reconstructed 3D
voxel array (human calf scanning, about 90M voxels, $10mm^{3}$ each).

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