# Some homological properties of ideals with cohomological dimension one

G. Pirmohammadi, K. Ahmadi Amoli, and K. Bahmanpour∗ Gholamreza Pirmohammadi;
Payame Noor University, Po Box 19395-3697, Tehran, Iran.
<EMAIL_ADDRESS>Khadijeh Ahmadi Amoli; Payame Noor University, Po
Box 19395-3697, Tehran, Iran<EMAIL_ADDRESS>Kamal Bahmanpour; Faculty of
Sciences, Department of Mathematics, University of Mohaghegh Ardabili,
56199-11367, Ardabil, Iran; and School of Mathematics, Institute for Research
in Fundamental Sciences (IPM), P.O. Box. 19395-5746, Tehran, Iran.
<EMAIL_ADDRESS>
###### Abstract.

Let $R$ denote a commutative Noetherian ring and let $I$ be an ideal of $R$
such that $H^{i}_{I}(R)=0$, for all integers $i\geq 2$. In this paper we shall
prove some results concerning the homological properties of $I$.

###### Key words and phrases:

Abelian category, cofinite modules, cohomological dimension, local cohomology,
Noetherian ring.

2010 Mathematics Subject Classification: 13D45, 14B15, 13E05.
∗Corresponding author<EMAIL_ADDRESS>(Kamal Bahmanpour)

##  1\. Introduction

Let $R$ denote a commutative Noetherian ring and let $I$ be an ideal of $R$.
In [11], Hartshorne defined an $R$-module $L$ to be $I$-cofinite, if
$\operatorname{Supp}L\subseteq V(I)$ and ${\rm Ext}^{i}_{R}(R/I,L)$ is a
finitely generated module for all $i$. He posed the following question:

Whether the category $\mathscr{C}(R,I)_{cof}$ of $I$-cofinite modules is an
Abelian subcategory of the category of all $R$-modules? That is, if
$f:M\longrightarrow N$ is an $R$-homomorphism of $I$-cofinite modules, are
$\ker f$ and ${\rm coker}f$ $I$-cofinite?

Hartshorne gave a counterexample to show that this question has not an
affirmative answer in general, (see [11, §3]). On the positive side,
Hartshorne proved that if $I$ is a prime ideal of dimension one in a complete
regular local ring $R$, then the answer to his question is yes. On the other
hand, in [7], Delfino and Marley extended this result to arbitrary complete
local rings. Kawasaki in [18] generalized the Delfino and Marley’s result for
an arbitrary ideal $I$ of dimension one in a local ring $R$. Finally,
Melkersson in [23] generalized the Kawasaki’s result for all ideals of
dimension one of any arbitrary Noetherian ring $R$. More recently, in [5] as a
generalization of Melkersson’s result it is shown that for any ideals $I$ in a
commutative Noetherian ring $R$, the category of all $I$-cofinite $R$-modules
$M$ with $\dim M\leq 1$ is an Abelian subcategory of the category of all
$R$-modules.

Recall that, for an $R$-module $M$, the cohomological dimension of $M$ with
respect to $I$, denoted by $\operatorname{cd}(I,M)$, is the smallest integer
$n\geq 0$ such that $H^{i}_{I}(M)=0$ for all $i>n$.

The cohomological dimension have been studied by several authors; see, for
example, Faltings [9], Hartshorne [12], Huneke-Lyubeznik [16], Divaani-Aazar
et al [8], Hellus [13], Hellus-Stückrad [14], Mehrvarz et al [22], and Ghasemi
et al [10].

Recall that, for any proper ideal $I$ of $R$, the _arithmetic rank_ of $I$,
denoted by $\operatorname{ara}(I)$, is the least number of elements of $I$
required to generate an ideal which has the same radical as $I$.

Now, let $I$ be an ideal of an arbitrary Noetherian ring $R$. Kawasaki in [17,
Theorem 2.1] proved that if $\operatorname{ara}(I)=1$ then the category
$\mathscr{C}(R,I)_{cof}$ of $I$-cofinite modules is an Abelian subcategory of
the category of all $R$-modules.

It is well known that, for any proper ideal $I$ of a Noetherian ring $R$ we
have $\operatorname{cd}(I,R)\leq\operatorname{ara}(I)$. (See [6, Corollary
3.3.3]). In particular, for any ideal $I$ with $\operatorname{ara}(I)=1$ we
have $\operatorname{cd}(I,R)\leq 1$. So, as a generalization of Kawasaki’s
interesting result, it is more natural that we ask the following question:

Question 1: Let $R$ be a Noetherian ring and $I$ be an ideal of $R$ with
$\operatorname{cd}(I,R)\leq 1$. Whether the category $\mathscr{C}(R,I)_{cof}$
of $I$-cofinite modules is an Abelian subcategory of the category of all
$R$-modules?

In section 2 of this paper we present an affirmative answer to the Question 1,
whenever $R$ is a local Noetherian ring. In section 3 we present some
equivalent conditions for the exactness of ideal transform functors. Using
these results, for any ideal $I$ generated by two elements, we provide some
necessary and sufficient conditions for the non-vanishing of the local
cohomology module $H^{2}_{I}(R)$. Finally, in section 4 we prepare some
vanishing conditions for the Bass and Betti numbers of some special local
cohomology modules. Also, we give a formula for the cohomological dimension of
some special ideals in Noetherian domains.

Throughout this paper, $R$ will always be a commutative Noetherian ring and
$I$ will be an ideal of $R$. Also, for an $R$-module $M$, $\Gamma_{I}(M)$
denotes the submodule of $M$ consisting of all elements annihilated by some
power of $I$, i.e., $\bigcup_{n=1}^{\infty}(0:_{M}I^{n})$. For every
$R$-module $L$, we denote by $\operatorname{mAss}_{R}L$ the set of minimal
elements of $\operatorname{Ass}_{R}L$ with respect to inclusion. Also, for any
ideal $\mathfrak{a}$ of $R$, we denote $\\{\mathfrak{p}\in{\rm
Spec}\,R:\,\mathfrak{p}\supseteq\mathfrak{a}\\}$ by $V(\mathfrak{a})$.
Finally, for any ideal $\mathfrak{b}$ of $R$, the radical of $\mathfrak{b}$,
denoted by $\operatorname{Rad}(\mathfrak{b})$, is defined to be the set
$\\{x\in R\,:\,x^{n}\in\mathfrak{b}$ for some $n\in\mathbb{N}\\}$. For any
unexplained notation and terminology we refer the reader to [6] and [21].

##  2\. A category of modules which is Abelian

The following well known lemma plays a key role in the proof of Proposition
2.2.

######  Lemma 2.1.

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $T$ be an $R$-module.
Then the following statements are equivalent:

  * (i)

The $R$-module $H^{i}_{I}(T)$ is Artinian, for $i\geq 0$.

  * (ii)

The $R$-module $\operatorname{Ext}^{i}_{R}(R/I,T)$ is Artinian, for $i\geq 0$.

  * (iii)

The $R$-module $\operatorname{Ext}^{i}_{R}(R/I,T)$ is Artinian, for $0\leq
i\leq\operatorname{cd}(I,R)$.

###### Proof.

See [24, Theorem 5.5] and [2, Theorem 2.9].∎

######  Proposition 2.2.

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring, $I$ be an
ideal of $R$ and $M$ be an $R$-module. Then the following conditions are
equivalent:

  * (i)

The $R$-modules $\operatorname{Tor}^{R}_{i}(R/I,M)$ are finitely generated for
all $i\geq 0$.

  * (ii)

The $R$-modules $\operatorname{Tor}^{R}_{i}(R/I,M)$ are finitely generated for
all $0\leq i\leq\operatorname{cd}(I,R)$.

###### Proof.

(i)$\Rightarrow$(ii) Is clear.

(ii)$\Rightarrow$(i) Let $T:=D(M)$, where $D(-):={\rm Hom}_{R}(-,E)$ denotes
the Matlis dual functor and $E:=E_{R}(R/\mathfrak{m})$ is the injective hull
of the residue field $R/\mathfrak{m}$. Then by the adjointness we have
$\operatorname{Ext}^{i}_{R}(R/I,T)\simeq
D(\operatorname{Tor}^{R}_{i}(R/I,M))$, for all $0\leq
i\leq\operatorname{cd}(I,R)$. In particular, the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/I,T)$ are Artinian for all $0\leq
i\leq\operatorname{cd}(I,R)$. So, it follows from Lemma 2.1 and adjointness,
that the $R$-modules $\operatorname{Ext}^{i}_{R}(R/I,T)\simeq
D(\operatorname{Tor}^{R}_{i}(R/I,M))$ are Artinian for all $i\geq 0$. Now, it
follows from [19, Lemma 1.15(a)] that, the $R$-modules
$\operatorname{Tor}^{R}_{i}(R/I,M)$ are finitely generated for all $i\geq 0$,
as required.∎

The following easy consequence of Proposition 2.2 plays a key role in the
proof of Theorem 2.4.

######  Corollary 2.3.

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring, $I$ be an
ideal of $R$ with $\operatorname{cd}(I,R)=n$ and let $M$ be an $R$-module with
$\operatorname{Supp}M\subseteq V(I)$. Then the $R$-module $M$ is $I$-cofinite,
if and only if, the $R$-modules $\operatorname{Tor}^{R}_{i}(R/I,M)$ are
finitely generated for all $0\leq i\leq n$.

###### Proof.

The assertion follows from Proposition 2.2 and [24, Theorem 2.1].∎

The following result gives an affirmative answer to the Question 1, for the
local case.

######  Theorem 2.4.

Let $I$ be an ideal of a Noetherian local ring
$(R,\operatorname{\mathfrak{m}})$ such that $\operatorname{cd}(I,R)\leq 1$.
Let $\mathscr{C}(R,I)_{cof}$ denote the category of $I$-cofinite $R$-modules.
Then $\mathscr{C}(R,I)_{cof}$ is an Abelian subcategory of the category of all
$R$-modules.

###### Proof.

Let $M,N\in\mathscr{C}(R,I)_{cof}$ and let $f:M\longrightarrow N$ be an
$R$-homomorphism. It is enough to show that the $R$-modules $\ker f$ and ${\rm
coker}f$ are $I$-cofinite.

To this end, the exact sequence

$0\longrightarrow\ker f\longrightarrow M\longrightarrow{\rm
im}f\longrightarrow 0,$

induces an exact sequence

$\operatorname{Tor}^{R}_{0}(R/I,M)\rightarrow\operatorname{Tor}^{R}_{0}(R/I,{\rm
im}f)\rightarrow 0,$

which using [24, Theorem 2.1], implies that the $R$-module
$\operatorname{Tor}^{R}_{0}(R/I,{\rm im}f)$ is finitely generated. Now, the
exact sequence

$0\longrightarrow{\rm im}f\longrightarrow N\longrightarrow{\rm
coker}f\longrightarrow 0.$

induces an exact sequence

$\operatorname{Tor}^{R}_{1}(R/I,N)\rightarrow\operatorname{Tor}^{R}_{1}(R/I,{\rm
coker}f)\rightarrow\operatorname{Tor}^{R}_{0}(R/I,{\rm im}f)$
$\rightarrow\operatorname{Tor}^{R}_{0}(R/I,N)\rightarrow\operatorname{Tor}^{R}_{0}(R/I,{\rm
coker}f)\rightarrow 0.$

By [24, Theorem 2.1] the modules $\operatorname{Tor}^{R}_{1}(R/I,N)$ and
$\operatorname{Tor}^{R}_{0}(R/I,N)$ are finitely generated $R$-modules, which
implies that the $R$-modules $\operatorname{Tor}^{R}_{0}(R/I,{\rm coker}f)$
and $\operatorname{Tor}^{R}_{1}(R/I,{\rm coker}f)$ are finitely generated.
Therefore, it follows from Corollary 2.3, that the $R$-module ${\rm coker}f$
is $I$-cofinite. Now, the assertion follows from the following exact sequences

$0\longrightarrow{\rm im}f\longrightarrow N\longrightarrow{\rm
coker}f\longrightarrow 0,$

and

$0\longrightarrow\ker f\longrightarrow M\longrightarrow{\rm
im}f\longrightarrow 0.$

∎

######  Corollary 2.5.

Let $I$ be an ideal of a Noetherian local ring
$(R,\operatorname{\mathfrak{m}})$ such that $\operatorname{cd}(I,R)\leq 1$.
Let $\mathscr{C}(R,I)_{cof}$ denote the category of $I$-cofinite modules over
$R$. Let

$X^{\bullet}:\cdots\longrightarrow X^{i}\stackrel{{\scriptstyle
f^{i}}}{{\longrightarrow}}X^{i+1}\stackrel{{\scriptstyle
f^{i+1}}}{{\longrightarrow}}X^{i+2}\longrightarrow\cdots,$

be a complex such that $X^{i}\in\mathscr{C}(R,I)_{cof}$ for all
$i\in\mathbb{Z}$. Then for each $i\in\mathbb{Z}$ the $i^{{\rm th}}$ cohomology
module $H^{i}(X^{\bullet})$ is in $\mathscr{C}(R,I)_{cof}$.

###### Proof.

The assertion follows from Theorem 2.4.∎

######  Corollary 2.6.

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring, $I$ be an
ideal of $R$ with $\operatorname{cd}(I,R)\leq 1$ and let $M$ be an
$I$-cofinite $R$-module. Then, the $R$-modules ${\rm Tor}_{i}^{R}(N,M)$ and
${\rm Ext}^{i}_{R}(N,M)$ are $I$-cofinite, for all finitely generated
$R$-modules $N$ and all integers $i\geq 0$.

###### Proof.

Since $N$ is finitely generated it follows that, $N$ has a free resolution
with finitely generated free $R$-modules. Now the assertion follows using
Corollary 2.5 and computing the $R$-modules ${\rm Tor}_{i}^{R}(N,M)$ and ${\rm
Ext}^{i}_{R}(N,M)$, using this free resolution. ∎

##  3\. Vanishing of the extension and torsion functors

In this section we present some equivalent conditions for the exactness of
ideal transform functors. Using these results, for any ideal $I$ generated by
two elements, we provide some necessary and sufficient conditions for the non-
vanishing of the local cohomology module $H^{2}_{I}(R)$.

The following lemma is needed in the proof of Proposition 3.2.

######  Lemma 3.1.

Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Let $M$ be an
$R$-module such that $\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for all integers
$i\geq 0$. Then $\operatorname{Hom}_{R}(R/I,M)=0$.

###### Proof.

Let $I=(x_{1},\dots,x_{n})$ and let

$K_{\bullet}(\underline{x},M):0\longrightarrow
M\overset{f_{0}}{\longrightarrow}\bigoplus_{k=1}^{C_{n}^{1}}M\overset{f_{1}}{\longrightarrow}\bigoplus_{k=1}^{C_{n}^{2}}M\longrightarrow\dots\longrightarrow\bigoplus_{k=1}^{C_{n}^{n-1}}M\overset{f_{n-1}}{\longrightarrow}M\longrightarrow
0,$

be the Koszul complex of $M$ with respect to
$\underline{x}=x_{1},\dots,x_{n}$.

We prove that $\operatorname{H}_{i}(\underline{x};M)=0$ for all $0\leq i\leq
n$. By the definition we have

$\operatorname{H}_{0}(\underline{x};M)=\operatorname{coker}f_{n-1}=M/IM\simeq
R/I\otimes_{R}M=0.$

So, we have $\operatorname{im}f_{n-1}=M$. Therefore, using the hypothesis it
follows that

$\operatorname{Tor}^{R}_{i}(R/I,\operatorname{im}f_{n-1})=0,\,\,{\rm
for}\,\,{\rm each}\,\,i\geq 0.$

Hence, the exact sequence

$0\longrightarrow\operatorname{ker}{f_{n-1}}\longrightarrow\bigoplus_{k=1}^{C_{n}^{n-1}}M\longrightarrow\operatorname{im}f_{n-1}\longrightarrow
0,$

implies that $\operatorname{Tor}^{R}_{i}(R/I,\operatorname{ker}f_{n-1})=0$,
for each $i\geq 0$. The exact sequence

$0\longrightarrow\operatorname{im}{f_{n-2}}\longrightarrow\operatorname{ker}f_{n-1}\longrightarrow\operatorname{H}_{1}(\underline{x};M)\longrightarrow
0,\,\,\,\,\,\,~{}(3.1.1)$

induces the exact sequence

$R/I\otimes_{R}\operatorname{ker}f_{n-1}\longrightarrow
R/I\otimes_{R}\operatorname{H}_{1}(\underline{x};M)\longrightarrow
0.\,\,\,\,\,\,~{}(3.1.2)$

Now as $R/I\otimes_{R}\operatorname{ker}f_{n-1}=0$, it follows from $(3.1.2)$
that

$R/I\otimes_{R}\operatorname{H}_{1}(\underline{x};M)=0.$

By the definition of the Koszul complex we have
$I\operatorname{H}_{1}(\underline{x};M)=0$. Therefore, we have
$\operatorname{H}_{1}(\underline{x};M)\simeq
R/I\otimes_{R}\operatorname{H}_{1}(\underline{x};M)=0$. Now it follows from
the exact sequence $(3.1.1)$ that
$\operatorname{Tor}^{R}_{i}(R/I,\operatorname{im}f_{n-2})=0$, for each $i\geq
0$. Moreover, the exact sequence

$0\longrightarrow\operatorname{ker}f_{n-2}\longrightarrow\bigoplus_{k=1}^{C_{n}^{n-2}}M\longrightarrow\operatorname{im}{f_{n-2}}\longrightarrow
0,$

implies that $\operatorname{Tor}^{R}_{i}(R/I,\operatorname{ker}f_{n-2})=0$,
for each $i\geq 0$. Proceeding in the same way we can see
$\operatorname{H}_{i}(\underline{x};M)=0$, for all $0\leq i\leq n$.

Now, since
$\operatorname{H}_{i}(\underline{x};M)\simeq\operatorname{H}^{n-i}(\underline{x};M)$
for all $0\leq i\leq n$, it follows that

$\operatorname{Hom}_{R}(R/I,M)\simeq(0:_{M}I)\simeq\operatorname{H}^{0}(\underline{x};M)\simeq\operatorname{H}_{n}(\underline{x};M)=0.$

∎

The following proposition plays a key role in the proof of our main results.

######  Proposition 3.2.

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $M$ be an $R$-module.
Then the following conditions are equivalent:

  * (i)

$\operatorname{Tor}^{R}_{n}(R/I,M)=0$, for all integers $n\geq 0$.

  * (ii)

$\operatorname{Ext}^{n}_{R}(R/I,M)=0$, for all integers $n\geq 0$.

###### Proof.

(i)$\Rightarrow$(ii) We argue by induction on $n$. For $n=0$, the assertion
holds by Lemma 3.1. We therefore assume, inductively, that $n>0$ and the
result has been proved for smaller values of $n$. Then there is an exact
sequence

$0\longrightarrow M\longrightarrow E_{R}(M)\longrightarrow
E_{R}(M)/M\longrightarrow 0.\,\,\,\,\,\,~{}(3.2.1)$

Since, by the hypothesis we have $(0:_{M}I)=0$ it follows that
$(0:_{E_{R}(M)}I)=0$ and hence $\Gamma_{I}(E_{R}(M))=0$. By [21, Theorem 18.5]
the $R$-module $E_{R}(M)$ is isomorph with a direct sum of a family of
indecomposable injective $R$-modules. Let $\operatorname{\mathfrak{p}}$ be a
prime ideal of $R$ such that $E_{R}(R/\operatorname{\mathfrak{p}})$ is a
direct summand of $E_{R}(M)$. Then we have
$\Gamma_{I}(E_{R}(R/\operatorname{\mathfrak{p}}))=0$. Therefore, form the fact
that
$\operatorname{Ass}_{R}E_{R}(R/\operatorname{\mathfrak{p}})=\\{\operatorname{\mathfrak{p}}\\}$
it follows that $I\not\subseteq\operatorname{\mathfrak{p}}$. So there exists
an element $a\in I$ such that $a\not\in\operatorname{\mathfrak{p}}$. Then by
[21, Theorem 18.4(iii)], multiplication by $a$ is an automorphism on
$E_{R}(R/\operatorname{\mathfrak{p}})$. Therefore, multiplication by $a$ is an
automorphism on
$\operatorname{Tor}^{R}_{i}(R/I,E_{R}(R/\operatorname{\mathfrak{p}}))$, for
all $i\geq 0$. But, since $a\in I$ it follows that, multiplication by $a$ on
$\operatorname{Tor}^{R}_{i}(R/I,E_{R}(R/\operatorname{\mathfrak{p}}))$ is the
zero map, for all $i\geq 0$. Thus,
$\operatorname{Tor}^{R}_{i}(R/I,E_{R}(R/\operatorname{\mathfrak{p}}))=0$ for
all $i\geq 0$. Since for each $i\geq 0$, the torsion functor
$\operatorname{Tor}^{R}_{i}(R/I,-)$ commutes with the direct sums it follows
that $\operatorname{Tor}^{R}_{i}(R/I,E_{R}(M))=0$, for all $i\geq 0$. So, from
the exact sequence $(3.2.1)$ it follows that
$\operatorname{Tor}^{R}_{i}(R/I,E_{R}(M)/M)=0$, for all $i\geq 0$. Hence, by
applying the inductive hypothesis to the $R$-module $E_{R}(M)/M$ we have

$\operatorname{Ext}^{i+1}_{R}(R/I,M)\simeq\operatorname{Ext}^{i}_{R}(R/I,E_{R}(M)/M)=0\,\,\,\,{\rm
for}\,\,0\leq i\leq n-1.$

So, we have $\operatorname{Ext}^{n}_{R}(R/I,M)=0$. This completes the
inductive step.∎

(ii)$\Rightarrow$(i) Assume the opposite. Then there is an integer $j\geq 0$
such that $\operatorname{Tor}^{R}_{j}(R/I,M)\neq 0$. Let
$\operatorname{\mathfrak{p}}\in\operatorname{Supp}\operatorname{Tor}^{R}_{j}(R/I,M)$.
Then localizing at $\operatorname{\mathfrak{p}}$, without loss of generality,
we may assume that $(R,\operatorname{\mathfrak{m}})$ is a Noetherian local
ring. Let $T:=D(M)$, where $D(-)$ denotes the Matlis dual functor. Then by the
adjointness we have

$\operatorname{Tor}^{R}_{i}(R/I,T)\simeq
D(\operatorname{Ext}^{i}_{R}(R/I,M))=0,$

for all $i\geq 0$. Therefore, by the previous part of the proof we have
$\operatorname{Ext}^{i}_{R}(R/I,T)=0,$ for each $i\geq 0$. So, by the
adjointness we have

$D(\operatorname{Tor}^{R}_{j}(R/I,M))\simeq\operatorname{Ext}^{j}_{R}(R/I,T)=0,$

which implies that $\operatorname{Tor}^{R}_{j}(R/I,M)=0$. This is the desired
contradiction.∎

######  Lemma 3.3.

Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Let $E$ be an
injective $R$-module and $K$ be a submodule of $E$. Then

$\frac{L}{\Gamma_{I}(L)}\otimes_{R}\frac{R}{I}=0,\,\,\,\,\,\,\,{\rm
where}\,\,L:=\frac{E}{K}.$

###### Proof.

In view of [6, Proposition 2.1.4] the $R$-module $E_{1}:=\Gamma_{I}(E)$ is
injective. Therefore, there exists an injective submodule $E_{2}$ of $E$ such
that $E_{1}+E_{2}=E$ and $E_{1}\cap E_{2}=0$. Since $\frac{E_{1}+K}{K}$ is a
submodule of $\Gamma_{I}(L)$ it follows that the $R$-module
$\frac{L}{\Gamma_{I}(L)}$ is a homomorphic image of the $R$-module $E_{2}$.
So, in order to prove the assertion it is enough to prove that
$E_{2}\otimes_{R}\frac{R}{I}=0$. So, we must prove that $E_{2}=IE_{2}$. Since
$\Gamma_{I}(R)E_{2}\subseteq\Gamma_{I}(E_{2})=0$, it follows that
$\Gamma_{I}(R)E_{2}=0$. Therefore, we have

$E_{2}=(0:_{E_{2}}\Gamma_{I}(R))\simeq\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2}).$

On the other hand, by [6, Lemma 2.1.1] there is an exact sequence

$0\longrightarrow R/\Gamma_{I}(R)\stackrel{{\scriptstyle
a}}{{\longrightarrow}}R/\Gamma_{I}(R),$

for some element $a\in I$, which effecting the $R$-linear exact functor
$\operatorname{Hom}_{R}(-,E_{2})$ induces the exact sequence

$\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})\longrightarrow
0.$

Therefore, we have
$\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})=a\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})$.
Hence, we have

$\displaystyle E_{2}/aE_{2}$ $\displaystyle\simeq$ $\displaystyle
E_{2}\otimes_{R}R/Ra$ $\displaystyle\simeq$
$\displaystyle\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})\otimes_{R}R/Ra$
$\displaystyle\simeq$
$\displaystyle\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})/a\operatorname{Hom}_{R}(R/\Gamma_{I}(R),E_{2})$
$\displaystyle\simeq$ $\displaystyle 0.$

So, we have $E_{2}=aE_{2}$ and hence $E_{2}=IE_{2}$.∎

######  Lemma 3.4.

Let $R$ be a Noetherian ring and let $I\subseteq J$ be two ideals of $R$. Let
$M$ be an $R$-module and $t\geq 2$ be an integer such that
$\operatorname{Tor}^{R}_{j}(R/J,H^{i}_{I}(M))=0$ for all $i>t$ and all $j\geq
0$. Then we have $\operatorname{Tor}^{R}_{j}(R/J,H^{t}_{I}(M))=0$ for $j=0,1$.

###### Proof.

Let

$0\longrightarrow
M\stackrel{{\scriptstyle\varepsilon}}{{\longrightarrow}}E_{0}\stackrel{{\scriptstyle
f_{0}}}{{\longrightarrow}}E_{1}\stackrel{{\scriptstyle
f_{1}}}{{\longrightarrow}}E_{2}\stackrel{{\scriptstyle
f_{2}}}{{\longrightarrow}}\cdots$

be a minimal injective resolution for $M$. Set $K_{i}:={\rm ker}f_{i}$ for
$i\geq 0$. By splitting this minimal injective resolution to some short exact
sequences we get the isomorphisms

$H^{t}_{I}(M)\simeq H^{t-1}_{I}(K_{1})\simeq
H^{t-2}_{I}(K_{2})\simeq\cdots\simeq H^{2}_{I}(K_{t-2})\simeq
H^{1}_{I}(K_{t-1})\simeq H^{1}_{I}(K_{t-1}/\Gamma_{I}(K_{t-1})).$

Set $N:=K_{t-1}/\Gamma_{I}(K_{t-1})$. Then we have $\Gamma_{I}(N)=0$ and

$K_{t-1}={\rm ker}f_{t-1}={\rm im}f_{t-2}\simeq E_{t-2}/{\rm
ker}f_{t-2}=E_{t-2}/K_{t-2}.$

So, there exists a submodule $K$ of $E:=E_{t-2}$ such that $N\simeq E/K$. By
[6, Remark 2.2.7] there is an exact sequence

$0\longrightarrow N\longrightarrow D_{I}(N)\longrightarrow
H^{1}_{I}(N)\longrightarrow 0.\,\,\,\,\,(3.4.1)$

For each $i\geq 2$ we have

$H^{i}_{I}(N)\simeq H^{i}_{I}(K_{t-1})\simeq H^{t+i-1}_{I}(M).$

Therefore, by the hypothesis we have

$\operatorname{Tor}^{R}_{j}(R/J,H^{i}_{I}(N))\simeq\operatorname{Tor}^{R}_{j}(R/J,H^{t+i-1}_{I}(M))=0$

for all $i\geq 2$ and all $j\geq 0$. Thus, by Proposition 3.2 we have
$\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(N))=0$ for all $i\geq 2$ and all
$j\geq 0$. The exact sequence $(3.4.1)$ yields the isomorphisms

$H^{i}_{I}(D_{I}(N))\simeq H^{i}_{I}(N),$

for $i\geq 2$. So, we have

$\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(D_{I}(N)))\simeq\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(N))=0$

for all $i\geq 2$ and all $j\geq 0$. Also, by [6, Corollary 2.2.8] we have
$H^{i}_{I}(D_{I}(N))=0$, for $i=0,1$. So
$\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(D_{I}(N)))=0$ for all $i\geq 0$ and
all $j\geq 0$. Therefore, by [10, Lemma 2.1] we have
$\operatorname{Ext}^{j}_{R}(R/J,D_{I}(N))=0$, for all integers $j\geq 0$. Thus
by Proposition 3.2 we have $\operatorname{Tor}^{R}_{i}(R/J,D_{I}(N))=0$, for
all integers $i\geq 0$. On the other hand, in view of Lemma 3.3 we have
$N\otimes_{R}R/I=0$ and hence it follows from the hypothesis $I\subseteq J$
that $N\otimes_{R}R/J=0$. Hence, using the long exact sequence induced by the
exact sequence $(3.4.1)$ it follows that
$\operatorname{Tor}^{R}_{j}(R/J,H^{1}_{I}(N))=0$, for $j=0,1$. Therefore,
$\operatorname{Tor}^{R}_{j}(R/J,H^{t}_{I}(M))\simeq\operatorname{Tor}^{R}_{j}(R/J,H^{1}_{I}(N))=0$,
for $j=0,1$.∎

######  Corollary 3.5.

Let $R$ be a Noetherian ring and $I\subseteq J$ be two ideals of $R$. Let $K$
be an $R$-module such that $2\leq{\rm cd}(I,K)=t$. Then
$\operatorname{Tor}^{R}_{i}(R/J,H^{t}_{I}(K))=0$, for $i=0,1$.

###### Proof.

The assertion follows from Lemma 3.4. ∎

######  Proposition 3.6.

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $M$ be an $R$-module.
Then the following conditions are equivalent:

  * (i)

$\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for all $i\geq 0$.

  * (ii)

$\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for all $0\leq
i\leq\operatorname{cd}(I,R)$.

###### Proof.

(i)$\Rightarrow$(ii) Is clear.

(ii)$\Rightarrow$(i) Assume the opposite. Then there is an integer
$j>\operatorname{cd}(I,R)$ such that $\operatorname{Tor}^{R}_{j}(R/I,M)\neq
0$. Let
$\operatorname{\mathfrak{p}}\in\operatorname{Supp}\operatorname{Tor}^{R}_{j}(R/I,M)$.
Then localizing at $\operatorname{\mathfrak{p}}$, without loss of generality,
we may assume that $(R,\operatorname{\mathfrak{m}})$ is a Noetherian local
ring. Let $T:=D(M)$, where $D(-)$ denotes the Matlis dual functor. Then by the
adjointness we have

$\operatorname{Ext}^{i}_{R}(R/I,T)\simeq
D(\operatorname{Tor}^{R}_{i}(R/I,M))=0,$

for all $0\leq i\leq\operatorname{cd}(I,R)$. Now, by [2, Theorem 2.9] we have
$H^{i}_{I}(T)=0$, for all $0\leq i\leq\operatorname{cd}(I,R)$. Hence,
$H^{i}_{I}(T)=0$, for all integers $i\geq 0$. So, in view of [2, Theorem 2.9]
we have $\operatorname{Ext}^{i}_{R}(R/I,T)=0,$ for all integers $i\geq 0$.
Consequently, by the adjointness, we have

$D(\operatorname{Tor}^{R}_{i}(R/I,M))\simeq\operatorname{Ext}^{i}_{R}(R/I,T)=0,$

for all $i\geq 0$. Thus, $\operatorname{Tor}^{R}_{i}(R/I,M)=0$ for all $i\geq
0$, which is a contradiction.∎

######  Theorem 3.7.

Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Then the following
conditions are equivalent:

  * (i)

${\rm cd}(I,R)\leq 1$.

  * (ii)

The ideal transform functor $D_{I}(-)$ is exact.

  * (iii)

For every $R$-module $M$, if $\operatorname{Tor}^{R}_{i}(R/I,M)=0$ for
$i=0,1$, then $\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for all integers $i\geq
0$.

  * (iv)

For every $R$-module $M$, if $\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for
$i=0,1$, then $\operatorname{Ext}^{i}_{R}(R/I,M)=0$, for all integers $i\geq
0$.

###### Proof.

(i)$\Leftrightarrow$(ii) See [6, Lemma 6.3.1].

(i)$\Rightarrow$(iii) Follows from Proposition 3.6.

(iii)$\Rightarrow$(i) Assume the opposite. Then we have
$\operatorname{cd}(I,R)\geq 2$. Let $t=\operatorname{cd}(I,R)$. Then by
Corollary 3.5 for the $R$-module $M:=H^{t}_{I}(R)$ we have
$\operatorname{Tor}^{R}_{i}(R/I,M)=0$ for $i=0,1$. So, by the hypothesis we
have $\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for all integers $i\geq 0$.
Therefore, by Proposition 3.2 we have $\operatorname{Ext}^{i}_{R}(R/I,M)=0$,
for all integers $i\geq 0$. Therefore
$(0:_{H^{t}_{I}(R)}I)\simeq\operatorname{Hom}_{R}(R/I,H^{t}_{I}(R))=0$. Hence
$H^{t}_{I}(R)=0$, which is a contradiction.

(i)$\Rightarrow$(iv) Let $M$ be an $R$-module such that
$\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for $i=0,1$. Then, by [2, Theorem 2.9]
we have $H^{i}_{I}(M)=0$ for $i=0,1$. On the other hand by [6, Lemma 6.3.1] we
have $H^{i}_{I}(M)=0$, for all integers $i\geq 2$. Therefore,
$H^{i}_{I}(M)=0$, for all integers $i\geq 0$. Hence, it follows from [2,
Theorem 2.9] that $\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for all integers
$i\geq 0$.

(iv)$\Rightarrow$(i) By [6, Corollary 2.2.8] we have $H^{i}_{I}(D_{I}(R))=0$,
for $i=0,1$. Hence, $\operatorname{Ext}^{i}_{R}(R/I,D_{I}(R))=0$ for $i=0,1$
by [2, Theorem 2.9]. So, by the hypothesis we have
$\operatorname{Ext}^{i}_{R}(R/I,D_{I}(R))=0$, for all integers $i\geq 0$.
Hence, by Proposition 3.2 it follows that $R/I\otimes_{R}D_{I}(R)=0$ and so
$D_{I}(R)=ID_{I}(R)$. Now the assertion follows from [6, Lemma 6.3.1 and
Proposition 6.3.5].∎

######  Theorem 3.8.

Let $R$ be a Noetherian ring and let $I=Ra_{1}+Ra_{2}$ be an ideal of $R$.
Then the following conditions are equivalent:

  * (i)

${\rm cd}(I,R)=2$.

  * (ii)

There exists an $R$-module $M$, such that
$\operatorname{Tor}^{R}_{i}(R/I,M)=0$ for $i=0,1$ and
$\operatorname{Tor}^{R}_{2}(R/I,M)\neq 0$.

  * (iii)

There exists an $R$-module $M$, such that
$\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for $i=0,1$ and
$\operatorname{Ext}^{2}_{R}(R/I,M)\neq 0$.

###### Proof.

(i)$\Rightarrow$(ii) If ${\rm cd}(I,R)=2$, then for the $R$-module
$M:=H^{2}_{I}(R)$ by Corollary 3.5 we have
$\operatorname{Tor}^{R}_{i}(R/I,M)=0$ for $i=0,1$. Now, we claim that
$\operatorname{Tor}^{R}_{2}(R/I,M)\neq 0$. Assume the opposite. Then, it
follows from Proposition 3.6 that $\operatorname{Tor}^{R}_{i}(R/I,M)=0$, for
all integers $i\geq 0$. Then it follows from Proposition 3.2 that
$\operatorname{Ext}^{i}_{R}(R/I,M)=0$, for all integers $i\geq 0$. Therefore,
$\operatorname{Hom}_{R}(R/I,H^{2}_{I}(R))=0$, which implies that
$H^{2}_{I}(R)=0$. This is a contradiction.

(ii)$\Rightarrow$(i) Under the given hypothesis it follows from Theorem 3.7
that ${\rm cd}(I,R)\geq 2$. On the other hand, by [6, Theorem 3.3.1] we have
${\rm cd}(I,R)\leq 2$. So, we have ${\rm cd}(I,R)=2$.

(i)$\Rightarrow$(iii) If ${\rm cd}(I,R)=2$, then for the $R$-module
$M:=D_{I}(R)$, by [6, Corollary 2.2.8] we have $H^{i}_{I}(M)=0$, for $i=0,1$.
Hence, by [2, Theorem 2.9] we have $\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for
$i=0,1$. Moreover, by [6, Remark 2.2.7] there is an exact sequence

$0\longrightarrow R/\Gamma_{I}(R)\longrightarrow D_{I}(R)\longrightarrow
H^{1}_{I}(R)\longrightarrow 0,$

which induces the isomorphisms

$H^{2}_{I}(M)\simeq H^{2}_{I}(R/\Gamma_{I}(R))\simeq H^{2}_{I}(R)\neq 0.$

Now, if $\operatorname{Ext}^{2}_{R}(R/I,M)=0$, then
$\operatorname{Ext}^{i}_{R}(R/I,M)=0$, for $i=0,1,2,$. So, it follows from [2,
Theorem 2.9] that $H^{i}_{I}(M)=0$ for $i=0,1,2$, which is a contradiction,
because $H^{2}_{I}(M)\neq 0$. So, for the $R$-module $M:=D_{I}(R)$ we have
$\operatorname{Ext}^{i}_{R}(R/I,M)=0$ for $i=0,1$ and
$\operatorname{Ext}^{2}_{R}(R/I,M)\neq 0$.

(iii)$\Rightarrow$(i) Under the given hypothesis it follows from Theorem 3.7
that ${\rm cd}(I,R)\geq 2$. On the other hand, by [6, Theorem 3.3.1] we have
${\rm cd}(I,R)\leq 2$. So, we have ${\rm cd}(I,R)=2$.∎

######  Remark 3.9.

For a given proper ideal $I$ of a Noetherian ring $R$, there are some other
well known equivalent conditions for $\operatorname{cd}(I,R)\leq 1$. For
instance, see [6, Lemma 6.3.1 and Proposition 6.3.5] and for more properties
of such ideals see [4].

##  4\. Vanishing of the Bass and Betti numbers of local cohomology modules

In this section we prepare some vanishing conditions for some of the Bass and
Betti numbers of special local cohomology modules. Also, we give a formula for
the cohomological dimension of special ideals over Noetherian domains.

######  Theorem 4.1.

Let $R$ be a Noetherian ring and let $I\subseteq J$ be two ideals of $R$ such
that $\operatorname{cd}(J,R)=1$. Then the following statements hold:

  * (i)

For every $R$-module $M$ we have
$\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(M))=0$, for all integers $i\geq 2$
and $j\geq 0$.

  * (ii)

For every $R$-module $M$ and every finitely generated $R$-module $N$ with
$\operatorname{Supp}N\subseteq V(J)$ we have
$\operatorname{Ext}^{j}_{R}(N,H^{i}_{I}(M))=0$, for all integers $i\geq 2$ and
$j\geq 0$.

  * (iii)

For every $R$-module $M$ and each prime ideal $\operatorname{\mathfrak{p}}\in
V(J)$ we have $\mu^{j}(\operatorname{\mathfrak{p}},H^{i}_{I}(M))=0$, for all
integers $i\geq 2$ and $j\geq 0$. $($Here
$\mu^{j}(\operatorname{\mathfrak{p}},H^{i}_{I}(M))$ denotes the j-th Bass
number of the $R$-module $H^{i}_{I}(M)$ with respect to
$\operatorname{\mathfrak{p}}$$)$.

  * (iv)

For every $R$-module $M$ and each prime ideal $\operatorname{\mathfrak{p}}\in
V(J)$ we have $\beta_{j}(\operatorname{\mathfrak{p}},H^{i}_{I}(M))=0$, for all
integers $i\geq 2$ and $j\geq 0$. $($Here
$\beta_{j}(\operatorname{\mathfrak{p}},H^{i}_{I}(M))$ denotes the j-th Betti
number of the $R$-module $H^{i}_{I}(M)$ with respect to
$\operatorname{\mathfrak{p}}$$)$.

###### Proof.

(i) Let $M$ be an arbitrary $R$-module. In order to prove the assertion we may
assume $\operatorname{cd}(I,M)=t\geq 2$. By Corollary 3.5 we have
$\operatorname{Tor}^{R}_{j}(R/J,H^{t}_{I}(M))=0$ for $j=0,1$. Now, it follows
from Proposition 3.6 that $\operatorname{Tor}^{R}_{i}(R/J,H^{t}_{I}(M))=0$,
for all integers $i\geq 0$. Therefore, by Proposition 3.2 we have
$\operatorname{Ext}^{i}_{R}(R/J,H^{t}_{I}(M))=0$, for all integers $i\geq 0$.
Now, if $t\geq 3$ then by Lemma 3.4 we have
$\operatorname{Tor}^{R}_{j}(R/J,H^{t-1}_{I}(M))=0$ for $j=0,1$. So, it follows
from Proposition 3.6 that $\operatorname{Tor}^{R}_{i}(R/J,H^{t-1}_{I}(M))=0$,
for all integers $i\geq 0$. Therefore, by Proposition 3.2 we have
$\operatorname{Ext}^{i}_{R}(R/J,H^{t-1}_{I}(M))=0$, for all integers $i\geq
0$. Proceeding in the same way we see that
$\operatorname{Ext}^{j}_{R}(R/J,H^{i}_{I}(M))=0$, for all integers $i\geq 2$
and $j\geq 0$.

(ii) Using [1, Lemma 2.2] follows from (i).

(iii) Follows from (ii).

(iv) Follows from (ii) using Proposition 3.2. ∎

######  Theorem 4.2.

Let $R$ be a Noetherian domain and let $I$ and $J$ be two non-zero proper
ideals of $R$ such that $\operatorname{cd}(J,R)=1$ and
$J\not\subseteq\operatorname{Rad}(I)$. Then we have

$\operatorname{cd}(I\cap
J,R)=\max\\{i\in\mathbb{Z}\,\,:\,\,\operatorname{Supp}H^{i}_{I}(R)\not\subseteq
V(J)\\}.$

###### Proof.

Since by the hypothesis we have $J\not\subseteq\operatorname{Rad}(I)$ it
follows that

$J\not\subseteq\bigcap_{\operatorname{\mathfrak{p}}\in\operatorname{mAss}_{R}R/I}\operatorname{\mathfrak{p}},$

which implies that $J\not\subseteq\operatorname{\mathfrak{q}}$ for some
$\operatorname{\mathfrak{q}}\in\operatorname{mAss}_{R}R/I$. Assume that
$k:=0pt\operatorname{\mathfrak{q}}$. Then by Grothendieck’s Non-vanishing
Theorem we have

$0\neq
H^{k}_{\operatorname{\mathfrak{q}}R_{\operatorname{\mathfrak{q}}}}(R_{\operatorname{\mathfrak{q}}})=H^{k}_{IR_{\operatorname{\mathfrak{q}}}}(R_{\operatorname{\mathfrak{q}}})\simeq(H^{k}_{I}(R))_{\operatorname{\mathfrak{q}}},$

which implies that
$\operatorname{\mathfrak{q}}\in\operatorname{Supp}H^{k}_{I}(R)$. Hence we have
$\operatorname{Supp}H^{k}_{I}(R)\not\subseteq V(J)$ and so we have

$k\in\\{i\in\mathbb{Z}\,\,:\,\,\operatorname{Supp}H^{i}_{I}(R)\not\subseteq
V(J)\\}.$

In particular, we have
$\\{i\in\mathbb{Z}\,\,:\,\,\operatorname{Supp}H^{i}_{I}(R)\not\subseteq
V(J)\\}\neq\emptyset.$

Now, set
$\ell:=\max\\{i\in\mathbb{Z}\,\,:\,\,\operatorname{Supp}H^{i}_{I}(R)\not\subseteq
V(J)\\}$ and $t:=\operatorname{cd}(I\cap J,R)$. Then as by [6, Corollary
3.3.3] we have $\ell\leq{\rm ara}(I)$ and $t\leq{\rm ara}(I\cap J)$ it follows
that $0\leq\ell<\infty$ and $0\leq t<\infty$. By the Mayer-Vietoris exact
sequence for each integer $i>t$ we have the exact sequence

$H^{i}_{I+J}(R)\longrightarrow H^{i}_{I}(R)\oplus H^{i}_{J}(R)\longrightarrow
H^{i}_{I\cap J}(R),$

which gives the exact sequence

$H^{i}_{I+J}(R)\longrightarrow H^{i}_{I}(R)\oplus H^{i}_{J}(R)\longrightarrow
0$

and hence we have

$\operatorname{Supp}H^{i}_{I}(R)\subseteq\operatorname{Supp}H^{i}_{I+J}(R)\subseteq
V(I+J)\subseteq V(J).$

So, it is clear that $\ell\leq t$. On the other hand, since by the hypothesis
$R$ is a domain and $I\cap J\neq 0$, it follows that $\ell\geq 1$ and $t\geq
1$. Hence, if $t=1$, then it is clear that $\ell=1=t$. Now, assume that $t\geq
2$ and $\ell<t$. Then, we have $\operatorname{Supp}H^{t}_{I}(R)\subseteq
V(J)$. Also, by the Mayer-Vietoris exact sequence we have the exact sequence

$H^{t}_{I}(R)\oplus H^{t}_{J}(R)\longrightarrow H^{t}_{I\cap
J}(R)\longrightarrow H^{t+1}_{I+J}(R),$

which implies that

$\operatorname{Supp}H^{t}_{I\cap
J}(R)\subseteq[\operatorname{Supp}H^{t}_{J}(R)\cup\operatorname{Supp}H^{t}_{I}(R)\cup\operatorname{Supp}H^{t+1}_{I+J}(R)]\subseteq
V(J).$

So, the non-zero $R$-module $H^{t}_{I\cap J}(R)$ is $J$-torsion and hence we
have

$\operatorname{Hom}_{R}(R/J,H^{t}_{I\cap J}(R))\neq 0.$

But, by Theorem 4.1 we have

$\operatorname{Hom}_{R}(R/J,H^{t}_{I\cap J}(R))=0,$

which is a contradiction. So, we have $\ell=t$, whenever $t\geq 2$. ∎

######  Proposition 4.3.

Let $(R,\operatorname{\mathfrak{m}})$ be a Noetherian local ring and let $I$
and $J$ be two proper ideals of $R$ such that $\operatorname{cd}(J,R)=1$. Let
$k\geq 2$ be an integer such that
$\operatorname{Supp}H^{k}_{I}(R)\not\subseteq V(J)$. Then, the $R$-module
$H^{k}_{I\cap J}(R)$ is not $I\cap J$-cofinite. In particular, $H^{k}_{I\cap
J}(R)\neq 0$.

###### Proof.

By the Mayer-Vietoris exact sequence we have the exact sequence

$H^{k}_{I+J}(R)\longrightarrow H^{k}_{I}(R)\oplus H^{k}_{J}(R)\longrightarrow
H^{k}_{I\cap J}(R),$

which considering that fact that $\operatorname{Supp}H^{k}_{I+J}(R)\subseteq
V(I+J)\subseteq V(J)$ implies that $\operatorname{Supp}H^{k}_{I\cap
J}(R)\not\subseteq V(J).$ (Note that by the hypothesis we have
$\operatorname{Supp}H^{k}_{I}(R)\not\subseteq V(J)$.) In particular, we have
$H^{k}_{I\cap J}(R)\neq 0$. In order to prove the assertion, assume the
opposite and assume that $\dim\operatorname{Supp}H^{k}_{I\cap J}(R)=d$. Then,
in view of [20, Theorem 2.9] we have
$H^{d}_{\operatorname{\mathfrak{m}}}(H^{k}_{I\cap J}(R))\neq 0$. On the other
hand, by Theorem 4.1 we have
$\operatorname{Ext}^{j}_{R}(R/\operatorname{\mathfrak{m}},H^{k}_{I\cap
J}(R))=0$, for all integers $j\geq 0$. Hence it follows from [2, Theorem 2.9]
that $H^{j}_{\operatorname{\mathfrak{m}}}(H^{k}_{I\cap J}(R))=0$, for all
integers $j\geq 0$. Therefore, we have
$H^{d}_{\operatorname{\mathfrak{m}}}(H^{k}_{I\cap J}(R))=0$, which is a
contradiction. ∎

######  Remark 4.4.

There are examples of Noetherian local rings $(R,\operatorname{\mathfrak{m}})$
with proper ideals $I$, for which $\operatorname{cd}(I,R)=1$ and
$\operatorname{ara}(I)\geq 2$. For instance, the following example is given by
Hellus and Stückrad in [15].

######  Example 4.5.

Let $k$ be a field and let $S=k[[x,y,z,w]]$, where $x,y,z,w$ are independent
indeterminacies over $k$. Let $f=xw-yz$, $g=y^{3}-x^{2}z$ and
$h=z^{3}-w^{2}y$. Let $R=S/fS$ and $I=(f,g,h)S/fS$. Then $R$ is a Noetherian
local ring of dimension $3$ with maximal ideal
$\operatorname{\mathfrak{m}}=(x,y,z,w)S/fS$. Also, for the ideal $I$ of $R$,
we have $\operatorname{cd}(I,R)=1$ and $\operatorname{ara}(I)=2$. $($See [15,
Remark 2.1(ii)]$)$.

Acknowledgments

The authors are deeply grateful to the referee for a very careful reading of
the manuscript and many valuable suggestions. Also, we would like to thank
Professors Reza Naghipour and Kamran Divaani-Aazar for their careful reading
of the first draft and many helpful suggestions.

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# Diffusion Models Beat GANs on Topology Optimization

François Mazé1, Faez Ahmed1

###### Abstract

Structural topology optimization, which aims to find the optimal physical
structure that maximizes mechanical performance, is vital in engineering
design applications in aerospace, mechanical, and civil engineering.
Generative adversarial networks (GANs) have recently emerged as a popular
alternative to traditional iterative topology optimization methods. However,
these models are often difficult to train, have limited generalizability, and
due to their goal of mimicking optimal structures, neglect manufacturability
and performance objectives like mechanical compliance. We propose TopoDiff — a
conditional diffusion-model-based architecture to perform performance-aware
and manufacturability-aware topology optimization that overcomes these issues.
Our model introduces a surrogate model-based guidance strategy that actively
favors structures with low compliance and good manufacturability. Our method
significantly outperforms a state-of-art conditional GAN by reducing the
average error on physical performance by a factor of eight and by producing
eleven times fewer infeasible samples. By introducing diffusion models to
topology optimization, we show that conditional diffusion models have the
ability to outperform GANs in engineering design synthesis applications too.
Our work also suggests a general framework for engineering optimization
problems using diffusion models and external performance with constraint-aware
guidance. We publicly share the data, code, and trained models here:
https://decode.mit.edu/projects/topodiff/.

##  1 Introduction

Structural topology optimization (TO) of solid structures involves generating
the optimal shape of a material by minimizing an objective function, for
instance, mechanical compliance, within a given domain and under a given set
of constraints (volume fraction, boundary conditions, and loads). TO is
therefore becoming an essential design tool and is now included in most
professional design software, such as Autodesk’s Fusion 360 and Solidworks. It
is the driving force behind Autodesk’s generative design toolset, where
designers input design goals into the software, along with parameters such as
performance or spatial requirements, materials, manufacturing methods, and
cost constraints. The software quickly generates design alternatives. Most
methods to solve TO rely on gradient-based approaches, the most common method
being the Solid Isotropic Material with Penalization method (Bendsøe and
Kikuchi 1988; Rozvany, Zhou, and Birker 1992). Despite their wide adoption,
these traditional methods have two major pitfalls: their iterative nature
makes them computationally expensive and they may generate non-optimal
designs, for example, when penalization and filtering augmentations are used
to avoid grayscale pixels in SIMP (Sigmund and Maute 2013).

Several deep learning methods have been developed in recent years to improve
and speed up the TO process (Yu et al. 2018; Sharpe and Seepersad 2019; Nie et
al. 2021; Behzadi and Ilieş 2021) by learning from large datasets of optimized
structures. The latest and most promising results were obtained with deep
generative models (DGMs) and notably with conditional generative adversarial
networks (cGANs) trained for image synthesis, which take as input the boundary
conditions and directly generate images of optimized structures. Although
popular, most of these models optimize a loss function which does not align
with the primary goals of topology optimization — getting high-performance and
feasible structures. They often train for loss functions related to image
reconstruction to achieve visual similarity and ignore modeling the physical
performance of the generated structures. Most of them produce disconnected,
floating material that seriously affects the manufacturability of the
generated design. They also suffer from limited generalizability, especially
for out-of-distribution boundary conditions.

We hypothesize that the absence of explicit methods to generate designs with
low compliance and good manufacturability causes these issues. We further
hypothesize that the reliance of the optimization objective on the sole cGAN
prompts the model to only mimic pixel-wise the ground truth produced by
traditional TO methods. As a result, two images with comparable pixel-wise
similarity may still have significantly different performance values. The
absence of explicit external guidance is even more problematic since the
ground truth data is not guaranteed to be optimal, as explained above.

This paper introduces TopoDiff, a conditional diffusion-model-based method for
TO. Dhariwal and Nichol (2021) have shown that diffusion models can outperform
GANs for image generation, are easier to train, and are thus more readily
adaptable to other tasks. We show that by introducing performance and
constraints to diffusion models, they outperform GANs on topology optimization
problems too. In addition, the sequential nature of diffusion models makes
them compatible with external guidance strategies that assist with performance
and feasibility goals. By creating surrogate models to estimate performance,
we thus introduce external guidance strategies to minimize mechanical
compliance and improve manufacturability in diffusion models.

Our main contributions include proposing: (1) TopoDiff — a diffusion model
based end-to-end Topology Optimization framework that achieves an eight-times
reduction in average physical performance errors and an eleven-times reduction
in infeasibility compared to a state-of-art conditional GAN, (2) a new
guidance strategy for diffusion models to perform physical performance
optimization with enhanced manufacturability constraint satisfaction, and (3)
a generalized framework to solve inverse problems in engineering using
diffusion models, when sample feasibility and performance are a high priority.

##  2 Background and Related Work

###  2.1 Topology Optimization

Topology Optimization (TO) finds an optimal subset of material $\Omega_{opt}$
included in the full design domain $\Omega$ under a set of displacement
boundary conditions and loads applied on the nodes of the domain and a volume
fraction condition. A structure is optimal when it minimizes an objective
function, such as mechanical compliance, subject to constraints. Fig. 1
summarizes the principle of TO.

Figure 1: Topology Optimization aims to find the optimal structure that
minimizes objectives such as compliance for a given set of load, boundary
conditions, and volume fraction.

Traditional TO methods rely on Finite Elements Analysis (FEA) using gradient-
based (Bendsøe and Kikuchi 1988) or gradient-free methods (Ahmed,
Bhattacharya, and Deb 2013). One popular gradient-based method is the Solid
Isotropic Material with Penalization (SIMP) method (Rozvany, Zhou, and Birker
1992).

SIMP associates every element of the mesh with a continuous density to perform
gradient-based methods (Sigmund 2001). However, because intermediary densities
make no physical sense, SIMP uses a penalization factor to encourage binary
densities.

This penalization strategy (with $p>1$) is efficient but introduces non-
convexity in the objective function, as stated by Sigmund and Maute (2013). As
a result, SIMP is likely to converge towards local optima. Other techniques to
encourage binary densities include filters, but they also introduce non-
convexity (Sigmund and Maute 2013).

###  2.2 Deep Learning for Topology Optimization

Traditional TO methods are often slow due to the iterative FEA steps they
include (Amir and Sigmund 2011). Many deep learning methods (Regenwetter,
Nobari, and Ahmed 2022; Guo et al. 2018; Lin et al. 2018; Sosnovik and
Oseledets 2019) have recently been developed to improve the speed and quality
of topology generation or address issues such as non-convexity.

A group of deep learning approaches, to which our work belongs, consists in
proposing an end-to-end TO framework from the constraints to the optimal
topology (Oh et al. 2019; Sharpe and Seepersad 2019; Chandrasekhar and Suresh
2021; Parrott, Abueidda, and James 2022). Below, we review a few
representative works. Yu et al. (2018) propose an iteration-free method, which
predicts a low-resolution solution with a CNN encoder-decoder, which is then
passed into a GAN to increase the resolution. In line with the work by Rawat
and Shen (2019), Li et al. (2019) use two GANs to solve the TO problem and
then predict the refined structure at high resolution. Sharpe and Seepersad
(2019) introduce conditional GANs as a means of generating a compact latent
representation of structures resulting from TO. Their work is improved by Nie
et al. (2021), who greatly extend the generalizability in their model named
TopologyGAN, which is trained on more diverse conditions and uses physical
fields as input to represent loads and boundary conditions. In parallel, Wang
et al. (2021) develop a U-Net to perform TO for improved generalization. These
promising models nevertheless show a limited generalization ability, notably
regarding boundary conditions outside the training distribution. They are also
both prone to the problem of disconnected material. To solve this issue,
Behzadi and Ilieş (2021) propose a conditional GAN architecture that includes
a topological measure of connectivity in its loss function. Their results seem
to improve generalizability and connectivity; however, they set the volume
fraction to a constant value, limiting the problem’s scope.

It is crucial to note that none of these methods explicitly include a process
to minimize compliance, which is the goal of TO. The minimization of
compliance is expected indirectly through the GAN training, which is
challenging to control. As incorporating some measure of predicted structural
performance inside a conditional model seems necessary, we propose explicit
guidance methods in diffusion models for low-compliance and good-feasibility
structures.

###  2.3 Diffusion Models

Diffusion models are a new type of deep generative models (DGMs) introduced by
Sohl-Dickstein et al. (2015). They have received much attention recently
because Dhariwal and Nichol (2021) showed that diffusion models outperform
GANs for image synthesis. Diffusion models are increasingly being applied to
various fields: image generation (Nichol and Dhariwal 2021), segmentation
(Amit et al. 2021), image editing (Meng et al. 2021), text-to-image (Nichol et
al. 2022; Kim and Ye 2021), etc.

The idea behind diffusion models is to train a neural network to reverse a
noising process that maps the data distribution to a white noise distribution.
The forward noising process, which is fixed, consists of progressively adding
noise to the samples following the Markov chain:

$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{\alpha_{t}}x_{t-1},(1-\alpha_{t})I)$
(1)

where $(\alpha_{t})_{t=1}^{T}$ is a variance schedule.

To reverse this noising process, we approximate the true posterior with the
parametric Gaussian process:

$p_{\theta}(x_{t-1}|x_{t})=\mathcal{N}(\mu_{\theta}(x_{t}),\Sigma_{\theta}(x_{t})).$
(2)

We then generate new data by sampling an image from $\mathcal{N}(0,I)$ and
gradually denoising it using Eq. 2.

Training a diffusion model, therefore, involves training two neural networks,
$\mu_{\theta}(x_{t})$ and $\Sigma_{\theta}(x_{t})$, to predict the mean and
the variance of the denoising process respectively. Let us note that Ho, Jain,
and Abbeel (2020) showed that $\Sigma_{\theta}(x_{t})$ might be fixed to a
constant instead of being learned.

###  2.4 Guidance Methods in Diffusion Models

In many machine learning applications, a model is expected to generate samples
conditioned on some input conditions. For example, popular text-to-image
models such as DALL-E are conditioned on text input. Researchers have
developed a few guidance methods to perform conditional image generation, such
as including class labels when the model tries to generate an image
corresponding to a given class.

#### Including conditioning information inside the denoising networks

A method to condition a diffusion model consists in adding the conditioning
information (for example, a class label) as an extra input to the denoising
networks $\mu_{\theta}$ and $\Sigma_{\theta}$. In practice, the conditioning
information can be added as an extra channel to the input image. Similarly,
Dhariwal and Nichol (2021) suggest adding conditioning information into an
adaptive group normalization layer in every residual block.

#### Classifier guidance

Additional methods have been developed to guide the denoising process using
classifier output. In line with Sohl-Dickstein et al. (2015) and Song et al.
(2020), who have used external classifiers to guide the denoising process,
Dhariwal and Nichol (2021) introduce classifier guidance to perform class-
conditional image generation. In classifier guidance, a separate classifier is
trained on noisy data (with different levels of noise) to predict the
probability $p_{\phi}(y|x_{t})$ that an image $x_{t}$ at noise level $t$
corresponds to the class $y$. Let $p_{\theta}(x_{t}|x_{t+1})$ be an
unconditional reverse noising process. Classifier guidance consists of
sampling from:

$p_{\theta,\phi}(x_{t}|x_{t+1},y)=Zp_{\theta}(x_{t}|x_{t+1})p_{\phi}(y|x_{t})$
(3)

instead of $p_{\theta}(x_{t}|x_{t+1})$, where $Z$ denotes a normalizing
constant. Under reasonable assumptions, Dhariwal and Nichol (2021) show that
sampling from $p_{\theta,\phi}(x_{t}|x_{t+1},y)$ is equivalent to perturbing
the mean with the gradient of the probability predicted by the classifier.
Specifically, the perturbed mean is:

$\hat{\mu_{\theta}}(x_{t})=\mu_{\theta}(x_{t})+s\Sigma_{\theta}(x_{t})\nabla_{x_{t}}\log
p_{\phi}(y|x_{t})$ (4)

where $s$ is a scale hyperparameter that needs to be tuned. A variant called
classifier-free guidance is proposed by Ho and Salimans (2021). This technique
is theoretically close to classifier guidance but does not require training a
separate classifier on noisy data.

However, none of these methods provide guidance for both continuous values
(such as performance obtained from regression models) and discrete values
(such as class labels obtained from classification models), which is important
for TO to achieve feasible, high-performing samples. To overcome these issues,
we propose a regressor and classifier guidance strategy that penalizes low-
compliance and infeasible structures at every step.

##  3 Method

###  3.1 Architecture and General Pipeline

TopoDiff’s diffusion architecture consists of a UNet (Ronneberger, Fischer,
and Brox 2015)-based denoiser at every step with attention layers. We add
conditioning to this architecture by including information on constraints and
boundary conditions as additional channels to the input image given to the
denoiser, as shown in Figure 2. The UNet model uses these extra channels as
additional information to denoise the first channel of the input in a way that
respects the constraints and is optimal for the given boundary conditions.
Similarly to TopologyGAN (Nie et al. 2021), we use physical fields, namely
strain energy density and von Mises stress, to represent constraints and
boundary conditions. The physical fields are computed using a finite element
method (Guarín-Zapata and Gómez 2020) and help avoid the sparsity problem
caused by raw constraints and boundary condition matrices. The final input to
our conditional diffusion model has four channels representing the volume
fraction, the strain energy density, the von Mises stress, and the loads
applied to the domain’s boundary.

Figure 2: TopoDiff: Proposed constrained guided conditional diffusion model
architecture for TO. $(x_{t})_{t=0,\cdots,T}$ is the gradually denoised
topology; $g_{c}$ and $g_{fm}$ are the guidance gradients; $l$ is the load
applied, represented with an arrow on the topology; $v$ is the volume
fraction; $f$ are the physical fields, and $bc$ are the boundary conditions,
represented with color lines and dots.

###  3.2 Minimizing compliance

Most deep learning models used for TO rely on minimizing the pixel-wise error
between the output topology and the ground truth obtained with traditional
methods. For instance, the reference model TopologyGAN (Nie et al. 2021)
attempts to mimic the ground truth topology and is encouraged to do so by the
L2 loss function of its generator. GANs for TO are often evaluated using mean
absolute error (MAE) between the ground truth topology and the topology
predicted by their model. However, we hypothesize that setting the
minimization of a pixel-wise error as an objective does not properly address
the aim of TO: generate manufacturable structures that minimize mechanical
compliance. We pose this hypothesis for two main reasons:

  1. 1.

The topology used as ground truth may be sub-optimal due to penalization
factor and filters (Sec. 2.1);

  2. 2.

A small pixel-wise error is compatible with a large compliance error if the
material is missing at critical places.

Without additional guidance, our conditional diffusion model is prone to the
same problem. To solve that issue, we introduce a new type of guidance called
regressor guidance, inspired by the classifier guidance used by Dhariwal and
Nichol (2021).

Consider a conditional diffusion model as presented in Sec. 3.1:
$p_{\theta}(x_{t}|x_{t+1},v,f,l)$, where $v$ is the volume fraction, $f$ are
the physical fields (strain energy density and von Mises stress) and $l$ are
the loads applied. Regressor guidance consists in sampling each transition
according to:

$p_{\theta,\phi}(x_{t}|x_{t+1},v,f,l,bc)=\\\
Zp_{\theta}(x_{t}|x_{t+1},v,f,l)e^{-c_{\phi}(x_{t},v,f,l,bc)}$ (5)

where $c_{\phi}$ is a surrogate neural network predicting the compliance of
the topology under given constraints, $bc$ are the boundary conditions
applied, and $Z$ is a normalizing constant. It is worth noting that $c_{\phi}$
must be able to predict compliance on noisy images of structures. To perform
this task, we use the encoder of a UNet architecture modified for regression
values.

We show in Sec. 3.5 that under simple assumptions, adding regressor guidance
simply amounts to shifting the mean predicted by the diffusion model by
$-\Sigma\nabla_{x_{t}}c_{\phi}(x_{t},v,f,l,bc)$ where $\Sigma$ is the variance
of the Gaussian distribution representing $p_{\theta}(x_{t}|x_{t+1},v,f,l)$.
This method thus modifies the distribution according to that which we sample
at each step by penalizing structures with high compliance. The resulting
algorithm is Alg. 1.

Algorithm 1 Regressor guidance for TO, given a conditional diffusion model
$(\mu_{\theta}(x_{t}|x_{t+1},v,f,l),\Sigma_{\theta}(x_{t}|x_{t+1},v,f,l))$ and
a regressor $c_{\phi}(x_{t},v,f,l,bc)$.

$v,l,bc$ $\triangleright$ Volume, loads and boundary conditions

$f$ $\triangleright$ Physical fields

$\lambda_{c}$ $\triangleright$ Regressor gradient scale

$x_{T}\leftarrow\textrm{sample from }\mathcal{N}(0,\,I)$

for $t$ from $T$ to 1 do

$\mu,\Sigma\leftarrow\mu_{\theta}(x_{t}|x_{t+1},v,f,l),\Sigma_{\theta}(x_{t}|x_{t+1},v,f,l)$

$x_{t-1}\leftarrow\textrm{sample from
}\mathcal{N}(\mu-\lambda_{c}\Sigma\nabla_{x_{t}}c_{\phi}(x_{t},v,f,l,bc)|_{x_{t}=\mu},\,\Sigma)$
return $x_{0}$

###  3.3 Avoiding Floating Material

Disconnected pixels in predicted structures are a serious problem because this
phenomenon leads to floating material and therefore affects the
manufacturability of the predicted topology. This problem is generally ignored
in deep learning models for TO, and is notably not considered by the pixel-
wise error because a small pixel-wise error is compatible with the presence of
floating material.

Similar to what has been exposed in Sec. 3.2, we further modify the sampling
distribution at each step by penalizing structures that contain floating
material. To do so, we train a classifier $p_{\gamma}$ that returns the
probability that the topology does _not_ contains floating material. We then
use this classifier to perform classifier guidance, as introduced in Sec. 2.4.
Eventually, this amounts to shifting the mean predicted by the diffusion model
by $+\Sigma\nabla_{x_{t}}\log p_{\gamma}(x_{t})$.

###  3.4 Combining Guidance Strategies

Our model ultimately consists of one conditional diffusion model
$p_{\theta}(x_{t}|x_{t+1},v,f,l)$ and two surrogate models used for guidance
when sampling: $c_{\phi}(x_{t},v,f,l,bc)$ for compliance and
$p_{\gamma}(x_{t})$ for floating material.

One challenge is to find a way to combine these two guidance strategies. To
combine them, we sample at every step according to:

$p_{\theta,\phi,\gamma}(x_{t}|x_{t+1},v,f,l,bc)=\\\
Zp_{\theta}(x_{t}|x_{t+1},v,f,l)e^{-c_{\phi}(x_{t},v,f,l,bc)}p_{\gamma}(x_{t}).$
(6)

This amounts to shifting the mean predicted by the diffusion model by:

$-\lambda_{c}\Sigma\nabla_{x_{t}}c_{\phi}(x_{t},v,f,l,bc)+\lambda_{fm}\Sigma\nabla_{x_{t}}\log
p_{\gamma}(x_{t})$ (7)

where $\lambda_{c}$ and $\lambda_{fm}$ are gradient scale hyperparameters.

However, as is, this approach has two pitfalls:  1. The gradients are always
computed at the same point $\mu$ (the mean predicted by the diffusion model),
even though this mean is shifted by the previous guidance strategy; and  2.
The gradients are computed at every denoising step, even if we might want to
favor one guidance over the other at a given denoising step.  We modify the
point at which the second gradient is computed to address these issues by
considering the shift induced by the previous gradient. In addition, we
determine a maximum level of noise (MLN) beyond which the classifier/regressor
guidance should not be included for every classifier and regressor. We then
introduce classifier/regressor guidance only if the image is denoised enough
to have a noise level below the MLN of the given classifier or regressor.

The final guidance algorithm resulting from the combination of these guidance
strategies is Alg. 2. Fig. 2 also summarizes the overall architecture.

Algorithm 2 Guidance strategy for TO using Conditional Diffusion Model.

$v,l,bc$ $\triangleright$ Volume, loads and boundary conditions

$f$ $\triangleright$ Physical fields

$\lambda_{c},\lambda_{fm}$ $\triangleright$ Regressor and classifier gradient
scale

$MLN_{c},MLN_{fm}$ $\triangleright$ Maximum levels of noise

$x_{T}\leftarrow\textrm{sample from }\mathcal{N}(0,\,I)$

for $t$ from $T$ to 1 do

$\mu,\Sigma\leftarrow\mu_{\theta}(x_{t}|x_{t+1},v,f,l),\Sigma_{\theta}(x_{t}|x_{t+1},v,f,l)$

if $t<MLN_{fm}$ then

$\mu\leftarrow\mu+\lambda_{fm}\Sigma\nabla_{x_{t}}\log
p_{\gamma}(x_{t})|_{x_{t}=\mu}$

if $t<MLN_{c}$ then

$\mu\leftarrow\mu-\lambda_{c}\Sigma\nabla_{x_{t}}c_{\phi}(x_{t},v,f,l,bc)|_{x_{t}=\mu}$

$x_{t-1}\leftarrow\textrm{sample from }\mathcal{N}(\mu,\,\Sigma)$ return
$x_{0}$

It should be noted that we also considered applying regressor and classifier
guidance for other constraints (volume, load position), but the conditional
diffusion model already sufficiently respects these constraints and makes
guidance unnecessary.

###  3.5 Mathematical Motivations for Regressor Guidance

Similarly to Dhariwal and Nichol (2021) about classifier guidance, we show in
this section the mathematical motivations behind regressor guidance, and in
particular, we prove that adding regressor guidance, i.e., sampling each
transition according to the modified distribution, amounts to a shift in the
mean predicted by the diffusion model.

Let $p_{\theta}(x_{t}|x_{t+1},v,f,l)$ be our conditional diffusion model.
Regressor guidance consists in sampling according to:

$p_{\theta,\phi}(x_{t}|x_{t+1},v,f,l,bc)=\\\
Zp_{\theta}(x_{t}|x_{t+1},v,f,l)e^{-c_{\phi}(x_{t},v,f,l,bc)}$ (8)

where $Z$ is a normalizing constant and all variables are the ones defined in
Sec. 3.2.

Let $\mu$ and $\Sigma$ be the mean and variance of the Gaussian distribution
representing $p_{\theta}(x_{t}|x_{t+1},v,f,l)$.

$\log p_{\theta}(x_{t}|x_{t+1},v,f,l)=\\\
-\frac{1}{2}(x_{t}-\mu)^{T}\Sigma^{-1}(x_{t}-\mu)+C$ (9)

where $C$ is a constant.

By doing a Taylor expansion of the regressor predicting the compliance
$c_{\phi}(x_{t},v,f,l,bc)$, we obtain:

$c_{\phi}(x_{t},v,f,l,bc)\approx c_{\phi}(\mu,v,f,l,bc)\\\
+(x_{t}-\mu)\nabla_{x_{t}}c_{\phi}(x_{t},v,f,l,bc)|_{x_{t}=\mu}.$ (10)

In Eq. 10, we neglect the terms of second order and above because we make the
assumption that the curvature of $c_{\phi}(x_{t})$ is low compared to
$\Sigma^{-1}$, to which it will be summed in Eq. 11. This assumption is
reasonable in the limit of infinite diffusion steps, where
$||\Sigma||\rightarrow 0$, as stated by Dhariwal and Nichol (2021).

Hence, Eq. 10 can be rewritten
$c_{\phi}(x_{t},v,f,l,bc)\approx(x_{t}-\mu)g_{c}+D$, where $g_{c}$ is the
gradient of the compliance regressor evaluated in $\mu$ and $D$ is a constant.

This gives:

$\log(p_{\theta}(x_{t}|x_{t+1},v,f,l)e^{-c_{\phi}(x_{t},v,f,l,bc)})\approx\\\
-\frac{1}{2}(x_{t}-\mu)^{T}\Sigma^{-1}(x_{t}-\mu)-(x_{t}-\mu)g_{c}+C+D.$ (11)

Hence:

$\log(p_{\theta}(x_{t}|x_{t+1},v,f,l)e^{-c_{\phi}(x_{t},v,f,l,bc)})\approx\\\
-\frac{1}{2}(x_{t}-\mu+\Sigma g_{c})^{T}\Sigma^{-1}(x_{t}-\mu+\Sigma
g_{c})+\frac{1}{2}g_{c}^{T}\Sigma g_{c}+C+D.$ (12)

The last three terms of Eq. 12 are all constant and are encapsulated in the
normalizing constant $Z$ from Eq. 8. Therefore, we have shown that
$p_{\theta,\phi}(x_{t}|x_{t+1},v,f,l,bc)$ can be approximated by a Gaussian
with a mean shifted by $-\Sigma g_{c}$.

##  4 Empirical Evaluation

We created three datasets to train the proposed models, which we make public
to provide a standard benchmark for future research in this area.

###  4.1 Dataset

The main dataset used to train, validate and test TopoDiff consists of 33000
64x64 2D images corresponding to optimal structures for diverse input
conditions. Every data sample contains six channels:

  1. 1.

The first channel is the black and white image representing the optimal
topology;

  2. 2.

The second channel is uniform and includes the prescribed volume fraction;

  3. 3.

The third channel is the von Mises stress of the full domain under the given
load constraints and boundary conditions, defined as
$\sigma_{vm}=\sqrt{\sigma_{11}^{2}-\sigma_{11}\sigma_{22}+\sigma_{22}^{2}+3\sigma_{12}^{2}}$;

  4. 4.

The fourth channel is the strain energy density of the full domain under the
given load constraints and boundary conditions, defined as
$W=\frac{1}{2}(\sigma_{11}\epsilon_{11}+\sigma_{22}\epsilon_{22}+2\sigma_{12}\epsilon_{12})$;

  5. 5.

The fifth channel represents the load constraints in the x-direction. Every
node is given the value of the force applied in the x-direction on this load
($0$ if no force is applied on the load);

  6. 6.

The sixth channel similarly represents the load constraints in the
y-direction;

where $(\sigma_{11},\sigma_{22},\sigma_{12})$ and
$(\epsilon_{11},\epsilon_{22},\epsilon_{12})$ are respectively the components
of the stress and strain fields.

We randomly selected a combination of conditions (volume fraction, boundary
conditions, loads) to generate every structure and then computed the optimal
topology using the SIMP-based TO library ToPy (Hunter et al. 2017). We defined
the possible conditions in a similar way to what was done in previous works,
namely:  1. The volume fraction is chosen in the interval $[0.3,0.5]$, with a
step of $0.02$;  2. The displacement boundary conditions are chosen among 42
scenarios for training and five additional scenarios only used for testing;
3. The loads are applied on unconstrained nodes randomly selected on the
domain’s boundary. The direction is selected in the interval $[0,\pi]$, with a
step of $\frac{\pi}{6}$.

The main dataset is divided into training, validation, and testing as follows:

  1. 1.

The training data consist of 30,000 combinations of constraints containing 42
of the 47 boundary conditions;

  2. 2.

The validation data consist of 200 new combinations of constraints containing
the same 42 boundary conditions;

  3. 3.

The _level 1_ test data consist of 1800 new combinations of constraints
containing the same 42 boundary conditions;

  4. 4.

The _level 2_ test data consist of 1000 new combinations of constraints
containing five out-of-distribution boundary conditions.

In all test data, the combination of constraints is unseen. While level 1
dataset contains boundary conditions that are also in the training data, we
introduce more difficult conditions in level 2 to rigorously compare the
TopoDiff model’s generalization ability with existing methods. In addition to
the main dataset, two other datasets consisting of 12,000 and 30,000 non-
optimal structures are used to train regressor and classifier guidance models.

###  4.2 Evaluation Metrics

Using relevant evaluation metrics is vital for every scientific study. It is
particularly critical for mechanical design generation because most metrics
used in DGMs do not correspond to the physical objective one wants a design to
achieve. In this work, contrary to most generative models applied to TO in
previous works, we do not use pixel-wise error as a final evaluation metric
because it does not guarantee low compliance, which is the objective we are
trying to achieve.

Hence, we define and use four evaluation metrics that reflect the compliance
minimization objective, as well as the constraints that the generated
structures have to respect:

  1. 1.

Compliance error (CE) relative to the ground truth, defined as:
$CE={(C(\hat{y})-C(y))}/{C(y)}$ where $C(y)$ and $C(\hat{y})$ are,
respectively, the compliance of the SIMP-generated topology and the topology
generated by our diffusion model. It should be noted that a negative
compliance error means that our model returns a topology with lower compliance
than the ground truth;

  2. 2.

Volume fraction error (VFE) relative to the input volume fraction, defined as:
$VFE=|VF(\hat{y})-VF(y)|/VF(y)$ where $VF(y)$ and $VF(\hat{y})$ are,
respectively, the prescribed volume fraction and the volume fraction of the
topology generated by our diffusion model;

  3. 3.

Load violation (LV), defined as a boolean that is 1 if there is no material at
a place where a load is applied and 0 if there is always material where loads
are applied;

  4. 4.

Presence of floating material (FM), defined as a boolean that is 1 if the
topology contains floating material and 0 otherwise.

A model which generates samples with high scores on these metrics is expected
to yield high-performance manufacturable designs.

###  4.3 Choice of hyperparameters

One of the most crucial hyperparameters is the gradient scales in our guidance
strategies. These parameters quantify the relative importance of compliance
minimization and floating material avoidance. As explained in Sec. 4.1, a
validation dataset of 200 structures was used to perform hyperparameter
tuning. We used a grid search method to decide the hyperparameters using
compliance error and floating material presence as evaluation metrics.
Topology generation and FEA were used to evaluate the results.

##  5 Results and Discussions

| Level 1 test data | Level 2 test data
---|---|---
Model | TopologyGAN | Unguided TopoDiff | Guided TopoDiff | TopologyGAN | Unguided TopoDiff | Guided TopoDiff
Average Compliance Error (%) | 48.51 $\pm$ 16.38 |  4.10 $\pm$ 0.88 | 4.39 $\pm$ 0.94 | 143.08 $\pm$ 38.50 | 22.13 $\pm$ 8.52 |  18.40 $\pm$ 5.88
Median Compliance Error (%) | 2.06 | 0.80 | 0.83 | 6.82 | 1.88 | 1.82
Proportion of Compliance Error >30% (%) | 10.11 | 2.33 | 2.56 | 24.10 | 8.20 | 8.10
Average Volume Fraction Error (%) | 11.87 $\pm$ 0.52 | 1.86 $\pm$ 0.03 |  1.85 $\pm$ 0.03 | 14.31 $\pm$ 0.75 | 1.81 $\pm$ 0.04 |  1.80 $\pm$ 0.04
Proportion of Load Violation (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
Proportion of Floating Material (%) | 46.78 | 6.64 | 5.54 | 67.90 | 7.53 | 6.21

Table 1: Comparison of performance between TopologyGAN and TopoDiff (guided
and not guided) on the two level test sets. Values after $\pm$ indicate the 95
% confidence interval around averages. The values in bold are the best ones
for each level.

###  5.1 Evaluation of the full diffusion model

To evaluate the performance, we use the two test sets described in Sec. 4.1,
corresponding to two difficulty levels. We run every test nine times and then
compute the results’ average. We compare the performance of our model on all
evaluation metrics (Sec. 4.2) to a state-of-art cGAN model, named TopologyGAN
(Nie et al. 2021), which performs the same task as our model.

Figure 3: Comparison of generated structures on randomly selected samples from
both test datasets. GT stands for ground truth, CE is the compliance error
relative to the GT, and FM indicates the presence or not of floating material.

Fig. 3 shows examples of a few structures obtained with the SIMP method
(ground truth), with TopologyGAN, and with TopoDiff for randomly selected
constraints from level 1 and level 2 test sets. Qualitatively, we notice that
TopologyGAN tries to mimic pixel-wise the topology obtained from SIMP but
neglects both the compliance and the manufacturability of the generated
structures, which almost all have some floating material and high compliance
error. The ten structures generated by TopoDiff, on the other hand, may
visually differ more from the SIMP results but have better physical properties
than TopologyGAN. Only one of the TopoDiff-generated structures have no
floating material, and all ten outperform the TopologyGAN structures in terms
of compliance error. To confirm these qualitative observations, Table 1
summarizes the performance of the structures obtained with all test sets.
TopoDiff outperforms TopologyGAN on all the metrics.

On the level 1 test set, TopoDiff notably reduces the average CE by a factor
of eleven and the proportion of FM by more than a factor of eight. The
proportion of non-manufacturable designs thus drops from 46.8% with cGAN to
5.5% with TopoDiff. It also significantly reduces the average VFE from 11.9%
to 1.9%.

On the level 2 test set, TopoDiff achieves demonstrates strong
generalizability performance. It performs an eight-times reduction in the
average CE, from 143.1% to 18.4%, and a four-times reduction in the median CE.
Non-manufacturability drops from 67.9% to 6.2%, while the VFE is reduced by a
factor of eight, from 14% to less than 2%. A paired one-tailed t-test confirms
a reduction of the average CE and of the average VFE with a p-value of $9\cdot
10^{-12}$ and $5\cdot 10^{-160}$ respectively. These results show the efficacy
of diffusion models in learning to generate high-performing and manufacturable
structures for a wide set of testing conditions.

###  5.2 Efficiency of guidance strategy

#### Surrogate models

Guidance can only work if the regressors and classifiers can perform well on
the challenging task of predicting compliance and floating material for noisy
images. Table 2 shows the compliance regressor and floating material
classifier performance according to the noise level. These results show that
both surrogate models are very reliable on low-noise structures, and as
expected, their performance decreases with an increase in noise.

| 0-25% noise | 25-75% noise | 75-100% noise | Global
---|---|---|---|---
Regressor R2 (%) | 82.4 | 82.4 | 61.8 | 77.3
Classifier accuracy (%) | 98.8 | 76.8 | 54.6 | 76.8

Table 2: Performance (R2-score and accuracy) of both surrogate models on
validation data with respect to noise level.

#### Ablation study

To evaluate the impact of our guidance strategy on the performance of
TopoDiff, we also tested it without guidance on the two test sets, as shown in
Table 1.

With _in-distribution_ boundary conditions (level 1), our guidance strategy
has no significant impact on compliance error (both on average and median). A
two-tailed paired t-test does not reject the null hypothesis ($p=0.1$). We
believe that this happens because the diffusion model has implicitly learned
to perform well with these boundary conditions and does not need explicit
compliance guidance. In contrast, our guidance strategy significantly impacts
the proportion of floating material, with decreases from $6.6\%$ to $5.5\%$.

With _out-of-distribution_ boundary conditions (level 2), the positive impact
of our guidance strategy is evident both on the average compliance error and
on the proportion of floating material. A paired one-tailed t-test confirms a
reduction of the average CE with a p-value of $0.05$. The average compliance
error is reduced by $17\%$ and the average proportion of floating material by
$18\%$. As expected, guidance seems to have no effect on load respect and on
volume fraction error. More interestingly, guidance seems to have no
significant effect on the median of the compliance error, which suggests that
compliance regressor guidance primarily reduces the number of structures with
very high compliance errors.

###  5.3 Limitations and future work

TopoDiff shows good performance and good generalization to out-of-distribution
boundary conditions. The proposed guidance strategy is beneficial to
minimizing compliance and ensuring constraints, such as manufacturability.
However, several challenges still need to be addressed. The most significant
limitation is the computation time, diffusion models being slower than GANs.
It takes 0.06 seconds for TopologyGAN to generate one topology, while TopoDiff
needs 21.59 seconds. Reducing the computation time of diffusion models has
recently seen significant successes (Ma et al. 2022), which will directly
improve TO-based diffusion models. Other potential directions for future
research include applying TopoDiff to more complex TO problems, notably 3D
problems, and scaling it to higher resolutions and more boundary conditions.
Reducing dependency on mesh size and large training datasets is also critical.
We provide a framework for conditioning a diffusion model with constraints,
training it on optimal data, and guiding it with a regressor predicting
physical performance and some classifiers predicting the respect of
constraints. This is a general method that should allow for solving similar
design generation problems involving performance objectives and constraints,
such as aerodynamic design (Heyrani, Chen, and Ahmed 2021) and bicycle
synthesis (Regenwetter, Curry, and Ahmed 2022). Future work also includes
expanding the TopoDiff framework to solve many inverse problems in engineering
domains with multi-modal inputs.

##  6 Conclusion

Diffusion Models have been extremely successful in modeling high-dimensional
multi-modal distributions with astonishing results in high-fidelity image
generation. We propose TopoDiff — a conditional diffusion model to perform
end-to-end topology optimization. By introducing conditional diffusion models
for topology optimization, we show that diffusion models can outperform GANs
in engineering design applications too. Our model is augmented with an
explicit guidance strategy to ensure performance maximization and avoidance of
non-manufacturable designs. TopoDiff achieves an eight-times reduction in the
average compliance error and produces 11-times fewer non-manufacturable
designs compared with a state-of-art conditional GAN. It also achieves an
eight-times reduction in volume fraction error and generalizes well to out-of-
distribution boundary conditions. More generally, we provide a diffusion
model-based framework to solve many physical optimization problems in
engineering with performance objectives and constraints.

## Acknowledgments

The authors gratefully acknowledge the funding received from Mines Paris
Foundation. They would also like to thank MISTI France for supporting this
research.

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##  A1 Additional datasets used for surrogate models

The main dataset used to train, validate and test the diffusion model was
presented in the paper. This section provides more information about the two
other datasets that were used to train and validate the two surrogate models,
namely the regressor predicting compliance and the classifier predicting the
presence of floating material.

For the regressor predicting compliance, we used a dataset of 72000 labeled
samples, with the label indicating the compliance of the topology under the
given constraints. Every sample is an image containing eight channels (black
and white topology, volume fraction, von Mises stress, strain energy density,
load in x-direction, load in y-direction, x-displacement boundary condition
and y-displacement boundary condition). In order to give our regressor good
generalization abilities, we not only used optimal topologies (generated by
SIMP), but also negative samples, corresponding to non optimal topologies.
This dataset includes:

  * •

30000 images from the diffusion model main dataset, corresponding to optimal
topologies (25000 for training, 5000 for validation);

  * •

12000 images corresponding to non-optimal topologies, generated using the
fake-load method, explained below (10000 for training, 2000 for validation);

  * •

30000 images corresponding to non-optimal topologies, generated by the
unguided conditional diffusion model by using the same constraints as in the
main dataset (25000 for training, 5000 for validation).

It should be noted that the final dataset contains less than 72000 samples
because it was filtered to remove outlier structures (with compliance higher
than 50 for the first 42000 and with compliance higher than 25 for the
diffusion-generated structures). To generate non-optimal structures, we
notably used the fake-load method, which consists in adding an extra load to
the input constraints given to SIMP for generating the structure. SIMP thus
generates a topology that is optimal for the extra-loaded constraints but not
for the real ones. This method allows to generate non-optimal data while still
respecting basic requirements like the presence of material where loads are
applied.

For the classifier predicting the presence of floating material, we used a
dataset of 70000 labeled samples (58000 for training and 12000 for
validation), with the label being a boolean indicating if floating material is
present on the topology or not. Every sample is an image containing only one
channel: the black and white topology. This dataset includes:

  * •

the first 15000 images from the diffusion model main dataset, on which
floating material was added;

  * •

the first 14000 images from a dataset containing topologies at diverse volume
fractions, on which floating material was added;

  * •

the first 6000 images from the non-optimal topologies dataset generated with
the fake-load method, on which floating material was added;

  * •

the next 15000 images from the diffusion model main dataset, on which floating
material was _not_ added;

  * •

the next 14000 images from a dataset containing topologies at diverse volume
fractions, on which floating material was _not_ added;

  * •

the next 6000 images from the non-optimal topologies dataset generated with
the fake-load method, on which floating material was _not_ added.

It should be noted that while some datasets were used to train several
different models, we have paid careful attention to avoiding leakage of data
from training datasets to validation datasets. All models were validated on
data that they had not been trained on. The final TopoDiff model (diffusion
model + surrogate models) was tested on data that was never used for training
or validating any of its component models.