# Lower bounds of certain general local cohomology modules

Mahmoud Behrouzian  Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran<EMAIL_ADDRESS>and
Moharram Aghapournahr  Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$,
$\mathfrak{a}\in\Phi$, $M$ an arbitrary $R$-module and $t$ a non-negative
integer. Let $\mathcal{S}$ be a Melkersson subcategory of $R$-modules. Among
other things, we prove that if $\operatorname{H}^{i}_{\Phi}(M)$ is in
$\mathcal{S}$ for all $i<t$ then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is
in $\mathcal{S}$ for all $i<t$ and for all $\mathfrak{a}\in\Phi$. If
$\mathcal{S}$ is the class of $R$-modules $N$ with $\dim N\leq k$ where
$k\geq-1$, is an integer, then $\operatorname{H}^{i}_{\Phi}(M)$ is in
$\mathcal{S}$ for all $i<t$ (if and only if)
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is in $\mathcal{S}$ for all $i<t$ and
for all $\mathfrak{a}\in\Phi$. As consequences we study and compare vanishing,
Artinianness and support of general local cohomology and ordinary local
cohomology supported at ideals of its system of ideals at initial points
$i<t$. We show that $\operatorname{Supp}_{R}(\operatorname{H}^{\dim
M-1}_{\Phi}(M))$ is not necessarily finite whenever $(R,\mathfrak{m})$ is
local and $M$ a finitely generated $R$-module.

###### Key words and phrases:

Local cohomology, system of ideals, Serre subcategories, $ETH$-cofinite
modules

###### 2010 Mathematics Subject Classification:

13D45, 13E05, 14B15.

##  1\. Introduction

Throughout this paper $R$ is a commutative Noetherian ring with non-zero
identity and $\mathfrak{a}$ an ideal of $R$. For an $R$-module $M$, the
$i^{th}$ local cohomology module $M$ with respect to ideal $\mathfrak{a}$ is
defined as

$\operatorname{H}^{i}_{\mathfrak{a}}(M)\cong\underset{n}{\varinjlim}\operatorname{Ext}^{i}_{R}(R/{\mathfrak{a}}^{n},M).$

There are some generalizations of the theory of ordinary local cohomology
modules. The following is introduced by Bijan-Zadeh in [10].

Let $\Phi$ be a non-empty set of ideals of $R$. We call $\Phi$ a system of
ideals of $R$ if, whenever $\mathfrak{a}_{1},\mathfrak{a}_{2}\in\Phi$, then
there is an ideal $\mathfrak{b}\in\Phi$ such that
$\mathfrak{b}\subseteq\mathfrak{a}_{1}{\mathfrak{a}_{2}}$. For such a system,
for every $R$-module $M$, one can define

$\operatorname{\Gamma}_{\Phi}(M)=\\{~{}x\in M\mid{\mathfrak{a}}x=0\textmd{\ \
for some}~{}\mathfrak{a}\in\Phi\\}.$

Then $\operatorname{\Gamma}_{\Phi}(-)$ is a functor from $\mathscr{C}(R)$ to
itself (where $\mathscr{C}(R)$ denotes the category of all $R$-modules and all
$R$-homomorphisms). The functor $\operatorname{\Gamma}_{\Phi}(-)$ is additive,
covariant, $R$-linear and left exact. In [11],
$\operatorname{\Gamma}_{\Phi}(-)$ is denoted by $L_{\Phi}(-)$ and is called
the “general local cohomology functor with respect to $\Phi$”. For each $i\geq
0$, the $i$-th right derived functor of $\operatorname{\Gamma}_{\Phi}(-)$ is
denoted by $\operatorname{H}_{\Phi}^{i}(-)$. The functor
$\operatorname{H}_{\Phi}^{i}(-)$ and
$\underset{\mathfrak{a}\in\Phi}{\varinjlim}\operatorname{H}_{\mathfrak{a}}^{i}(-)$
(from $\mathscr{C}(R)$ to itself) are naturally equivalent (see [10]). For an
ideal $\mathfrak{a}$ of $R$, if
$\Phi=\\{{\mathfrak{a}}^{n}|n\in\mathbb{N}_{0}\\}$, then the functor
$\operatorname{H}_{\Phi}^{i}(-)$ coincides with the ordinary local cohomology
functor $\operatorname{H}_{\mathfrak{a}}^{i}(-)$. It is shown that, the study
of torsion theory over $R$ is equivalent to study the general local cohomology
theory (see [11]). It is easy to see that the above definition of system of
ideals and general local cohomology modules is equivalent to the [12,
Definition 2.1.10 and Notation 2.2.2]. General local cohomology modules was
studied by several authors in [10, 11, 15, 4, 1, 2, 12].

Let $\mathcal{S}$ be a Serre subcategory of the category of all $R$-modules.
The crucial points of Section 2, are Theorems 2.1 and 2.9 which show when the
$R$-modules $\operatorname{Ext}_{R}^{s+t}(N,M)$ and
$\operatorname{Ext}_{R}^{s}(N,\operatorname{H}_{\Phi}^{t}(M))$ belong to
$\mathcal{S}$, where $s,t$ are non-negative integers. These two theorems,
which are frequently used through the paper, enable us to demonstrate some new
facts and improve some older facts about the extension functors of general
local cohomology modules. Our main result in this section are Corollaries 2.6,
2.7 and 2.14 as follows:

######  Corollary 1.1.

(See Corollaries 2.6 and 2.7) Let $M$ be an arbitrary $R$-module,
$\mathcal{S}$ a Melkersson subcategory with respect to the any ideal
$\mathfrak{a}\in\Phi$ and $N$ be a finitely generated $R$-module with
$\operatorname{Supp}_{R}N=V(\mathfrak{a})$ for some $\mathfrak{a}\in\Phi$. Let
$t$ be a non-negative integer such that
$\operatorname{Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i<t$ and all $j<t-i$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all $i<t$. In
particular, if $\operatorname{H}^{i}_{\Phi}(M)$ is in $\mathcal{S}$ for all
$i<t$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all
$i<t$.

######  Corollary 1.2.

(See Corollary 2.14) Let $\mathcal{S}$ be the class of all $R$-modules $N$
with $\dim_{R}N\leq k$, where $k$ is an integer and $M$ an arbitrary
$R$-module. Then the following statements are equivalent:

  * (i)

$\operatorname{H}^{i}_{\Phi}(M)\in\mathcal{S}$ for all $i<t$ (for all $i\geq
0$);

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all $i<t$ (for all
$i\geq 0$) and all $\mathfrak{a}\in\Phi$.

In section 3, first of all we generalize and improve [20, Theorem 2.2] (Local-
global Principle for the Artinianness of local cohomology modules). Then we
prove the following main result:

######  Corollary 1.3.

(See Corollary 3.4) Let $M$ be an $\mathfrak{a}$-$ETH$-cofinite $R$-module for
all ideal $\mathfrak{a}\in\Phi$ or finitely generated and $t$ a positive
integer. Then the following conditions are equivalent:

  * (i)

$\operatorname{Supp}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\operatorname{Max}(R)$
for all $i<t$ (for all $i\geq 0$);

  * (ii)

$\operatorname{Supp}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for all $i<t$ (for all $i\geq 0$) and all $\mathfrak{a}\in\Phi$;

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian $R$-module for all $i<t$
(for all $i\geq 0$) and all $\mathfrak{a}\in\Phi$;

when $R$ is a semi-local ring these conditions are also equivalent to:

  * (iv)

$\operatorname{H}^{i}_{\Phi}(M)$ is Artinian for all $i<t$ (for all $i\geq
0$).

As an application of Corollary 1.3, we prove the following theorem:

######  Theorem 1.4.

(See Theorem 3.6) Let $R$ be a semi-local Noetherian ring, $\Phi$ a system of
ideals of $R$ and $M$ a finitely generated $R$-module. If $\dim
R/{\mathfrak{a}}\leq 0$ for all $\mathfrak{a}\in\Phi$, then
$\operatorname{H}^{i}_{\Phi}(M)$ is $\Phi$-cofinite Artinian for all $i\geq
0.$

In Example 3.7, we show that implication $(iii)\Leftrightarrow(iv)$ of
Corollary 1.3 is not true in general. In Corollary 3.8, we show that the
vanishing of $\operatorname{H}^{i}_{\Phi}(M)$ for all $i<t$ (for all $i\geq
0$) is equivalent to the vanishing of $\operatorname{H}^{i}_{\mathfrak{a}}(M)$
for all $i<t$ (for all $i\geq 0$) and all $\mathfrak{a}\in\Phi$. Using this
vanishing result in Corollary 3.9, we prove the following equality:

$\underset{i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))=\underset{\mathfrak{a}\in\Phi,i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M)).$

In Section 4, we get some applications of theorems 2.1 and 2.9. Then for non-
negative integers $s$ and $t$, we find some sufficient conditions for validity
of the isomorphism:

$\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\cong\operatorname{Ext}^{s+t}_{R}(N,M)\cong\operatorname{H}^{s+t}_{\mathfrak{a}}(N,M)\cong\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\mathfrak{a}}(M))$

As a consequence, we prove the following corollary:

######  Corollary 1.5.

(See Corollary 4.9) Let $M$, $N$ be two finitely generated $R$-modules such
that $\operatorname{Supp}_{R}(N)=\operatorname{V}(\mathfrak{a})$ for some
$\mathfrak{a}\in\Phi$. If $\operatorname{grade}(\Phi,M)=t$, then

$\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\cong\operatorname{Ext}^{t}_{R}(N,M)\cong\operatorname{H}^{t}_{\mathfrak{a}}(N,M)\cong\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\mathfrak{a}}(M)).$

We close the paper with the example 4.10 that shows, for a local ring
$(R,\mathfrak{m})$ and a finitely generated $R$-module $M$, with $\dim M=d$,
the set $\operatorname{Supp}_{R}(\operatorname{H}^{d-1}_{\Phi}(M))$ is not
necessarily finite.

Even though we can show some of our results by using spectral sequences, we
are avoiding the use of this technique completely in this work and we provide
more elementary proofs for the results.

Throughout this paper, $R$ will always be a commutative Noetherian ring with
non-zero identity, $\Phi$ a system of ideals of $R$ and $\mathfrak{a}$ will be
an ideal of $R$. We denote $\\{\mathfrak{p}\in{\rm
Spec}\,R:\,\mathfrak{p}\supseteq\mathfrak{a}\\}$ by $V(\mathfrak{a})$. As a
convention we consider $\dim 0=-1$. For any unexplained notation and
terminology we refer the reader to [12], [18] and [13].

##  2\. Lower bounds of general local cohomology

Recall that a Serre subcategory $\mathcal{S}$ of the category of $R$–modules
is a subclass of $R$–modules such that for any short exact sequence

$0\longrightarrow X^{\prime}\longrightarrow X\longrightarrow
X^{\prime\prime}\longrightarrow 0,$

the module $X$ is in $\mathcal{S}$ if and only if $X^{\prime}$ and
$X^{\prime\prime}$ are in $\mathcal{S}$. In other words, it is closed under
taking submodules, quotients and extensions.

We begin the paper with a useful theorem that is a generalization of [7,
Theorem 2.1].

######  Theorem 2.1.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Serre subcategory and $N$
be a $\Phi$-torsion $R$-module. If
$\operatorname{Ext}^{t-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))\in\mathcal{S}$
for all $i\leq t$. Then $\operatorname{Ext}^{t}_{R}(N,M)\in\mathcal{S}$.

###### Proof.

We prove the result by induction on $t$. The case $t=0$ is clear, because if
$t=0$ then, $i=0$ so

$\operatorname{Ext}^{0}_{R}(N,\operatorname{H}^{0}_{\Phi}(M))=\operatorname{Hom}_{R}(N,\Gamma_{\Phi}(M))\cong\operatorname{Hom}_{R}(N,M).$

Suppose that $t>0$ and $t-1$ is settled. Let
$\overline{M}=M/\Gamma_{\Phi}(M)$, $L=E(\overline{M})/\overline{M}$ where
$E(\overline{M})$ is the injective hull of $\overline{M}$. Since
$\Gamma_{\Phi}(\overline{M})=0=\Gamma_{\Phi}(E(\overline{M}))$, by applying
the derived functors $\Gamma_{\Phi}(-)$ and $\operatorname{Hom}_{R}(N,-)$ on
the short exact sequence

$0\longrightarrow\overline{M}\longrightarrow E(\overline{M})\longrightarrow
L\longrightarrow 0$

we obtain for all $i>0$, the isomorphisms,

$\operatorname{H}^{i-1}_{\Phi}(L)\cong\operatorname{H}^{i}_{\Phi}(\overline{M})(\cong\operatorname{H}^{i}_{\Phi}(M))~{}~{},~{}~{}\operatorname{Ext}^{i-1}_{R}(N,L)\cong\operatorname{Ext}^{i}_{R}(N,\overline{M}).$

From the above isomorphisms for all $0\leq i\leq t-1$ we have,

$\operatorname{Ext}^{(t-1)-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(L))\cong\operatorname{Ext}^{t-(i+1)}_{R}(N,\operatorname{H}^{i+1}_{\Phi}(M))$

which is in $\mathcal{S}$ by assumptions. Thus, from the induction hypothesis
on $L$, $\operatorname{Ext}^{t-1}_{R}(N,L)\in\mathcal{S}$.
Now by the short exact sequence
$0\longrightarrow\Gamma_{\Phi}(M)\longrightarrow
M\longrightarrow\overline{M}\longrightarrow 0$ we get the long exact sequence

$\cdots\longrightarrow\operatorname{Ext}^{t}_{R}(N,\Gamma_{\Phi}(M))\longrightarrow\operatorname{Ext}^{t}_{R}(N,M)\longrightarrow\operatorname{Ext}^{t}_{R}(N,\overline{M})\longrightarrow\cdots$

which shows that $\operatorname{Ext}^{t}_{R}(N,M)\in\mathcal{S}$. ∎

######  Corollary 2.2.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Serre subcategory and $N$
be a $\Phi$-torsion $R$-module. Let $t$ be a non-negative integer such that
$\operatorname{Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))\in\mathcal{S}$
for all $i<t$ and all $j<t-i$ Then
$\operatorname{Ext}^{i}_{R}(N,M)\in\mathcal{S}$ for all $i<t$.

######  Corollary 2.3.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Serre subcategory and $N$
be a finite $\mathfrak{a}$-torsion $R$-module for some $\mathfrak{a}\in\Phi$.
If $\operatorname{Ext}^{t-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i\leq t$, then
$\operatorname{H}^{t}_{\mathfrak{a}}(N,M)\in\mathcal{S}$.

###### Proof.

The result follows by Theorem 2.1 and [21, theorem 2.5]. Note that
$N=\Gamma_{\mathfrak{a}}(N)\subseteq\Gamma_{\Phi}(N)$. ∎

######  Corollary 2.4.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Serre subcategory and $N$
be a finite $\mathfrak{a}$-torsion $R$-module for some $\mathfrak{a}\in\Phi$.
If $\operatorname{Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i<t$ and all $j<t-i$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(N,M)\in\mathcal{S}$ for all $i<t$.

The second author of present paper and Melkersson in [6, Definition 2.1]
defined Melkersson subcategory with respect to an ideal as follows:

######  Definition 2.5.

A full subcategory $\mathcal{S}$ of the category of $R$-modules is said to be
Melkersson subcategory with respect to the ideal $\mathfrak{a}$ if for any
$\mathfrak{a}$-torsion $R$-module $M$, $(0:_{M}\mathfrak{a})\in\mathcal{S}$
implies $M\in\mathcal{S}$.

To see some examples of Melkersson subcategories, we refer the reader to [6,
Examples 2.4 and 2.5].

The following two corollaries are our first main results of this paper.

######  Corollary 2.6.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Melkersson subcategory
with respect to the any ideal $\mathfrak{a}\in\Phi$ and $N$ be a finitely
generated $R$-module with $\operatorname{Supp}_{R}N=V(\mathfrak{a})$ for some
$\mathfrak{a}\in\Phi$. Let $t$ be a non-negative integer such that
$\operatorname{Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i<t$ and all $j<t-i$, then
$\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all $i<t$.

###### Proof.

It follows by Corollary 2.2 and [6, Theorem 2.9]. ∎

######  Corollary 2.7.

Let $M$ be an arbitrary $R$-module, $\mathcal{S}$ a Melkersson subcategory
with respect to the any ideal $\mathfrak{a}\in\Phi$ and $t$ a non-negative
integer. If $\operatorname{H}^{i}_{\Phi}(M)$ is in $\mathcal{S}$ for all
$i<t$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all
$i<t$.

######  Corollary 2.8.

Let $M$ be an arbitrary $R$-module and $\mathcal{S}$ a Melkersson subcategory
with respect to the any ideal $\mathfrak{a}\in\Phi$ then:

$\inf\\{i:\operatorname{H}^{i}_{\Phi}(M)\notin\mathcal{S}\\}\leq\inf\\{\inf\\{i:\operatorname{H}^{i}_{\mathfrak{a}}(M)\notin\mathcal{S}\\}:\mathfrak{a}\in\Phi\\}.$

The following theorem is a generalization of [7, Theorem 2.3].

######  Theorem 2.9.

Let $N$ be a $\Phi$-torsion $R$-module, $M$ be an arbitrary $R$-module,
$\mathcal{S}$ a Serre subcategory of $R$-modules and $s,t$ be non-negative
integers such that

  * (i)

$\operatorname{Ext}^{s+t}_{R}(N,M)$ is in $\mathcal{S}$;

  * (ii)

$\operatorname{Ext}^{s+1+i}_{R}(N,\operatorname{H}^{t-i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i$, $1\leq i\leq t$;

  * (iii)

$\operatorname{Ext}^{s-1-i}_{R}(N,\operatorname{H}^{t+i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i$, $1\leq i\leq s-1$.

Then
$\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\in\mathcal{S}$.

###### Proof.

We prove by induction on $t$. Let $t=0$ and set
$\overline{M}=M/\Gamma_{\Phi}(M)$. By hypothesis (iii)
$\operatorname{Ext}^{(s-1)-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(\overline{M}))$
is in $\mathcal{S}$ for all $i$, $0\leq i\leq s-1$. Thus from Theorem 2.1,
$\operatorname{Ext}^{s-1}_{R}(N,\overline{M})$ belongs to $\mathcal{S}$ .
Applying the derived functor $\operatorname{Hom}_{R}(N,-)$ to the short exact
sequence

$0\longrightarrow\Gamma_{\Phi}(M)\longrightarrow
M\longrightarrow\overline{M}\longrightarrow 0$

we obtain the long exact sequence

$\cdots\longrightarrow\operatorname{Ext}^{s-1}_{R}(N,\overline{M})\longrightarrow\operatorname{Ext}^{s}_{R}(N,\Gamma_{\Phi}{M})\longrightarrow\operatorname{Ext}^{s}_{R}(N,M)\longrightarrow\cdots$

Since $\operatorname{Ext}^{s-1}_{R}(N,\overline{M})$ is in $\mathcal{S}$ and
by $t=0$ in (i), $\operatorname{Ext}^{s}_{R}(N,M)$ is in $\mathcal{S}$ which
shows that $\operatorname{Ext}^{s}_{R}(N,\Gamma_{\Phi}{M})\in\mathcal{S}$. Now
assume that $t>0$ and that $t-1$ is settled. Let
$\overline{M}=M/\Gamma_{\Phi}(M)$, $L=E(\overline{M})/\overline{M}$ where
$E(\overline{M})$ is the injective hull of $\overline{M}$. By applying the
derived functors $\Gamma_{\Phi}(-)$ and $\operatorname{Hom}_{R}(N,-)$ on the
short exact sequence

$0\longrightarrow\overline{M}\longrightarrow E(\overline{M})\longrightarrow
L\longrightarrow 0$

the proof sufficiently similar to that of Theorem 2.1 to be omitted. We leave
the proof to the reader. ∎

The following corollary is a generalization of [7, Corollary 2.4].

######  Corollary 2.10.

Let $M$ an arbitrary $R$-module and $N$ a $\Phi$-torsion $R$-module. Let
$\mathcal{S}$ be a Serre subcategory of $R$-modules and $t$ a non-negative
integer. If $\operatorname{Ext}^{t+1-i}_{R}(N,H^{i}_{\Phi}(M))\in\mathcal{S}$
for all $i<t$ and $\operatorname{Ext}^{t}_{R}(N,M)\in\mathcal{S}$ then
$\operatorname{Hom}_{R}(N,H^{t}_{\Phi}(M))\in\mathcal{S}$.

###### Proof.

Put $s=0$ in Theorem 2.9. ∎

######  Proposition 2.11.

Let $\mathcal{S}$ be the class of all $R$-modules $N$ with $\dim_{R}N\leq k$,
where $k$ is an integer and $M$ an arbitrary $R$-module. Let $t$ be a non-
negative integer such that
$\operatorname{Ext}^{t+1-i}_{R}(R/\mathfrak{a},\operatorname{H}^{i}_{\Phi}(M))\in\mathcal{S}$
for all $i<t$ and all $\mathfrak{a}\in\Phi$ and
$\operatorname{Ext}^{t}_{R}(R/\mathfrak{a},M)\in\mathcal{S}$ for all
$\mathfrak{a}\in\Phi$, then $\operatorname{H}^{t}_{\Phi}(M)\in\mathcal{S}$.

###### Proof.

We have
$\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M))\in\mathcal{S}$
for all $\mathfrak{a}\in\Phi$ by Corollary 2.10 . Also, hence
$\operatorname{H}^{t}_{\Phi}(M)$ is a $\Phi$-torsion $R$-modules, thus

$\operatorname{H}^{t}_{\Phi}(M)=\bigcup_{\mathfrak{a}\in\Phi}(0:_{\operatorname{H}^{t}_{\Phi}(M)}\mathfrak{a})=\bigcup_{\mathfrak{a}\in\Phi}\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M)).$

Thus,
$\operatorname{Supp}_{R}\operatorname{H}^{t}_{\Phi}(M)\subseteq\bigcup_{\mathfrak{a}\in\Phi}\operatorname{Supp}_{R}\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M))$,
and therefore $\operatorname{H}^{t}_{\Phi}(M)\in\mathcal{S}$. ∎

######  Corollary 2.12.

Let $\mathcal{S}$ be the class of all $R$-modules $N$ with $\dim_{R}N\leq k$,
where $k$ is an integer and $M$ an arbitrary $R$-module. Let
$\operatorname{Ext}^{t}_{R}(R/\mathfrak{a},M)\in\mathcal{S}$ for all $i<t$ and
all $\mathfrak{a}\in\Phi$. Then $\operatorname{H}^{i}_{\Phi}(M)\in\mathcal{S}$
for all $i<t$.

###### Proof.

By a similar proof of Proposition 2.11, we can show that
$\Gamma_{\Phi}(M)\in\mathcal{S}$. Now it follows from Proposition 2.11 that
$\operatorname{H}_{\Phi}^{1}(M)\in\mathcal{S}$. By keeping this process we
have $\operatorname{H}_{\Phi}^{i}(M)\in\mathcal{S}$ for all $i<t$. ∎

######  Corollary 2.13.

Let $M$ be an arbitrary $R$-module. If $k\geq-1$ be an integer such that
$\dim_{R}\operatorname{H}^{i}_{\mathfrak{a}}(M)\leq k$ for all $i<t$ (resp.
for all $i$) and all $\mathfrak{a}\in\Phi$, then
$\dim_{R}\operatorname{H}^{i}_{\Phi}(M)\leq k$ for all $i<t$ (resp. for all
$i$).

###### Proof.

The result follows by Corollary 2.13 and [6, Theorem 2.9]. ∎

The following result is one of the main results of this paper.

######  Corollary 2.14.

Let $\mathcal{S}$ be the class of all $R$-modules $N$ with $\dim_{R}N\leq k$,
where $k$ is an integer, and $M$ an arbitrary $R$-module. Then the following
statements are equivalent:

  * (i)

$\operatorname{H}^{i}_{\Phi}(M)\in\mathcal{S}$ for all $i<t$ (for all $i\geq
0$);

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)\in\mathcal{S}$ for all $i<t$ (for all
$i\geq 0$) and all $\mathfrak{a}\in\Phi$.

###### Proof.

It follows by Corollaries 2.13 and 2.7. ∎

######  Corollary 2.15.

Let $\mathcal{S}$ be the class of all $R$-modules $N$ with $\dim_{R}N\leq k$,
where $k$ is an integer, and $M$ an arbitrary $R$-module. Then:

$\displaystyle inf\\{i:\operatorname{H}^{i}_{\Phi}(M)\notin\mathcal{S}\\}$
$\displaystyle=inf\\{inf\\{i:\operatorname{H}^{i}_{\mathfrak{a}}(M)\notin\mathcal{S}\\}:\mathfrak{a}\in\Phi\\}$
$\displaystyle=inf\\{inf\\{i:\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)\notin\mathcal{S}\\}:\mathfrak{a}\in\Phi\\}.$

###### Proof.

By Corollary 2.14, [6, Theorem 2.9]. ∎

##  3\. Cofiniteness, Artinianness and vanishing

As a generalization of definition of cofinite modules with respect to an ideal
([19, Definition 2.1]), it is introduced in [3, Definition 2.2] the following
definition.

######  Definition 3.1.

Let $\mathfrak{a}$ be an ideal of $R$, then the $R$-module $M$ (not necessary
$\mathfrak{a}$-torsion) is called $ETH$-cofinite with respect to
$\mathfrak{a}$ or $\mathfrak{a}$-$ETH$-cofinite, if
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is a finitely generated
$R$-module for all $i$.

To see the properties of this class see [4, Remark 2.2 and Example 2.3].

The following theorem which is Local-global Principle for the Artinianness of
local cohomology modules generalizes and improves [20, Theorem 2.2]. Tang in
[20, Theorem 2.2] proved the following theorem when $M$ is finitely generated,
whereas here, with a completely different proof, we prove it for the class of
$\mathfrak{a}$-$ETH$-cofinite modules. The implication $(iv)\Rightarrow(ii)$
is also a generalization of [9, Proposition 2.2 and Corokkary 2.3].

######  Theorem 3.2.

(Local-global Principle for the Artinianness of local cohomology modules). Let
$\mathfrak{a}$ be a proper ideal of $R$, $M$ an $\mathfrak{a}$-ETH-cofinite
$R$-module and $t$ a positive integer. Then the following statements are
equivalent:

  * (i)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian for all $i<t$ (for all
$i$);

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian and
$\mathfrak{a}$-cofinite for all $i<t$ (for all $i$);

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{p}}$ is Artinian for all
$i<t$ (for all $i$) and all prime ideal
$\mathfrak{p}\in\operatorname{Spec}(R)$;

  * (iv)

$\operatorname{Supp}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for all $i<t$ (for all $i$).

###### Proof.

The implications $(i)\Rightarrow(iii)$, $(ii)\Rightarrow(i)$ and
$(ii)\Rightarrow(iv)$ are obviously true.

To prove implication $(iii)\Rightarrow(ii)$, we use induction on $t$. Let
$t=1$. Since

$\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))=\operatorname{Hom}_{R}(R/\mathfrak{a},M),$

it follows easily that the set
$\operatorname{Ass}_{R}(\operatorname{\Gamma}_{\mathfrak{a}}(M))$ is finite.
Let
$\operatorname{Ass}_{R}(\operatorname{\Gamma}_{\mathfrak{a}}(M))=\\{\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}\\}$.
So we have

$\operatorname{E}(\operatorname{\Gamma}_{\mathfrak{a}}(M))=\oplus_{i=1}^{n}\mu^{0}({\mathfrak{p}}_{i},\operatorname{\Gamma}_{\mathfrak{a}}(M))\operatorname{E}(R/{\mathfrak{p}}_{i})$.

By localizing of the above equality at each $\mathfrak{p}_{i}$ of
$\operatorname{Ass}_{R}(\operatorname{\Gamma}_{\mathfrak{a}}(M))$ we get
$\operatorname{Ass}_{R}(\operatorname{\Gamma}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$

and $\mu^{0}({\mathfrak{p}}_{i},\operatorname{\Gamma}_{\mathfrak{a}}(M))$ is
finite. So it follows
$\operatorname{E}(\operatorname{\Gamma}_{\mathfrak{a}}(M))$ and therefore
$\operatorname{\Gamma}_{\mathfrak{a}}(M)$ is an Artinian $R$-module. Hence
$\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))$ has finite
lengths. Now, it follows by [19, Proposition 4.1], $\Gamma_{\mathfrak{a}}(M)$
is Artinian and $\mathfrak{a}$-cofinite $R$-module. Suppose $t>1$ and the case
$t-1$ is settled. Since by induction hypothesis
$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian and
$\mathfrak{a}$-cofinite for all $i<t-1$. It follows by [5, Corollary 4.4] that
$\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t-1}_{\mathfrak{a}}(M))$
is a finitely generated $R$-module. Therefore the set
$\operatorname{Ass}_{R}(\operatorname{H}^{t-1}_{\mathfrak{a}}(M))$ is finite.
The rest of proof is the same as the case $t=1$. Therefor
$\operatorname{H}^{t-1}_{\mathfrak{a}}(M)$ is an Artinian and
$\mathfrak{a}$-cofinite $R$-module, this complete the inductive step.

To prove the implication $(iv)\Rightarrow(ii)$ again we use induction on $t$.
Let $t=1$. Since
$\operatorname{Supp}_{R}(\Gamma_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$,
thus
$\operatorname{Supp}_{R}(\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M)))\subseteq\operatorname{Max}(R)$.
On the other hand since
$\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))\cong\operatorname{Hom}_{R}(R/\mathfrak{a},M)$,
so $\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))$ is
finitely generated $R$-module. It is easy to see that
$\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))$ is an
Artinian $R$-module and therefore
$\operatorname{Hom}_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(M))$ has finite
lengths. It follows by [19, Proposition 4.1] that $\Gamma_{\mathfrak{a}}(M)$
is Artinian and $\mathfrak{a}$-cofinite $R$-module. Now, assume that $t>1$ and
the case $t-1$ is settled. Consider $\overline{M}=M/\Gamma_{\mathfrak{a}}(M)$.
Since $\Gamma_{\mathfrak{a}}(M)=\operatorname{H}^{0}_{\mathfrak{a}}(M)$ is
Artinian and $\mathfrak{a}$-cofinite $R$-module, it is easy to see that $M$ is
$\mathfrak{a}$-$ETH$-cofinite if and only if $\overline{M}$ is
$\mathfrak{a}$-$ETH$-cofinite. Further, note that
$\operatorname{H}^{i}_{\mathfrak{a}}(\overline{M})\cong\operatorname{H}^{i}_{\mathfrak{a}}(M),~{}~{}i>0$.
Thus we may assume $M$ is $\mathfrak{a}$-torsion-free and so
$\operatorname{\Gamma}_{\mathfrak{a}}(M)=0$. Let $E$ be an injective hall of
$M$ and put $L=E/M$. Then also $\Gamma_{\mathfrak{a}}(E)=0$. Therefore the
exact sequence $0\longrightarrow M\longrightarrow E\longrightarrow
L\longrightarrow 0,$ implies
$\operatorname{H}^{i}_{\mathfrak{a}}(L)\cong\operatorname{H}^{i+1}_{\mathfrak{a}}(M)$
and
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},L)\cong\operatorname{Ext}^{i+1}_{R}(R/\mathfrak{a},M)$
for all $i\geq 0$. It follows that $L$ is an $\mathfrak{a}$-$ETH$-cofinite
$R$-module and
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(L))\subseteq\operatorname{Max}(R)$
for all $i<t-1$. Therefore by induction hypothesis
$\operatorname{H}^{i}_{\mathfrak{a}}(L)$ is Artinian and
$\mathfrak{a}$-cofinite $R$-module for all $i<t-1$. The isomorphism
$\operatorname{H}^{i}_{\mathfrak{a}}(L)\cong\operatorname{H}^{i+1}_{\mathfrak{a}}(M)$
shows that $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian and
$\mathfrak{a}$-cofinite $R$-module for all $i<t$. This complete the inductive
step. ∎

######  Corollary 3.3.

Let $\mathfrak{b}\subseteq\mathfrak{a}$ be two ideals of a Noetherian ring
$R$, $M$ be a $\mathfrak{b}$-cofinite $R$-module and $t$ be a positive
integer. Then the above theorem holds.

###### Proof.

Note that
$\operatorname{Supp}_{R}(R/{\mathfrak{a}})=\operatorname{V}(\mathfrak{a})\subseteq\operatorname{V}(\mathfrak{b})$,
therefore ${\operatorname{Ext}}^{i}_{R}(R/\mathfrak{a},M)$ is finitely
generated for all $i\geq 0$ by [19, Corollary 2.5]. Now the assertion follows
by Theorem 3.2. ∎

The following corollary that is one of our main results, shows that the
Artinianness of general local cohomology at initial points is not necessarily
equivalent to the Artinianness of its ordinary local cohomology in general and
it needs some more conditions.

######  Corollary 3.4.

Let $M$ be an $\mathfrak{a}$-$ETH$-cofinite $R$-module for all ideal
$\mathfrak{a}\in\Phi$ or finitely generated and $t$ a positive integer. Then
the following statements are equivalent:

  * (i)

$\operatorname{Supp}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\operatorname{Max}(R)$
for all $i<t$ (for all $i\geq 0$);

  * (ii)

$\operatorname{Supp}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subseteq\operatorname{Max}(R)$
for all $i<t$ (for all $i\geq 0$) and all $\mathfrak{a}\in\Phi$;

  * (iii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is Artinian $R$-module for all $i<t$
(for all $i\geq 0$) and all $\mathfrak{a}\in\Phi$;

when $R$ is a semi-local ring these conditions are also equivalent to:

  * (iv)

$\operatorname{H}^{i}_{\Phi}(M)$ is Artinian for all $i<t$ (for all $i\geq
0$).

###### Proof.

$(i)\Leftrightarrow(ii)$ follows by Corollary 2.14.
$(ii)\Leftrightarrow(iii)$ follows by Theorem 3.2.
$(iii)\Leftrightarrow(iv)$ follows by [14, Lemma 3.2], [15, Lemma 3.2] and [6,
Theorem 2.9 $(i)\Leftrightarrow(ii)$]. ∎

The following definition is a generalization of the concept
$\mathfrak{a}$-cofinite modules which was introduced by Hartshorne.

######  Definition 3.5 ([8]).

Let $\Phi$ be a system of ideals of $R$ and $X$ an $R$-module. The general
local cohomology module $\operatorname{H}^{j}_{\Phi}(X)$ is called to be
$\Phi$-cofinite if there exists an ideal $\mathfrak{a}\in\Phi$ such that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},\operatorname{H}^{j}_{\Phi}(X))$ is
finitely generated for all $i.$

######  Theorem 3.6.

Let $R$ be a semi-local Noetherian ring, $\Phi$ a system of ideals of $R$ and
$M$ a finitely generated $R$-module. If $\dim R/{\mathfrak{a}}\leq 0$ for all
$\mathfrak{a}\in\Phi$, then $\operatorname{H}^{i}_{\Phi}(M)$ is
$\Phi$-cofinite Artinian for all $i\geq 0.$

###### Proof.

Since by [10, Lemma 2.1],

$\operatorname{H}^{i}_{\Phi}(M)\cong\underset{\mathfrak{a}\in\Phi}{\varinjlim}\operatorname{H}^{i}_{\mathfrak{a}}(M),$

it is easy to see that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\underset{\mathfrak{a}\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
and therefore

$\operatorname{dimSupp}\operatorname{H}^{i}_{\Phi}(M)\leq\sup\\{\operatorname{dimSupp}\operatorname{H}^{i}_{\mathfrak{a}}(M)|\mathfrak{a}\in\Phi\\}\leq
0,$

thus
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\operatorname{Max}(R)$.
Now, it follows by Corollary 3.4 that $\operatorname{H}^{i}_{\Phi}(M)$ is an
Artinian $R$-module for all $i\geq 0$. Therefore [4, Corollary 2.10] implies
that $\operatorname{H}^{i}_{\Phi}(M)$ is $\Phi$-cofinite for all $i\geq 0$. ∎

The following example shows that in Corollary 3.4 the Artinianness
of$~{}\operatorname{H}_{\mathfrak{a}}^{i}(M)$ for all $\mathfrak{a}\in\Phi$
and $i<t$ for a non-negative integer $t$ or for all $i\geq 0$, may not be
equivalent to the Artinianness of general local cohomology module
$\operatorname{H}_{\Phi}^{i}(M)$ for all $i<t$ or for all $i\geq 0$.

######  Example 3.7.

Let $R$ be a Gorenstein ring of finite dimension $d$ such that has infinite
maximal ideal with $\operatorname{ht}\mathfrak{m}=d$. Let

$\Psi=\\{\mathfrak{m}\in\operatorname{Max}(R)|\operatorname{ht}\mathfrak{m}=d\\}$
and $\Phi=\\{\mathfrak{a}|\mathfrak{a}\text{ is an ideal of }R\text{ and
}\dim{R/{\mathfrak{a}}}\leq 0\\}.$

Then, it is easy to see that $\Phi$ is a system of ideals of $R$ and
$\Psi\subseteq\Phi$. By [18, Theorem 18.8], the minimal injective resolution
of $R$ has the form

$0\longrightarrow\underset{\operatorname{ht}_{R}{\mathfrak{p}}=0}{\bigoplus}\operatorname{E}(R/\mathfrak{p})\longrightarrow\underset{\operatorname{ht}_{R}{\mathfrak{p}}=1}{\bigoplus}\operatorname{E}(R/\mathfrak{p})\longrightarrow\dots\longrightarrow\underset{\operatorname{ht}_{R}{\mathfrak{p}}=d}{\bigoplus}\operatorname{E}(R/\mathfrak{p})\longrightarrow
0$

Applying the functor $\operatorname{\Gamma}_{\Phi}(-)$ to the above injective
resolution, we can deduce that

$\operatorname{H}^{i}_{\Phi}(R)=\left\\{\begin{array}[]{cc}\underset{\mathfrak{m}\in\Psi}{\bigoplus}E(R/\mathfrak{m})&\text{if}~{}~{}~{}~{}i=d,\\\
0&\text{if}~{}~{}~{}~{}i\neq d.\end{array}\right.$

Clearly $\operatorname{H}_{\Phi}^{d}(R)$ is not Artinian while it is easy to
see that $\operatorname{H}_{\mathfrak{a}}^{i}(R)$ is Artinian for all
$\mathfrak{a}\in\Phi$ and all $i\geq 0$.

The following corollary shows unlike the Artinianness, vanishing of general
local cohomology modules in initial points is equivalent to the vanishing of
its ordinary local cohomology.

######  Corollary 3.8.

Let $M$ be an arbitrary $R$-module, $\Phi$ be a system of ideals of $R$ and
$t$ be a non-negative integer. Then the following statements are equivalent:

  * (i)

$\operatorname{H}^{i}_{\Phi}(M)=0$ for all $i<t$ (for all $i\geq 0$);

  * (ii)

$\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i<t$ (for all $i\geq 0$)
and all $\mathfrak{a}\in\Phi$.

###### Proof.

Take $k=-1$ in Corollary 2.14. ∎

The following corollary shows that union of supports of general local
cohomology modules at initial points $i<t$ is equal to union of supports of
its all ordinary local cohomology modules at initial points $i<t$ while for
each non-negative integer $i$, Since
$\operatorname{H}^{i}_{\Phi}(M)\cong\underset{\mathfrak{a}\in\Phi}{\varinjlim}\operatorname{H}^{i}_{\mathfrak{a}}(M),$
by [10, Lemma 2.1], it is easy to see that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\underset{\mathfrak{a}\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$.

######  Corollary 3.9.

Let $M$ be an be an arbitrary $R$–module and $t$ be a non-negative integer.
Then

$\underset{i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))=\underset{\mathfrak{a}\in\Phi,i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M)).$

###### Proof.

By Corollary 3.8, we have

$\begin{matrix}\mathfrak{p}\notin\underset{i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))&\Leftrightarrow&\forall
i<t;&\operatorname{H}^{i}_{\Phi}(M)_{\mathfrak{p}}=0\\\
&\Leftrightarrow&\forall
i<t;&\operatorname{H}^{i}_{{\Phi}_{\mathfrak{p}}}(M_{\mathfrak{p}})=0\\\
&\Leftrightarrow&\forall{\mathfrak{a}}R_{\mathfrak{p}}\in\Phi_{\mathfrak{p}},i<t;&\operatorname{H}^{i}_{{\mathfrak{a}}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0\\\
&\Leftrightarrow&\forall\mathfrak{a}\in\Phi,i<t;&\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{p}}=0\\\
&\Leftrightarrow&&\mathfrak{p}\notin\underset{\mathfrak{a}\in\Phi,i<t}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M)),\end{matrix}$

as we desired. ∎

##  4\. Some applications and identities

As applicatios of Theorems 2.1 and 2.9, we can state Corollaries 4.1, 4.2 and
4.3.

The following corollary is a generalization of [5, Theorem 4.1(c)].

######  Corollary 4.1.

Let $N$ be a $\Phi$-torsion and finitely generated $R$-module. Let $M$ be an
arbitrary $R$-module and $t$ a non-negative integer such that
$\operatorname{Ext}^{j-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))$ is in
$\mathcal{S}$ for all $i,j$ with $0\leq i\leq t-1$ and $j=t,t+1$. Then
$\operatorname{Ext}^{t}_{R}(N,M)$ is in $\mathcal{S}$ if and only if
$\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))$ is in
$\mathcal{S}$.

###### Proof.

Follows by Theorems 2.1 and 2.10. ∎

######  Corollary 4.2.

Let $N$ be a $\Phi$-torsion and finitely generated $R$-module. Let $M$ be an
arbitrary $R$-module and $s,t$ non-negative integers such that
$\operatorname{H}^{i}_{\Phi}(M)$ is in $\mathcal{S}$ for all $i$ with $0\leq
i\leq t-1$ or $t+1\leq i\leq s+t$. Then $\operatorname{Ext}^{s+t}_{R}(N,M)$ is
in $\mathcal{S}$ if and only if
$\operatorname{Ext}_{R}^{s}(N,\operatorname{H}^{t}_{\Phi}(M))\in\mathcal{S}$.

###### Proof.

Follows by Theorems 2.1 and 2.9. ∎

The following Corollary is a generalization of [7, Theorem 2.8].

######  Corollary 4.3.

Let $M$ be an $R$-module and $s,t$ be non-negative integers such that $t\leq
s$. Assume also that $\operatorname{H}^{i}_{\Phi}(M)$ is in $\mathcal{S}$, for
all $i$, $i\neq t$ (respectively $0\leq i\leq t-1$ or $t+1\leq i\leq s$). Then
for all $i$, $i\geq t$ ( respectively $0\leq i\leq t-s$ )
$\operatorname{Ext}_{R}^{i}(N,\operatorname{H}^{t}_{\Phi}(M))$ is in
$\mathcal{S}$ if and only if
$\operatorname{Ext}_{R}^{i+t}(N,M)\in\mathcal{S}$.

###### Proof.

Easily shown with Theorems 2.1 and 2.9. ∎

In the following theorem, for non-negative integers $s$ and $t$, we find some
sufficient conditions for validity of the isomorphism
$\operatorname{Ext}^{s+t}_{R}(N,M)\cong\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))$.
It is a generalization of [7, Theorem 3.5].

######  Theorem 4.4.

Let $N$ be a $\Phi$-torsion and finitely generated $R$-module. Let $M$ be an
arbitrary $R$-module and $s,t$ be non-negative integers. Assume also as
follows:

  * (i)

$\operatorname{Ext}^{s+t-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))=0$ for all
$i,0\leq i<t$ or $t+1\leq i\leq s+t$;

  * (ii)

$\operatorname{Ext}^{s+1+i}_{R}(N,\operatorname{H}^{t-i}_{\Phi}(M))=0$ for all
$i,0\leq i\leq t$;

  * (iii)

$\operatorname{Ext}^{s-1-i}_{R}(N,\operatorname{H}^{t+i}_{\Phi}(M))=0$ for all
$i,0\leq i\leq s-1$.

Then we have
$\operatorname{Ext}^{s+t}_{R}(N,M))\cong\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))$.

###### Proof.

We prove by using induction on $t$. Let $t=0$. We have

$\operatorname{Ext}^{s-1}_{R}(N,M/\Gamma_{\Phi}(M))=0=\operatorname{Ext}^{s}_{R}(N,M/\Gamma_{\Phi}(M))$

from hypothesis $(iii)$ and $(i)$, and Theorem 2.1 with $\mathcal{S}=\\{0\\}$.
Now, the assertion follows by the exact sequence

$\operatorname{Ext}_{R}^{s-1}(N,M/\Gamma_{\Phi}(M))\longrightarrow\operatorname{Ext}^{s}_{R}(N,\Gamma_{\Phi}(M))\longrightarrow\operatorname{Ext}^{s}_{R}(N,M)\longrightarrow\operatorname{Ext}^{s}_{R}(N,M/\Gamma_{\Phi}(M))$

obtained from the short exact sequence

$0\longrightarrow\Gamma_{\Phi}(M)\longrightarrow M\longrightarrow
M/\Gamma_{\Phi}(M)\longrightarrow 0$.

Assume that $t>0$ and that $t-1$ is settled. By considering the short exact
sequence

$0\rightarrow\overline{M}\rightarrow E(\overline{M})\rightarrow L\rightarrow
0,$

the rest of proof is similar to that of Theorem 2.1. ∎

The following corollary is a generalization of [7, Corollary 3.8].

######  Corollary 4.5.

Let $M$ be an arbitrary $R$-module, $\mathfrak{a}$ an deal of $R$ and
$\mathfrak{p}\in\bigcup_{\mathfrak{a}\in\Phi}\operatorname{V}(\mathfrak{a})$.
Suppose that
$\operatorname{Ext}^{s+t-i}_{R}(R/\mathfrak{p},\operatorname{H}^{i}_{\Phi}(M))=0$
for all $i$ with $0\leq i<t$ or $t+1\leq i\leq s+t$,
$\operatorname{Ext}^{s++1+i}_{R}(R/\mathfrak{p},\operatorname{H}^{i}_{\Phi}(M))=0$
for all $i,0\leq i\leq t$, and
$\operatorname{Ext}^{s-1-i}_{R}(R/\mathfrak{p},\operatorname{H}^{i}_{\Phi}(M))=0$
for all $i,0\leq i\leq s-1$. Then
$\mu^{s+t}(\mathfrak{p},M)=\mu^{s}(\mathfrak{p},\operatorname{H}^{t}_{\Phi}(M))$.

###### Proof.

Note that by Theorem 4.4,
$\operatorname{Ext}^{s+t}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},M_{\mathfrak{p}})\cong\operatorname{Ext}^{s}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},\operatorname{H}^{t}_{\Phi}(M)_{\mathfrak{p}})$
. Therefore
$\mu^{s+t}(\mathfrak{p},M)=\mu^{s}(\mathfrak{p},\operatorname{H}^{t}_{\Phi}(M))$.

∎

######  Corollary 4.6.

Let $M$ be an arbitrary $R$-module and $N$ be a finitely generated
$\mathfrak{a}$-torsion $R$-module for some $\mathfrak{a}\in\Phi$. Suppose that
$\operatorname{Ext}^{s+t-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))=0$ for all
$i,0\leq i<t$ or $t+1\leq i\leq s+t$,
$\operatorname{Ext}^{s+1+i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))=0$ for all
$i,0\leq i\leq t$, and
$\operatorname{Ext}^{s-1-i}_{R}(N,\operatorname{H}^{i}_{\Phi}(M))=0$ for all
$i,0\leq i\leq s-1$. Then:

$\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\cong\operatorname{Ext}^{s+t}_{R}(N,M)\cong\operatorname{H}^{s+t}_{\mathfrak{a}}(N,M)\cong\operatorname{Ext}^{s}_{R}(N,\operatorname{H}^{t}_{\mathfrak{a}}(M))$

###### Proof.

It follows by Theorem 4.4 and [21, Lemma 2.5(c) and Theorem 2.21]. Note that
$N=\Gamma_{\mathfrak{a}}(N)\subseteq\Gamma_{\Phi}(N)$. ∎

######  Corollary 4.7.

Let $M$ be an arbitrary $R$-module and $N$ be a finitely generated $R$-module
with $\operatorname{Supp}_{R}(N)=\operatorname{V}(\mathfrak{a})$ for some
$\mathfrak{a}\in\Phi$. If $\operatorname{H}^{i}_{\Phi}(M)=0$ for all $0\leq
i<t$, then:

$\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\cong\operatorname{Ext}^{t}_{R}(N,M)\cong\operatorname{H}^{t}_{\mathfrak{a}}(N,M)\cong\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\mathfrak{a}}(M)).$

###### Proof.

Put $s=0$ in Corollary 4.6. ∎

######  Definition 4.8.

([10, Definition 5.3]) Let $M$ be a finitely generated $R$-module and $\Phi$ a
system of ideals of $R$. We define grade of $\Phi$ on $M$ as:

$\operatorname{grade}(\Phi,M)=\inf\\{\operatorname{grade}(\mathfrak{a},M)|\mathfrak{a}\in\Phi\\}$.

It is easy to see that, if for each $\mathfrak{a}\in\Phi$, $M={\mathfrak{a}}M$
then, $\operatorname{grade}(\Phi,M)=+\infty$, otherwise we have
$\operatorname{grade}(\Phi,M)<+\infty$.

######  Corollary 4.9.

Let $M$, $N$ be two finitely generated $R$-modules such that
$\operatorname{Supp}_{R}(N)=\operatorname{V}(\mathfrak{a})$ for some
$\mathfrak{a}\in\Phi$. If $\operatorname{grade}(\Phi,M)=t$, then

$\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\Phi}(M))\cong\operatorname{Ext}^{t}_{R}(N,M)\cong\operatorname{H}^{t}_{\mathfrak{a}}(N,M)\cong\operatorname{Hom}_{R}(N,\operatorname{H}^{t}_{\mathfrak{a}}(M)).$

###### Proof.

Let $\mathfrak{a}\in\Phi$, then $\operatorname{grade}(\mathfrak{a},M)>t$.
Therefore $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i<t$. Now the
assertion follows by Corollaries 3.8 and 4.7. ∎

A subset $\mathcal{Z}$ of $\operatorname{Spec}(R)$ is said to be stable under
specialization if $\operatorname{V}(\mathfrak{p})\subseteq\mathcal{Z}$ for all
$\mathfrak{p}\in\mathcal{Z}$. Let $M$ be an $R$-module, then
$\Gamma_{\mathcal{Z}}(M)$ is defined by $\Gamma_{\mathcal{Z}}(M):=\\{~{}x\in
M\mid\operatorname{Supp}_{R}(Rx)\subseteq\mathcal{Z}\\}$. So
$\operatorname{\Gamma}_{\Phi}(-)$ is a functor from $\mathscr{C}(R)$ to itself
(where $\mathscr{C}(R)$ denotes the category of all $R$-modules and all
$R$-homomorphisms). The functor $\operatorname{\Gamma}_{\mathcal{Z}}(-)$ is
additive, covariant, $R$-linear and left exact. For each $i\geq 0$, the $i$-th
right derived functor of $\operatorname{\Gamma}_{\mathcal{Z}}(-)$ is denoted
by $\operatorname{H}_{\mathcal{Z}}^{i}(-)$. By [15, Lemma 3.2], for any stable
under specialization subset $\mathcal{Z}$ of $\operatorname{Spec}(R)$, there
is a system of ideals $\Phi$ such that for each $R$-module $M$, the
$R$-modules $\operatorname{H}_{\mathcal{Z}}^{i}(M)$ and
$\operatorname{H}^{i}_{\Phi}(M)$ are isomorphic.

Let $(R,\mathfrak{m})$ be a local ring and $\mathfrak{a}$ be an ideal of $R$.
Let $M$ be a finitely generated $R$-module with $\dim M=d$. It is well-known
that the set
$\operatorname{Supp}_{R}(\operatorname{H}^{d-1}_{\mathfrak{a}}(M))$ is finite.
We close this paper with the following example. It shows that the similar
result is not true for the local cohomology modules with respect to a system
of ideals or with respect to an specialization closed subset.

######  Example 4.10.

Let $(R,\mathfrak{m})$ be a Noetherian local domain such that $\dim(R)=2$.
Since $\\{0\\}$ is a prime ideal of $R$ and $\\{0\\}\subset\mathfrak{m}$, so
there is infinitely many prime ideals of $R$ between $\\{0\\}$ and
$\mathfrak{m}$ such that $\operatorname{ht}_{R}{\mathfrak{p}}=1$. Let
$\mathcal{Z}=\operatorname{Spec}(R)$ and
$\mathcal{Z}_{n}=\\{\mathfrak{p}\in\mathcal{Z}\mid\operatorname{ht}_{R}{\mathfrak{p}}\geq
n\\}$ for $n=0,1,2$. Then $\mathcal{Z}_{2}=\\{\mathfrak{m}\\}$ and
$\mathcal{Z}_{1}=\\{\mathfrak{p}\in\mathcal{Z}\mid\operatorname{ht}_{R}{\mathfrak{p}}=1\\}\cup\\{\mathfrak{m}\\}$.
According to [16, Corollary 4.3 and Lemma 4.2], for $n=2$, $i=1$ and $X=R$,
there is the following exact sequence

$\operatorname{H}^{1}_{\mathfrak{m}}(R)\longrightarrow\operatorname{H}^{1}_{\mathcal{Z}_{1}}(R)\longrightarrow\underset{{\operatorname{ht}_{R}{\mathfrak{p}}=1}}{\bigoplus}\operatorname{H}^{1}_{{\mathfrak{p}}R_{\mathfrak{p}}}(R_{\mathfrak{p}})\longrightarrow\operatorname{H}^{2}_{\mathfrak{m}}(R)$.

By localizing this exact sequence at every $\mathfrak{p}$ with
$\operatorname{ht}_{R}{\mathfrak{p}}=1$, we get
$\operatorname{H}^{1}_{\mathfrak{m}}(R)_{\mathfrak{p}}=0=\operatorname{H}^{2}_{\mathfrak{m}}(R)_{\mathfrak{p}}$
and
$\operatorname{H}^{1}_{\mathcal{Z}_{1}}(R)_{\mathfrak{p}}\cong\operatorname{H}^{1}_{{\mathfrak{p}}R_{\mathfrak{p}}}(R_{\mathfrak{p}})$.
Since
$\operatorname{H}^{1}_{{\mathfrak{p}}R_{\mathfrak{p}}}(R_{\mathfrak{p}})\neq
0$ by Grothendieck non-vanishing theorem [12, Theorem 6.1.4], thus
$\mathfrak{p}\in\operatorname{Supp}_{R}(\operatorname{H}^{1}_{\mathcal{Z}_{1}}(R))$.
Also as we mentioned before there is a system of ideals $\Phi$ such that the
$R$-modules $\operatorname{H}_{\mathcal{Z}_{1}}^{1}(R)$ and
$\operatorname{H}^{1}_{\Phi}(R)$ are isomorphic. This shows that
$\operatorname{Supp}_{R}(\operatorname{H}^{1}_{\mathcal{Z}_{1}}(R))=\operatorname{Supp}_{R}(\operatorname{H}^{1}_{\Phi}(R))$
is an infinite set.

Acknowledgments

The authors would like to thank Dr. Majid Rahro Zargar for his reading of the
first draft and useful discussions.

## References

  * [1] M. Aghapournahr, Weakly cofiniteness of local cohomology modules, J. Algebra Appl., 48 (2019), 1950090.
  * [2] M. Aghapournahr, Cominimaxness of certain general local cohomology modules, Hacet. J. Math. Stat., 18 (2019), 1-13.
  * [3] M. Aghapournahr, Cofiniteness of local cohomology modules for a pair of ideals for small dimensions, J. Algebra Appl., 17 (2018), 1850020.
  * [4] M. Aghapournahr and K. Bahmanpour, Cofiniteness of general local cohomology modules for small dimensions, Bull. Korean Math. Soc., 53 (2016), 1341-1352.
  * [5] M. Aghapournahr, L. Melkersson, A natural map in local cohomology, Ark. Mat., 48 (2010), 243–251.
  * [6] M. Aghapournahr, L. Melkersson, Local cohomology and Serre subcategories, J. Algebra., 320 (2008), 1275–1287.
  * [7] M. Aghapournahr, A. J. Taherizadeh, A. Vahidi, Extension functors of local cohomology modules, Bull. Iranian Math. Soc., 37 (2011), 117-134.
  * [8] J. Asadollahi, K. Khashyarmanesh and Sh. Salarian, _A generalization of the cofiniteness problem in local cohomology modules_ , J. Aust. Math. Soc., 75 (2003), 313–324.
  * [9] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321 (2009), 1997–2011.
  * [10] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21(1980), 173-181.
  * [11] M. H. Bijan-Zadeh, Torsion theory and local cohomology over commutative Noetherian ring, J. London Math. Soc., 19(1979), 402-410.
  * [12] M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, in Cambridge Studies in Advanced Mathematics, Vol. 136, 2nd edn. (Cambridge University Press, Cambridge, 2013), xxii+491 pp.
  * [13] W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge Univ. Press, Cambridge, UK, 1993.
  * [14] K. Divaani-Aazar, H. Faridian and M. Tousi, Stable under specialization sets and cofiniteness, J. Algebra Appl., 18 (2019), 1950015.
  * [15] K. Divaani-Aazar, R. Naghipour and M. Tousi, The Lichtenbaum-Hartshorne theorem for generalized local cohomology and connectedness, Comm. Algebra., 30 (2002), 3687-3702.
  * [16] K. Divaani-Aazar and M. R. Zargar, The derived category analogues of Faltings Local-global Principle and annihilator theorems, J. Algebra Appl., 18 (2019), 1950140.
  * [17] M. Hatamkhani and K. Divaani-Aazar, The derived category analogue of the Hartshorne-Lichtenbaum vanishing theorem, Tokyo J. Math., 36 (2013), 195-205.
  * [18] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK, 1986.
  * [19] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005), 649–668.
  * [20] Z. Tang, _Local-global Principle for the Artinianness of local cohomology modules_ , Comm. Algebra, 40 (2012), 58-63.
  * [21] A. Vahidi and M. Aghapournahr, Some results on generalized local cohomology modules, Comm. Algebra, 43 (2015), 2214-2230.


# Pre-training with Large Language Model-based Document Expansion
for Dense Passage Retrieval

Guangyuan Ma1,2, Xing Wu1,2, Peng Wang1,2, Zijia Lin3, Songlin Hu1,2

###### Abstract

In this paper, we systematically study the potential of pre-training with
Large Language Model(LLM)-based document expansion for dense passage
retrieval. Concretely, we leverage the capabilities of LLMs for document
expansion, i.e. query generation, and effectively transfer expanded knowledge
to retrievers using pre-training strategies tailored for passage retrieval.
These strategies include contrastive learning and bottlenecked query
generation. Furthermore, we incorporate a curriculum learning strategy to
reduce the reliance on LLM inferences. Experimental results demonstrate that
pre-training with LLM-based document expansion significantly boosts the
retrieval performance on large-scale web-search tasks. Our work shows strong
zero-shot and out-of-domain retrieval abilities, making it more widely
applicable for retrieval when initializing with no human-labeled data.

## Introduction

Dense passage retrieval (Karpukhin et al. 2020) has broad real-world
applications, like web search (Liu et al. 2021; Zou et al. 2023), retrieval-
augmented generation (Lewis et al. 2020; Cai et al. 2022) and query answering
(Sakata et al. 2019). It utilizes well-trained language-model-based retrievers
to extract sentence representations and retrieve relevant passages with given
queries. Recent studies have made impressive progress in improving the
effectiveness of dense retrievers, such as hard negative mining (Qu et al.
2021), late interaction (Khattab and Zaharia 2020; Santhanam et al. 2022),
distillation (Ren et al. 2021; Lu et al. 2022), and ensembling (Gao and Callan
2022; Wu et al. 2023b). Moreover, the development of task-specific pre-
training (Gao and Callan 2021; Wu et al. 2023a; Liu and Shao 2022) pushes the
limits of retrieval tasks to new boundaries. Specifically, those studies
usually employ contrastive learning with span corruption (Gao and Callan 2022;
Izacard et al. 2021; Ma et al. 2022), or additional decoder with bottlenecked
structures (Gao and Callan 2021; Lu et al. 2021; Liu and Shao 2022; Wu et al.
2023a) for better representation learning.

Large language models (LLMs), like ChatGPT (Ouyang et al. 2022), PaLM
(Chowdhery et al. 2022), LLaMA (Touvron et al. 2023), and tk-Instruct (Wang et
al. 2022b), are pre-trained on large-scale web corpus and exhibit excellent
abilities in context generation and instruction following. There has been
growing interest in incorporating powerful LLMs into retrieval tasks. Existing
studies (Gao et al. 2023; Wang, Yang, and Wei 2023; Jagerman et al. 2023; Yu
et al. 2023) focus on query expansion with LLMs for enhancing the lexical
match of query-passage pairs. They utilize the LLM-generated relevant passages
as enriched query contexts. Those studies have yielded better retrieval
performances, especially for zero-shot scenarios. Nevertheless, conducting
query expansion still needs heavy online inferences with LLMs, which slows
down the retrieval speed.

While query expansion expands the query with generated passages, document
expansion, i.e., query generation, is also a popular technique to boost
retrieval performances. It exploits a fully fine-tuned model, like T5
(Nogueira et al. 2019) or BART (Cho et al. 2022), to generate relevant queries
of a given passage, which either enrich the context of the passage or serve as
additional fine-tuning corpus. Due to the excellent generation ability of
LLMs, huge potential lies in the utilization of LLMs as document expansion
models. However, we argue that several drawbacks still hinder such usage.
Firstly, document expansion relies on the online inference of LLM in open-
domain passage retrieval, particularly when dealing with candidate corpora
from new domains. To avoid the need for additional LLM inferences during
retrieval, a feasible solution is to pre-train or fine-tune an end-to-end
retriever. However, this approach lacks exploration and necessitates training
paradigms specifically designed for retrieval. Furthermore, document expansion
involves feeding a substantial corpus into LLMs to generate queries, resulting
in significant costs associated with LLM inferences. Unfortunately, there is a
shortage of methods to mitigate these inference costs.

To mitigate the high online inference costs of LLM document expansion, as is
presented in Figure 1, we prompt the LLM query generation for a series of pre-
training experiments tailored for dense retrieval. We emphasize that our work
only involves LLM inferences at the pre-training stage of retrievers, but not
the inference stage as traditional query (Gao et al. 2023; Wang, Yang, and Wei
2023) or document expansion (Nogueira et al. 2019). Two pre-training
paradigms, i.e., contrastive learning and bottlenecked query generation, are
explored in detail.

For contrastive pre-training, a direct contrastive loss of the generated
queries and passages is used to pull together their embeddings, while pushing
away in-batch negatives in the latent space. We follow the contrastive
architecture in (Gao and Callan 2022) for fair comparision, and we argue that
LLM-generated queries can serve as the better context for effective query-
passage alignment.

Figure 1:  Query Generation prompts for Alpaca-LLaMA and tk-Instruct.

Bottlenecked pre-training techniques are popular in recent works (Lu et al.
2021; Liu and Shao 2022; Wu et al. 2023a), which connect accessional decoders
solely through the encoder’s representation. To pre-train with bottlenecked
query generation, similar to (Wu, Ma, and Hu 2022), we adapt a single-layer
Transformers decoder and use the casual language model (CLM) task to generate
expanded queries with the assistance of the encoder’s embeddings. This
bottlenecked encoder-decoder structure first derives a compressed
representation through the encoder and then decompresses the context
information as LLM-expanded queries via the decoder. As a result, the sentence
embeddings contain enriched context information, providing effective
initialization for fine-tuning and inference. Especially, LLM-based document
expansion requires no human-labeled corpus as previous works (Wu, Ma, and Hu
2022; Cho et al. 2022) for training additional domain-specific generative
models like docT5query (Nogueira et al. 2019).

Furthermore, to mitigate the LLM inference costs for document expansion, we
interpolate a two-stage curriculum learning strategy for both pre-training
schemas. Span corruption is firstly used to randomly sample contextual pairs
from a long document. Then we leverage the generation abilities of LLMs to
produce a relatively small amount of queries for the next stage of pre-
training.

In our study, we use Alpaca-LLaMA (Wang et al. 2023) and tk-Instruct (Wang et
al. 2022b) with different parameter sizes for query generation. We conduct the
experiments on the large-scale MS-MARCO (Nguyen et al. 2016) datasets and test
on the in-domain MS-MARCO passage retrieval task, TREC-DL 2019 & 2020
(Craswell et al. 2020, 2021) and the out-of-domain BEIR (Thakur et al. 2021)
task. Several benefits are observed in our studies. 1) LLMs can generate a
large number of high-quality queries based on the world knowledge of LLM
itself, which requires no additional human labeling and is suitable for
scenarios lacking in manually annotated data. 2) Contrastive pre-training with
LLM-generated queries has stronger in-domain zero-shot retrieval performance
and on-par performance with the state-of-the-art (SOTA) methods after full
fine-tuning. It also shows better domain adaption abilities in out-of-domain
BEIR datasets. 3) Bottlenecked query generation shows better initialization
abilities after full fine-tuning. 4) With our two-stage curriculum learning
strategy, we reduce the number of MS-MARCO corpus involved in LLM inferences
from 8.8 million to 0.4 million, while keeping the minor performance
degeneration.

Our contributions are summarized as follows.

  1. •

We systematically study the potential of incorporating LLMs into the pre-
training stage of dense passage retrieval, suitable for the scarcity of human-
annotated data.

  2. •

We find stronger zero-shot and fine-tuned performances with contrastive
learning and good initialization abilities with bottlenecked query generation
pre-training.

  3. •

We design a two-stage curriculum learning strategy that greatly reduces the
usage of LLM-expanded queries while keeping the minor performance
degeneration.

Figure 2:  Pre-training with LLM-based document expansion for dense passage
retrieval. a) We utilize large language models (LLMs) to generate pseudo-
queries with zero-shot or few-shot prompts. b) Bottlenecked query generation
pre-training appends an auxiliary Transformers decoder to the encoder. Besides
the Masked Language Modelling (MLM) loss of the encoder, we connect the
encoder-decoder with merely the bottlenecked representation, i.e., the hidden
states of [CLS] token, and make the decoder generate whole LLM-expanded
queries with the Cross-Entropy (CE) loss. c) Contrastive pre-training pulls
together the representations of the passage and LLM-expanded queries and
pushes away in-batch negatives. To minimize reliance on LLM expansions, we
implement a two-stage curriculum learning strategy. It first utilizes randomly
sampled passages to fully initialize the encoders. And then we can use a
relatively small amount of LLM-expanded queries in the second phase.

## Methodology

In this section, we first introduce the definition of dense passage retrieval.
Then we introduce our method for LLM query generation, the detailed pre-
training designs of contrastive learning and bottlenecked query generation,
and the two-stage curriculum learning strategy for extended analyses.

### Preliminaries

Given a query $q$ and a set of passages $P_{n}$, the passage retrieval task
aims to find the relevant passages based on the similarity search. Dense
passage retrieval utilizes an encoder model $Enc$, e.g., a Transformers-based
model like BERT (Devlin et al. 2019), to yield the sentence representations
and measure query-passage similarities through inner product or cosine
distance. Formally, given a query $q$ and a passage $q$, we can use a query
encoder $Enc_{q}$ and a passage encoder $Enc_{p}$ to derive their
corresponding sentence representations, i.e., $v_{q}$ and $v_{p}$ from the
encoder hidden states of the last layer at CLS position
$h_{last}^{\texttt{[CLS]}}$. Then the similarity between $q$ and $p$, i.e.,
$Sim(q,p)$, can be calculated as the inner product of $v_{q}$ and $v_{p}$ for
simplicity as follows.

$Sim(q,p)=Enc_{q}(q)\cdot Enc_{p}(p)=v_{q}^{T}v_{p}$ (1)

The key to improving retrieval performances is to yield stronger
representations $v_{q}$, $v_{p}$ with better context alignment. The
representations can be regarded as the compression of full contexts. We
believe that incorporating the strong context-generation abilities of LLMs
into the pre-training stage with carefully designed pre-tasks can be a new way
for improving such alignment.

### LLM Query Generation

Given a passage $p$, we use a zero-shot prompt for Alpaca-LLaMA and a few-shot
prompt for tk-Instruct to expand queries, as illustrated in Figure 1. We
empirically find that Alpaca 7B and 13B models work well on the zero-shot
prompt, which helps save computation budgets. We manually write a few examples
for tk-Instruct, as we find that few-shot prompts make its query generation
more stable.

LLM-based document expansion enriches the pre-training corpus with additional
contextual information. Instead of directly appending the expanded queries
onto the passage, we seek to incorporate them into our pre-training stage for
better initialization of end-to-end retrievers. Our work only involves LLM
inference at the pre-training stage, but not the retrieval stage like
traditional query or document expansion works. Two pre-training paradigms are
involved to incorporate the LLM-generated queries into the dense model pre-
training.

| MS-MARCO | TREC DL 19 | TREC DL 20
---|---|---|---
Model / Zero-shot Evaluation | MRR@10 | Recall@50 | Recall@1k | nDCG@10 | nDCG@10
BM25 | 18.7 | 59.2 | 85.7 | 51.2 | 47.7
SimCSE (Gao, Yao, and Chen 2021)$\dagger$ | 8.7 | 33.7 | 64.6 | 24.5 | 17.9
coCondenser (Gao and Callan 2022)$\dagger$ | 7.5 | 31.3 | 58.1 | 22.1 | 20.7
Contriever (Izacard et al. 2021)$\dagger$ | 16.8 | 60.8 | 89.1 | 44.5 | 43.2
Contrastive Pre-training
Baseline | 12.5 | 49.0 | 82.3 | 36.0 | 38.4
\+ tk-inst 3b queries | 20.9+8.4 | 70.2+21.2 | 92.8+10.5 | 47.0+11.0 | 48.6+10.2
\+ Alpaca 7b queries | 22.6+10.1 | 70.7+21.7 | 93.8+11.5 | 51.0+15.0 | 48.9+10.5
\+ Alpaca 13b queries | 22.7+10.2 | 71.7+22.7 | 94.3+12.0 | 53.9+17.9 | 50.1+11.7
Table 1:  Zero-shot evaluation of contrastive pre-training with LLM-based
document expansion. $\dagger$ denotes our reproduced results. The best scores
are marked in bold. Results with the increment over the corresponding baseline
have been tested with two-tailed t-tests, demonstrating statistically
significant improvements ( p-value $\leq$ 0.01 ).

### Bottlenecked Query Generation Pre-training

Bottlenecked pre-training trains a monomeric encoder ($Enc$) with good
initialization abilities for subsequent fine-tuning. Given a tokenized
sentence $t\in T$ from the training corpus, we randomly select a certain ratio
of tokens, with the corresponding indices denoted as $M$, and replace them
with mask tokens [m]:

$mask(t)=\\{\texttt{[CLS]},t_{1},t_{2},\texttt{[m]},t_{4},...,t_{n},\texttt{[SEP]}\\}$
(2)

Cross-Entropy (CE) loss is then used to optimize as Masked Language Model
(MLM) loss for the encoder.

$\mathcal{L}_{enc}=-\sum_{t\in T}\sum_{i\in{M}}\log p(t_{i}|Enc(mask({t})))$
(3)

where $t_{i}$ is groundtruth tokens w.r.t corresponding mask tokens [m].

A single-layer accessional Transformers decoder ($Dec$) is further introduced,
which receives the input from the concatenation of the encoder representation
$h_{last}^{\texttt{[CLS]}}$ and contextual texts $x$, e.g., LLM-generated
queries.

${T}_{ctx}=\\{h_{last}^{\texttt{[CLS]}},x_{1},...,x_{N},\texttt{[SEP]}\\}$ (4)

Then the decoder uses the Casual Language Model (CLM) loss to generate the
whole input context with the assistance of encoder representation.

$\mathcal{L}_{dec}=-\sum_{x_{i}\in{T}_{ctx}}\log p(x_{i}|Dec(x[:i-1]))$ (5)

The final loss $\mathcal{L}$ is then formulated as follows.

$\mathcal{L}=\mathcal{L}_{enc}+\mathcal{L}_{dec}$ (6)

Through the bottlenecked encoder-decoder structure, we seek to compress the
context signal from LLM-generated queries into the encoder representations and
give strong initialization ability to the encoder.

### Contrastive Pre-training

For reproduction and fair comparison, we adapt the contrastive pre-training
architecture from coCondenser (Gao and Callan 2022). The passage $p$ and its
sampled or generated context ${p}_{ctx}$ are directly forwarded through the
encoder $Enc$. Besides the MLM loss $\mathcal{L}_{enc}$ of the encoder, an
extra Transformers decoder $Dec_{ext}$ is also introduced for representation
pre-training, which takes the concatenation of the encoder representation
$h_{last}^{\texttt{[CLS]}}$ and encoder hidden states $h_{l}^{i}$ from $l$-th
layer. Then a cross-entropy loss is used for the decoder’s pre-task.

$\mathcal{L}_{ext}=-\sum_{t\in T}\sum_{i\in{M}}\log
p(t_{i}|Dec_{ext}({h_{last}^{\texttt{[CLS]}},h_{l}^{1},...,h_{l}^{i}}))$ (7)

Differently, for pre-training with LLM-expanded queries, assuming $v_{p}$ and
$v_{ctx}$ denote encoders’ representations, a contrastive loss with in-batch
negatives is used as follows.

$\mathcal{L}_{CL}=-\log\frac{\exp(v_{p}\cdot v_{ctx}^{+})}{\exp({v_{p}\cdot
v_{ctx}^{+}})+\sum\exp({v_{p}\cdot v_{ctx}^{-}})}$ (8)

where $v_{ctx}^{+}$ is the context text, e.g. LLM-generated queries,
corresponding to $p$. And $v_{ctx}^{-}$ is the in-batch negatives, i.e., the
context texts of the other passages in the batch.

The final optimization objective is the sum of the above losses.

$\mathcal{L}=\mathcal{L}_{enc}+\mathcal{L}_{ext}+\mathcal{L}_{CL}$ (9)

Through contrastive pre-training, the representations of passage and LLM-
generated queries are directly pulled together in the same latent space, which
gives better query-passage alignment and zero-shot ability to encoders.

### Curriculum Learning

As discussed before, LLM-based document expansion faces the challenge of
costly inference due to large numbers of documents or passages. Since we
intend to pre-train our model with enriched contexts, inspired by the wisdom
of curriculum learning (Bengio et al. 2009), we consider 1) a randomly cropped
passage span as a coarse-grained context, while 2) the LLM-expanded queries as
fine-grained context, as depicted in Figure 2. Following the span corruption
strategies in the seed-encoder (Lu et al. 2021) and coCondenser (Gao and
Callan 2022), we use the coarse-grained context as the passage itself in the
bottlenecked generation pre-training, and the randomly sampled passage span in
contrastive pre-training. As we focus on LLM-based document expansion, other
span corruption strategies (Wu et al. 2023a) are left to our future work.
After pre-training on a large amount of randomly cropped contexts, we
initialize from the first stage and then use the fine-grained LLM-expanded
queries for the second-phrase pre-training. Experiments find that this
curriculum strategy greatly reduces the need for LLM inferences on MS-MARCO
passages, while still maintaining similar retrieval performances.

| MS-MARCO | TREC DL 19 | TREC DL 20
---|---|---|---
Model / Fine-tuned Results | MRR@10 | Recall@50 | Recall@1k | nDCG@10 | nDCG@10
Contriever (Izacard et al. 2021)$\dagger$ | 33.4 | 85.0 | 98.4 | 62.8 | 63.2
Condenser (Gao and Callan 2021) | 36.6 | 85.4$\dagger$ | 97.4 | 69.8 | 66.5$\dagger$
coCondenser (Gao and Callan 2022) | 38.2 | 86.5$\dagger$ | 98.4 | 71.7$\dagger$ | 68.4$\dagger$
SimLM (Wang et al. 2022a) | 39.1 | 87.3$\dagger$ | 98.6 | 68.9$\dagger$ | 68.8$\dagger$
RetroMAE (Liu and Shao 2022) | 39.3 | 87.0$\dagger$ | 98.5 | 69.1$\dagger$ | 70.0$\dagger$
CoT-MAE (Wu et al. 2023a) | 39.4 | 87.0 | 98.7 | 70.9$\dagger$ | 70.4
Contrastive Pre-training
Baseline | 38.8 | 87.8 | 98.8 | 71.1 | 68.4
\+ tk-instruct 3b queries | 39.6+0.8 | 88.8+1.0 | 99.0 | 72.9+1.8 | 71.1+2.7
\+ Alpaca 7b queries | 40.0+1.2 | 89.0+1.2 | 99.1 | 72.9+1.8 | 71.3+2.9
\+ Alpaca 13b queries | 39.6+0.8 | 88.8+1.0 | 98.9 | 72.6+1.5 | 72.3+3.9
Bottlenecked Query Generation
Baseline | 39.3 | 87.9 | 98.6 | 69.9 | 67.4
\+ tk-instruct 3b queries | 40.3+1.0 | 88.7+0.8 | 98.9 | 70.7+0.8 | 70.0+2.6
\+ Alpaca 7b queries | 39.9+0.6 | 88.2 | 98.7 | 69.6 | 70.7+3.3
\+ Alpaca 13b queries | 39.7 | 88.3 | 98.7 | 70.8+0.9 | 69.4+2.0
Table 2:  Fine-tuned results of pre-training with LLM-based document
expansion. $\dagger$ denotes our reproduced results. The best scores are
marked in bold. Results with the increment over the corresponding baseline
have been tested with two-tailed t-tests, demonstrating statistically
significant improvements ( p-value $\leq$ 0.01 ).

### Zero-shot evaluation and Fine-tuning

We conduct the zero-shot evaluation of the contrastive pre-trained encoder
without fine-tuning on MS-MARCO, TREC-DL, and BEIR datasets. We conduct fine-
tuning on both pre-training schemas to verify their retrieval initialization
ability. Following DPR (Karpukhin et al. 2020), a simple contrastive loss is
applied to optimize the retriever.

$\mathcal{L}=-\log\frac{\exp(q\cdot p^{+})}{\exp(q\cdot p^{+})+\sum\exp(q\cdot
p^{-})}$ (10)

where $q$ is a given query, $p^{+}$ and $p^{-}$ are their corresponding
positive passage and negative passages respectively.

## Experiments

This section introduces detailed experiment settings for pre-training and
fine-tuning. Then we present the main results.

### Pre-training

Following the pretraining settings in (Gao and Callan 2022), we choose the MS-
MARCO dataset (Nguyen et al. 2016) with 3.2M documents as our pre-training
corpus. LLMs with different types and parameter sizes, i.e. Alpaca 7B, 13B
(Wang et al. 2023), and tk-instruct 3B (Wang et al. 2022b) are used to
generate the queries for LLM-based document expansion. Nucleus sampling with
$top_{p}=0.95$, $top_{k}=50$, and $temperature=0.7$ is used for LLM
generation.

For bottlenecked query generation pre-training, the encoder is initialized
from the 12-layer BERT-base model (Devlin et al. 2019), while the single-layer
decoder is randomly initialized from scratch. We use the AdamW optimizer with
a learning rate of 3e-4, batch size of 2048, total steps of 80k, and a warmup
ratio of 0.1. The pre-training uses 8 Tesla A100 GPUs and trains for 19 hours.
For contrastive pre-training, we adapt the codes and architecture from (Gao
and Callan 2022) and initialize from (Gao and Callan 2021) by following their
settings. We use a learning rate of 1e-4, batch size of 2048, and total steps
of 120k and keep other hyper-parameters the same as above for training 50
hours. For curriculum learning, 75% of the total steps are used for the first
stage of pre-training with sampled spans, and the remaining 25% of the steps
are used for the second stage of pre-training with LLM-generated queries. We
use the cosine scheduler with the same hyper-parameter settings for the first
stage, and a constant learning rate for the second stage. All pre-training
seeds are set to 42 for reproducibility. The encoders are directly tested on
downstream tasks without fine-tuning for zero-shot evaluation.

Results / nDCG@10 | BM25 | coCondenser | Contriever | SimCSE | Baseline | \+ tk-Instruct 3b | \+ Alpaca 7b | \+ Alpaca 13b
---|---|---|---|---|---|---|---|---
TREC-COVID | 65.6 | 21.2 | 27.3 | 27.5 | 16.2 | 36.8+20.6 | 52.3+36.1 | 54.7+38.5
NFCorpus | 32.5 | 13.7 | 31.7 | 10.5 | 29.9 | 33.1+3.2 | 30.9+1.0 | 33.5+3.5
NQ | 32.9 | 10.7 | 25.4 | 16.3 | 9.3 | 34.3+25.0 | 31.8+22.5 | 31.9+22.6
HotpotQA | 60.3 | 22.3 | 48.1 | 23.8 | 24.2 | 56.2+32.0 | 51.5+27.3 | 51.8+27.6
FiQA-2018 | 23.6 | 7.2 | 24.5 | 9.7 | 19.6 | 29.8+10.3 | 27.2+7.6 | 28.6+9.0
ArguAna | 31.5 | 34.4 | 37.9 | 28.0 | 35.8 | 44.6+8.8 | 40.5+4.8 | 40.6+4.9
Touché-2020 | 36.7 | 5.8 | 16.7 | 13.4 | 8.1 | 16.3+8.2 | 13.7+5.5 | 16.9+8.7
CQADupStack | 29.9 | 10.5 | 28.4 | 13.5 | 18.2 | 30.9+12.8 | 32.4+14.2 | 33.3+15.1
Quora | 78.9 | 71.3 | 83.5 | 73.7 | 75.8 | 83.8+8.0 | 83.3+7.5 | 84.3+8.5
DBPedia | 31.3 | 16.3 | 29.2 | 16.7 | 22.5 | 30.2+7.7 | 28.8+6.3 | 29.6+7.1
SCIDOCS | 15.8 | 4.6 | 14.9 | 6.1 | 10.4 | 13.6+3.2 | 13.5+3.2 | 14.4+4.1
FEVER | 75.3 | 16.8 | 68.2 | 29.2 | 43.6 | 61.9+18.3 | 67.2+23.6 | 73.1+29.5
Climate-FEVER | 21.3 | 6.4 | 15.5 | 14.2 | 8.5 | 18.4+9.8 | 13.8+5.3 | 17.2+8.6
SciFact | 66.5 | 43.2 | 64.9 | 25.0 | 52.7 | 64.4+11.7 | 60.8+8.1 | 60.9+8.2
Average | 43.0 | 20.3 | 36.9 | 22.0 | 26.8 | 39.6+12.8 | 39.1+12.4 | 40.8+14.0

Table 3:  Out-of-domain zero-shot evaluation of contrastive pre-training with
LLM-based document expansion on BEIR benchmark. All baselines tested on
nDCG@10 are based on our reproduction. Results with the increment over the
corresponding baseline have been tested with two-tailed t-tests, demonstrating
statistically significant improvements ( p-value $\leq$ 0.01 ).

### Fine-tuning

The encoder is fine-tuned and tested on MS-MARCO Passage Ranking task (Nguyen
et al. 2016), TREC Deep Learning (DL) 2019 (Craswell et al. 2020) and 2020
(Craswell et al. 2021). MS-MARCO Passage Ranking dataset contains 8.8 million
passages and 500k human annotated query-passage pairs. Following (Gao and
Callan 2021), we report the performance metrics on MRR@10, Recall@50,
Recall@1K, and evaluate the models on its development set with 6,980 queries,
because its test set is not publicly available. TREC-DL 2019 and 2020 test
sets both contain 200 annotated queries. We adopt the Tevatron pipeline (Gao
et al. 2022) with the AdamW optimizer for a learning rate of 2e-5, a batch
size of 8, negative samples per passage of 15, a negative depth of 200, and
trains for 3 epochs. The performance metrics of TREC and BEIR are reported on
NDCG@10.

### Baselines

We compare to self-contained baselines without using LLM-expanded queries, but
only use randomly sampled spans as coarse-grained contexts. All other hyper-
parameters used in the pre-training remain the same as the main experiments
for fair comparison. In fine-tuned experiments, the contrastive pre-training
baselines are mainly from (Wu, Ma, and Hu 2022) by following their hyper-
parameter settings, and other baselines are based on our settings.

We also compare with other remarkable baselines, including the traditional
sparse retrieval BM25 (Robertson, Zaragoza et al. 2009), unsupervised sentence
similarity encoder SimCSE (Gao, Yao, and Chen 2021), unsupervised contrastive
pre-training method coCondenser (Gao and Callan 2022) and Contriever (Izacard
et al. 2021) for zero-shot evaluation. For fine-tuned results, we also compare
with the latest bottlenecked pre-training methods, including Condenser (Gao
and Callan 2021), SimLM (Wang et al. 2022a), RetroMAE (Liu and Shao 2022) and
CoT-MAE (Wu et al. 2023a). Note that the recent bottlenecked methods using
multi-task pre-training (Zhou et al. 2022) or hybrid retrieval (Liu et al.
2023; Wu et al. 2023b) are not compared, as they are beyond the scope of fair
comparison.

### Zero-shot Evaluation

Table 1 reports the in-domain zero-shot evaluation of contrastive pre-training
with LLM-based document expansion. Pre-training with LLM-expanded queries
shows clear improvements over its baselines that merely use randomly sampled
passages. This indicates that our method achieves strong zero-shot retrieval
abilities for in-domain evaluation on the MS-MARCO and TREC-DL 19 & 20
datasets.

### Fine-tuned Retrieval

The fine-tuned results of the two pre-training methods, i.e., contrastive
pretraining and bottlenecked query generation pretraining, are presented in
Table 2. Pre-training with LLM-expanded queries also gives a statistically
significant boost to their baselines and counterparts. In addition, we notice
that 1) Contrastive pre-training gives better results on the MS-MARCO passage
task (in Recall@50 and Recall@1k) and TREC-DL 19 & 20 (in nDCG@10). 2)
Bottlenecked query generation gives better initialization on MS-MARCO w.r.t
the official preferred metric MRR@10, but still lies behind contrastive pre-
training in other metrics.

### Out-of-domain Evaluation

We also evaluate the out-of-domain zero-shot BEIR benchmark for contrastive
pre-training with LLM-based document expansion and report the metric (nDCG@10)
in Table 3. BM25 is a very strong baseline w.r.t all the other contrastive
pre-training methods that do not go through human-labeled fine-tuning.
Nevertheless, our method still shows strong improvements over its contrastive
baseline. Specifically, compared with Contriever (Izacard et al. 2021), which
is an unsupervised contrastive method pre-trained on a much larger corpus
CCNET (Wenzek et al. 2020), pre-training with LLM expansion also shows
superior retrieval performances.

## Extended Analyses

In this section, we analyze the effect of scaling up LLMs and the curriculum
learning strategy with expanded queries generated by Alpaca 13b 111Alpaca 13b
is chosen because of better results in zero-shot and on-par performances in
fine-tuned retrieval..

Figure 3:  Effects of curriculum learning for fine-tuned bottlenecked pre-
training with expanded queries generated by Alpaca 13b. The dashed lines are
the corresponding baselines from Table 2.

### Effects of Scaling up LLMs

We use three LLMs with different parameter sizes ranging from 3b to 13b,
prompting them for document expansion and integrating the generated queries
into pre-training. As shown in Table 1, scaling up the LLMs shows better
retrieval performances in zero-shot contrastive pre-training. But this
observation is not valid after fine-tuning in Table 2. We hypothesize that for
fine-tuning with human labels, these LLMs are all capable enough for giving a
good initialization for retrieval.

### Effects of Curriculum Learning

To further reduce the need for LLM-expanded queries in pre-training, we
attempt to use a curriculum learning strategy as detailed before. We use
randomly sampled spans as the coarse-grained context in the first stage of
curriculum pre-training for 75% of the total training steps. Then we use a
small amount of LLM-expanded queries as the fine-grained context for the
remaining pre-training steps. Figure 3 and 4 show that both pre-training
schemas benefit from curriculum learning. Bottleneck query generation
outperforms its baseline with just 0.4 million LLM-expanded queries after
fine-tuning. Zero-shot contrastive pre-training surpasses the baselines and
continues to demonstrate sustainable improvements as the number of fine-
grained queries increases.

## Related Works

### Pre-training for Dense Retrieval

Dense passage retrieval has gained sustainable improvements with the recent
development of pre-training tasks. Some works focus on contrastive pre-
training with constructed span relationship (Chang et al. 2020), randomly
cropped spans (Gao and Callan 2022) or multiple granularity alignments (Ma et
al. 2022). And meanwhile, the others focus on pre-training with auxiliary
bottlenecked decoders, like pre-training with a weak generative decoder (Lu et
al. 2021), extreme masked ratio (Liu and Shao 2022), and contextual span
sampling (Wu et al. 2023a). Our method is similar to (Gao and Callan 2022) and
(Wu et al. 2023a), but our core contribution is the methodology of
incorporating expanded queries generated by LLMs into such pre-training
schemas, which brings better context alignment and stronger zero-shot and
fine-tuned performances.

Figure 4:  Effects of curriculum learning for zero-shot contrastive pre-
training with LLM-expanded queries.

### LLM-based Query and Document Expansion

Traditional query or document expansions generate additional context via query
rewriting (Lavrenko and Croft 2017), or with specially fine-tuned T5 (Nogueira
et al. 2019) or BART models (Cho et al. 2022). With the bloom of LLMs (Ouyang
et al. 2022; Touvron et al. 2023; Wang et al. 2022b), growing researches focus
on using LLMs as query expansion models (Gao et al. 2023; Wang, Yang, and Wei
2023; Jagerman et al. 2023; Yu et al. 2023), which enhance the lexical match
of query-passage pairs.

However, as discussed before, LLM-based document expansion is yet lacking
exploration due to expensive inference costs brought by the huge amount of
documents and the online inference issue. We propose to tackle those issues
with pre-training techniques and curriculum learning strategies tailored for
dense retrieval. Our method is also orthogonal to traditional query and
document expansion and can incorporate them into the retrieval stage.

## Conclusion

This paper systematically studies the potential of pre-training with Large
Language Model-based document expansion for dense passage retrieval. Strong
improvements in zero-shot and out-of-domain performances are observed in
contrastive pre-training with LLM-based document expansion. Moreover, both
contrastive pretraining and bottlenecked query generation pretraining achieve
good retrieval abilities after fine-tuning. We further propose a two-stage
curriculum learning strategy that can greatly reduce the need for LLM-expanded
queries in pre-training, while keeping the minor performance degeneration.
LLMs excel in expanding high-quality queries with enriched context
information, which is suitable for scenarios lacking in human annotations.
Researchers can thus deploy quick initialization of an unsupervised dense
retrieval system with the pre-training of LLM-based document expansion, with
even NO human labels provided.

## Limitation

We are also interested in testing more types of LLMs with different sizes,
such as ChatGPT (Ouyang et al. 2022), and LLaMA 2 (Touvron et al. 2023), or
different prompts for document expansion, but our experiment budget is limited
to support immediate investigations and we leave that to our future works.

## References

  * Bengio et al. (2009) Bengio, Y.; Louradour, J.; Collobert, R.; and Weston, J. 2009.  Curriculum learning.  In Danyluk, A. P.; Bottou, L.; and Littman, M. L., eds., _Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009, Montreal, Quebec, Canada, June 14-18, 2009_ , volume 382 of _ACM International Conference Proceeding Series_ , 41–48. ACM.
  * Cai et al. (2022) Cai, D.; Wang, Y.; Liu, L.; and Shi, S. 2022.  Recent advances in retrieval-augmented text generation.  In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_ , 3417–3419.
  * Chang et al. (2020) Chang, W.; Yu, F. X.; Chang, Y.; Yang, Y.; and Kumar, S. 2020.  Pre-training Tasks for Embedding-based Large-scale Retrieval.  In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.
  * Cho et al. (2022) Cho, S.; Jeong, S.; Yang, W.; and Park, J. C. 2022.  Query Generation with External Knowledge for Dense Retrieval.  In Agirre, E.; Apidianaki, M.; and Vulic, I., eds., _Proceedings of Deep Learning Inside Out: The 3rd Workshop on Knowledge Extraction and Integration for Deep Learning Architectures, DeeLIO@ACL 2022, Dublin, Ireland and Online, May 27, 2022_ , 22–32. Association for Computational Linguistics.
  * Chowdhery et al. (2022) Chowdhery, A.; Narang, S.; Devlin, J.; Bosma, M.; Mishra, G.; Roberts, A.; Barham, P.; Chung, H. W.; Sutton, C.; Gehrmann, S.; Schuh, P.; Shi, K.; Tsvyashchenko, S.; Maynez, J.; Rao, A.; Barnes, P.; Tay, Y.; Shazeer, N.; Prabhakaran, V.; Reif, E.; Du, N.; Hutchinson, B.; Pope, R.; Bradbury, J.; Austin, J.; Isard, M.; Gur-Ari, G.; Yin, P.; Duke, T.; Levskaya, A.; Ghemawat, S.; Dev, S.; Michalewski, H.; Garcia, X.; Misra, V.; Robinson, K.; Fedus, L.; Zhou, D.; Ippolito, D.; Luan, D.; Lim, H.; Zoph, B.; Spiridonov, A.; Sepassi, R.; Dohan, D.; Agrawal, S.; Omernick, M.; Dai, A. M.; Pillai, T. S.; Pellat, M.; Lewkowycz, A.; Moreira, E.; Child, R.; Polozov, O.; Lee, K.; Zhou, Z.; Wang, X.; Saeta, B.; Diaz, M.; Firat, O.; Catasta, M.; Wei, J.; Meier-Hellstern, K.; Eck, D.; Dean, J.; Petrov, S.; and Fiedel, N. 2022.  PaLM: Scaling Language Modeling with Pathways.  _CoRR_ , abs/2204.02311.
  * Craswell et al. (2021) Craswell, N.; Mitra, B.; Yilmaz, E.; and Campos, D. 2021.  Overview of the TREC 2020 deep learning track.  arXiv:2102.07662.
  * Craswell et al. (2020) Craswell, N.; Mitra, B.; Yilmaz, E.; Campos, D.; and Voorhees, E. M. 2020.  Overview of the TREC 2019 deep learning track.  arXiv:2003.07820.
  * Devlin et al. (2019) Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2019.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , 4171–4186. Minneapolis, Minnesota: Association for Computational Linguistics.
  * Gao and Callan (2021) Gao, L.; and Callan, J. 2021.  Condenser: a Pre-training Architecture for Dense Retrieval.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 981–993. Online and Punta Cana, Dominican Republic: Association for Computational Linguistics.
  * Gao and Callan (2022) Gao, L.; and Callan, J. 2022.  Unsupervised Corpus Aware Language Model Pre-training for Dense Passage Retrieval.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2843–2853. Dublin, Ireland: Association for Computational Linguistics.
  * Gao et al. (2022) Gao, L.; Ma, X.; Lin, J.; and Callan, J. 2022.  Tevatron: An efficient and flexible toolkit for dense retrieval.  _arXiv preprint arXiv:2203.05765_.
  * Gao et al. (2023) Gao, L.; Ma, X.; Lin, J.; and Callan, J. 2023.  Precise Zero-Shot Dense Retrieval without Relevance Labels.  In Rogers, A.; Boyd-Graber, J. L.; and Okazaki, N., eds., _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2023, Toronto, Canada, July 9-14, 2023_ , 1762–1777. Association for Computational Linguistics.
  * Gao, Yao, and Chen (2021) Gao, T.; Yao, X.; and Chen, D. 2021.  SimCSE: Simple Contrastive Learning of Sentence Embeddings.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 6894–6910. Online and Punta Cana, Dominican Republic: Association for Computational Linguistics.
  * Izacard et al. (2021) Izacard, G.; Caron, M.; Hosseini, L.; Riedel, S.; Bojanowski, P.; Joulin, A.; and Grave, E. 2021.  Towards Unsupervised Dense Information Retrieval with Contrastive Learning.  _CoRR_ , abs/2112.09118.
  * Jagerman et al. (2023) Jagerman, R.; Zhuang, H.; Qin, Z.; Wang, X.; and Bendersky, M. 2023.  Query Expansion by Prompting Large Language Models.  _CoRR_ , abs/2305.03653.
  * Karpukhin et al. (2020) Karpukhin, V.; Oguz, B.; Min, S.; Lewis, P.; Wu, L.; Edunov, S.; Chen, D.; and Yih, W.-t. 2020.  Dense Passage Retrieval for Open-Domain Question Answering.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , 6769–6781. Online: Association for Computational Linguistics.
  * Khattab and Zaharia (2020) Khattab, O.; and Zaharia, M. 2020.  ColBERT: Efficient and Effective Passage Search via Contextualized Late Interaction over BERT.  In Huang, J. X.; Chang, Y.; Cheng, X.; Kamps, J.; Murdock, V.; Wen, J.; and Liu, Y., eds., _Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval, SIGIR 2020, Virtual Event, China, July 25-30, 2020_ , 39–48. ACM.
  * Lavrenko and Croft (2017) Lavrenko, V.; and Croft, W. B. 2017.  Relevance-Based Language Models.  _SIGIR Forum_ , 51(2): 260–267.
  * Lewis et al. (2020) Lewis, P. S. H.; Perez, E.; Piktus, A.; Petroni, F.; Karpukhin, V.; Goyal, N.; Küttler, H.; Lewis, M.; Yih, W.; Rocktäschel, T.; Riedel, S.; and Kiela, D. 2020.  Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks.  In Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.; and Lin, H., eds., _Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual_.
  * Liu et al. (2021) Liu, Y.; Lu, W.; Cheng, S.; Shi, D.; Wang, S.; Cheng, Z.; and Yin, D. 2021.  Pre-trained Language Model for Web-scale Retrieval in Baidu Search.  In Zhu, F.; Ooi, B. C.; and Miao, C., eds., _KDD ’21: The 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Virtual Event, Singapore, August 14-18, 2021_ , 3365–3375. ACM.
  * Liu and Shao (2022) Liu, Z.; and Shao, Y. 2022.  RetroMAE: Pre-training Retrieval-oriented Transformers via Masked Auto-Encoder.  _arXiv preprint arXiv:2205.12035_.
  * Liu et al. (2023) Liu, Z.; Xiao, S.; Shao, Y.; and Cao, Z. 2023.  RetroMAE-2: Duplex Masked Auto-Encoder For Pre-Training Retrieval-Oriented Language Models.  In Rogers, A.; Boyd-Graber, J. L.; and Okazaki, N., eds., _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2023, Toronto, Canada, July 9-14, 2023_ , 2635–2648. Association for Computational Linguistics.
  * Lu et al. (2021) Lu, S.; He, D.; Xiong, C.; Ke, G.; Malik, W.; Dou, Z.; Bennett, P.; Liu, T.-Y.; and Overwijk, A. 2021.  Less is More: Pretrain a Strong Siamese Encoder for Dense Text Retrieval Using a Weak Decoder.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 2780–2791.
  * Lu et al. (2022) Lu, Y.; Liu, Y.; Liu, J.; Shi, Y.; Huang, Z.; Sun, S. F. Y.; Tian, H.; Wu, H.; Wang, S.; Yin, D.; et al. 2022.  Ernie-search: Bridging cross-encoder with dual-encoder via self on-the-fly distillation for dense passage retrieval.  _arXiv preprint arXiv:2205.09153_.
  * Ma et al. (2022) Ma, X.; Guo, J.; Zhang, R.; Fan, Y.; and Cheng, X. 2022.  Pre-train a Discriminative Text Encoder for Dense Retrieval via Contrastive Span Prediction.  _arXiv preprint arXiv:2204.10641_.
  * Nguyen et al. (2016) Nguyen, T.; Rosenberg, M.; Song, X.; Gao, J.; Tiwary, S.; Majumder, R.; and Deng, L. 2016.  MS MARCO: A Human Generated MAchine Reading COmprehension Dataset.  In Besold, T. R.; Bordes, A.; d’Avila Garcez, A. S.; and Wayne, G., eds., _Proceedings of the Workshop on Cognitive Computation: Integrating neural and symbolic approaches 2016 co-located with the 30th Annual Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain, December 9, 2016_ , volume 1773 of _CEUR Workshop Proceedings_. CEUR-WS.org.
  * Nogueira et al. (2019) Nogueira, R. F.; Yang, W.; Lin, J.; and Cho, K. 2019.  Document Expansion by Query Prediction.  _CoRR_ , abs/1904.08375.
  * Ouyang et al. (2022) Ouyang, L.; Wu, J.; Jiang, X.; Almeida, D.; Wainwright, C. L.; Mishkin, P.; Zhang, C.; Agarwal, S.; Slama, K.; Ray, A.; Schulman, J.; Hilton, J.; Kelton, F.; Miller, L.; Simens, M.; Askell, A.; Welinder, P.; Christiano, P. F.; Leike, J.; and Lowe, R. 2022.  Training language models to follow instructions with human feedback.  In _NeurIPS_.
  * Qu et al. (2021) Qu, Y.; Ding, Y.; Liu, J.; Liu, K.; Ren, R.; Zhao, W. X.; Dong, D.; Wu, H.; and Wang, H. 2021.  RocketQA: An Optimized Training Approach to Dense Passage Retrieval for Open-Domain Question Answering.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 5835–5847. Online: Association for Computational Linguistics.
  * Ren et al. (2021) Ren, R.; Qu, Y.; Liu, J.; Zhao, W. X.; She, Q.; Wu, H.; Wang, H.; and Wen, J.-R. 2021.  RocketQAv2: A Joint Training Method for Dense Passage Retrieval and Passage Re-ranking.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 2825–2835. Online and Punta Cana, Dominican Republic: Association for Computational Linguistics.
  * Robertson, Zaragoza et al. (2009) Robertson, S.; Zaragoza, H.; et al. 2009.  The probabilistic relevance framework: BM25 and beyond.  _Foundations and Trends® in Information Retrieval_ , 3(4): 333–389.
  * Sakata et al. (2019) Sakata, W.; Shibata, T.; Tanaka, R.; and Kurohashi, S. 2019.  FAQ Retrieval using Query-Question Similarity and BERT-Based Query-Answer Relevance.  In Piwowarski, B.; Chevalier, M.; Gaussier, É.; Maarek, Y.; Nie, J.; and Scholer, F., eds., _Proceedings of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR 2019, Paris, France, July 21-25, 2019_ , 1113–1116. ACM.
  * Santhanam et al. (2022) Santhanam, K.; Khattab, O.; Saad-Falcon, J.; Potts, C.; and Zaharia, M. 2022.  ColBERTv2: Effective and Efficient Retrieval via Lightweight Late Interaction.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 3715–3734. Seattle, United States: Association for Computational Linguistics.
  * Thakur et al. (2021) Thakur, N.; Reimers, N.; Rücklé, A.; Srivastava, A.; and Gurevych, I. 2021.  BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models.  _CoRR_ , abs/2104.08663.
  * Touvron et al. (2023) Touvron, H.; Lavril, T.; Izacard, G.; Martinet, X.; Lachaux, M.; Lacroix, T.; Rozière, B.; Goyal, N.; Hambro, E.; Azhar, F.; Rodriguez, A.; Joulin, A.; Grave, E.; and Lample, G. 2023.  LLaMA: Open and Efficient Foundation Language Models.  _CoRR_ , abs/2302.13971.
  * Wang et al. (2022a) Wang, L.; Yang, N.; Huang, X.; Jiao, B.; Yang, L.; Jiang, D.; Majumder, R.; and Wei, F. 2022a.  SimLM: Pre-training with Representation Bottleneck for Dense Passage Retrieval.  _CoRR_ , abs/2207.02578.
  * Wang, Yang, and Wei (2023) Wang, L.; Yang, N.; and Wei, F. 2023.  Query2doc: Query Expansion with Large Language Models.  _CoRR_ , abs/2303.07678.
  * Wang et al. (2023) Wang, Y.; Kordi, Y.; Mishra, S.; Liu, A.; Smith, N. A.; Khashabi, D.; and Hajishirzi, H. 2023.  Self-Instruct: Aligning Language Models with Self-Generated Instructions.  In Rogers, A.; Boyd-Graber, J. L.; and Okazaki, N., eds., _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2023, Toronto, Canada, July 9-14, 2023_ , 13484–13508. Association for Computational Linguistics.
  * Wang et al. (2022b) Wang, Y.; Mishra, S.; Alipoormolabashi, P.; Kordi, Y.; Mirzaei, A.; Naik, A.; Ashok, A.; Dhanasekaran, A. S.; Arunkumar, A.; Stap, D.; Pathak, E.; Karamanolakis, G.; Lai, H. G.; Purohit, I.; Mondal, I.; Anderson, J.; Kuznia, K.; Doshi, K.; Pal, K. K.; Patel, M.; Moradshahi, M.; Parmar, M.; Purohit, M.; Varshney, N.; Kaza, P. R.; Verma, P.; Puri, R. S.; Karia, R.; Doshi, S.; Sampat, S. K.; Mishra, S.; A, S. R.; Patro, S.; Dixit, T.; and Shen, X. 2022b.  Super-NaturalInstructions: Generalization via Declarative Instructions on 1600+ NLP Tasks.  In Goldberg, Y.; Kozareva, Z.; and Zhang, Y., eds., _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, EMNLP 2022, Abu Dhabi, United Arab Emirates, December 7-11, 2022_ , 5085–5109. Association for Computational Linguistics.
  * Wenzek et al. (2020) Wenzek, G.; Lachaux, M.; Conneau, A.; Chaudhary, V.; Guzmán, F.; Joulin, A.; and Grave, E. 2020.  CCNet: Extracting High Quality Monolingual Datasets from Web Crawl Data.  In Calzolari, N.; Béchet, F.; Blache, P.; Choukri, K.; Cieri, C.; Declerck, T.; Goggi, S.; Isahara, H.; Maegaard, B.; Mariani, J.; Mazo, H.; Moreno, A.; Odijk, J.; and Piperidis, S., eds., _Proceedings of The 12th Language Resources and Evaluation Conference, LREC 2020, Marseille, France, May 11-16, 2020_ , 4003–4012. European Language Resources Association.
  * Wu, Ma, and Hu (2022) Wu, X.; Ma, G.; and Hu, S. 2022.  Query-as-context Pre-training for Dense Passage Retrieval.  _CoRR_ , abs/2212.09598.
  * Wu et al. (2023a) Wu, X.; Ma, G.; Lin, M.; Lin, Z.; Wang, Z.; and Hu, S. 2023a.  ConTextual Masked Auto-Encoder for Dense Passage Retrieval.  In Williams, B.; Chen, Y.; and Neville, J., eds., _Thirty-Seventh AAAI Conference on Artificial Intelligence, AAAI 2023, Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence, IAAI 2023, Thirteenth Symposium on Educational Advances in Artificial Intelligence, EAAI 2023, Washington, DC, USA, February 7-14, 2023_ , 4738–4746. AAAI Press.
  * Wu et al. (2023b) Wu, X.; Ma, G.; Wang, P.; Lin, M.; Lin, Z.; Zhang, F.; and Hu, S. 2023b.  CoT-MAE v2: Contextual Masked Auto-Encoder with Multi-view Modeling for Passage Retrieval.  arXiv:2304.03158.
  * Yu et al. (2023) Yu, W.; Iter, D.; Wang, S.; Xu, Y.; Ju, M.; Sanyal, S.; Zhu, C.; Zeng, M.; and Jiang, M. 2023.  Generate rather than Retrieve: Large Language Models are Strong Context Generators.  In _The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023_. OpenReview.net.
  * Zhou et al. (2022) Zhou, K.; Liu, X.; Gong, Y.; Zhao, W. X.; Jiang, D.; Duan, N.; and Wen, J.-R. 2022\.  MASTER: Multi-task Pre-trained Bottlenecked Masked Autoencoders are Better Dense Retrievers.  _arXiv preprint arXiv:2212.07841_.
  * Zou et al. (2023) Zou, L.; Lu, W.; Liu, Y.; Cai, H.; Chu, X.; Ma, D.; Shi, D.; Sun, Y.; Cheng, Z.; Gu, S.; Wang, S.; and Yin, D. 2023.  Pre-trained Language Model-based Retrieval and Ranking for Web Search.  _ACM Trans. Web_ , 17(1): 4:1–4:36.