# Gorenstein homological dimension and some invariants of groups

Wei Ren, Gang Yang

###### Abstract.

For any group $G$, the Gorenstein homological dimension ${\rm Ghd}_{R}G$ is
defined to be the Gorenstein flat dimension of the coefficient ring $R$, which
is considered as an $RG$-module with trivial group action. We prove that ${\rm
Ghd}_{R}G<\infty$ if and only if the Gorenstein flat dimension of any
$RG$-module is finite, if and only if there exists an $R$-pure $RG$-monic
$R\rightarrow A$ with $A$ being $R$-flat and ${\rm Ghd}_{R}G={\rm fd}_{RG}A$,
where $R$ is a commutative ring with finite Gorenstein weak global dimension.
As applications, properties of ${\rm Ghd}$ on subgroup, quotient group,
extension of groups as well as Weyl group are investigated. Moreover, we
compare the relations between some invariants such as ${\rm sfli}RG$, ${\rm
silf}RG$, ${\rm spli}RG$, ${\rm silp}RG$, and Gorenstein projective,
Gorenstein flat and PGF dimensions of $RG$-modules; a sufficient condition for
Gorenstein projective-flat problem over group rings is given.

###### Key words and phrases:

Gorenstein homological dimension, Gorenstein flat, group ring

###### 2010 Mathematics Subject Classification:

18G20, 18G25, 20J05

E-mail: wren$\char 64\relax$cqnu.edu.cn; yanggang$\char 64\relax$mail.lzjtu.cn

##  1\. Introduction

Let $G$ be any group. Recall that the cohomological dimension ${\rm
cd}_{\mathbb{Z}}G$ and homological dimension ${\rm hd}_{\mathbb{Z}}G$ are
defined as the projective and flat dimension of the $\mathbb{Z}G$-module
$\mathbb{Z}$ respectively, where $G$ acts trivially. Studying groups through
these dimensions arose from both topological and algebraic sources, and has a
long history in group theory.

Let $R$ be a commutative ring of coefficients. For any group $G$, the
Gorenstein cohomological dimension ${\rm Gcd}_{R}G$ [3, 12], and the
Gorenstein homological dimension ${\rm Ghd}_{R}G$ [1, 21], are defined as the
Gorenstein projective and Gorenstein flat dimension of the trivial $RG$-module
$R$, respectively. The notions of Gorenstein projective, injective and flat
modules, which have the origin dating back to the study of $G$-dimension by
Auslander and Bridger [2] in 1960s, now form the basis of a version of
relative homological algebra known as Gorenstein homological algebra [14]. The
Gorenstein projective (resp. Gorenstein flat) dimension generalizes the
projective (resp. flat) dimension, in the sense that whenever the latter is
finite, then they coincide.

For any group $G$, there is an interesting result [3, Theorem 2.7] that ${\rm
Gcd}_{\mathbb{Z}}G<\infty$ if and only if there is a $\mathbb{Z}$-split
$\mathbb{Z}G$-exact sequence $0\rightarrow\mathbb{Z}\rightarrow A$, where $A$
is a $\mathbb{Z}$-projective ($\mathbb{Z}$-free) $\mathbb{Z}G$-module such
that the projective dimension ${\rm pd}_{\mathbb{Z}G}A$ of $A$ is finite; in
particular, ${\rm pd}_{\mathbb{Z}G}A={\rm Gcd}_{\mathbb{Z}}G$. This was
generalized in [12, Theorem 1.7] from the coefficient ring of integers
$\mathbb{Z}$ to any commutative ring with finite global dimension.

First, we are motivated to characterize the finiteness of ${\rm Ghd}_{R}G$,
and its relations with Gorenstein flat dimension of modules over the group
ring $RG$. However, the arguments are not simply analogous to those for
Gorenstein cohomological dimension ${\rm Gcd}_{R}G$. There are some evidences
for this phenomenon. Unlike the well-known fact that every projective module
is flat, the relation between Gorenstein projective and Gorenstein flat
modules is not fully understood, and it is still an open problem whether
Gorenstein projective modules are Gorenstein flat. There is a basic and
important property that the class of Gorenstein projective modules is closed
under extensions; see for example [18, Theorem 2.5]. However, the analogous
result for Gorenstein flat modules is not easy to prove. In [18, Theorem 3.7],
an additional assumption that the base ring is coherent is needed. Until
recently, the assertion was proved affirmatively over any ring by [25,
Corollary 4.12], where a notion of PGF-modules was invented.

In fact, projectivity connects closely to the splitness, however, flatness is
a bit inseparable from purity. We succeed in getting the following, which
extend the aforementioned results in [3, 12] to the case of flatness. Let $R$
be a commutative ring such that the supremum of flat dimension (length) of
injective $R$-modules ${\rm sfli}R$ is finite. We show in Theorem 3.12 that
the following conditions are equivalent:

  1. (1)

${\rm Ghd}_{R}G<\infty$.

  2. (2)

There exists an $R$-pure $RG$-exact sequence $0\rightarrow R\rightarrow A$,
where $A$ is an $R$-flat $RG$-module of finite flat dimension.

  3. (3)

Every $RG$-module has finite Gorenstein flat dimension.

In this case, we have an equality ${\rm Ghd}_{R}G={\rm fd}_{RG}A$, and the
following inequalities

${\rm sfli}R\leq{\rm sfli}RG={\rm G.wgldim}RG\leq{\rm Ghd}_{R}G+{\rm sfli}R,$

where ${\rm G.wgldim}RG$ denotes the Gorenstein weak global dimension of the
group ring $RG$. In particular, we show in Corollary 3.11 that if ${\rm
sfli}R<\infty$, then ${\rm Gfd}_{RG}M<\infty$ for any $RG$-module $M$ if and
only if ${\rm Ghd}_{R}G={\rm Gfd}_{RG}R<\infty$ for the trivial $RG$-module
$R$; moreover, $\mathrm{Gfd}_{RG}M$ is bounded by ${\rm Ghd}_{R}G+{\rm
Gfd}_{R}M$, i.e. $\mathrm{Gfd}_{RG}M\leq{\rm Ghd}_{R}G+{\rm Gfd}_{R}M$.

It is worth to remark that the assumption ${\rm sfli}R<\infty$ on the
coefficient ring $R$ is not too restrictive. It follows from [10, Theorem 5.3]
that ${\rm sfli}R<\infty$ if and only if every $R$-module has finite
Gorenstein flat dimension. In this case, $R$ is called a ring with finite
Gorenstein weak global dimension. In fact, we frequently concern modules with
finite, rather than infinite, Gorenstein flat dimension. Rings with finite
weak global dimension are strictly contained in those of finite Gorenstein
weak global dimension. The most interesting examples of coefficient rings in
dealing with applications of group rings in geometry and representation
theory, such as the ring of integers $\mathbb{Z}$, the field of rationals
$\mathbb{Q}$ and finite fields, are all of finite “sfli”.

For further applications, we strengthen the role of Gorenstein homological
dimension of groups by proving the following properties. As shown in Section
4, the arguments heavily rely on the above $R$-pure monomorphism $R\rightarrow
A$ . More precisely, for any commutative ring $R$ with ${\rm sfli}R<\infty$
and any group $G$, we show that

  1. (I)

${\rm Ghd}_{R}H\leq{\rm Ghd}_{R}G$ for any subgroup $H$ of $G$ (see
Proposition 4.1).

  2. (II)

If $1\to H\rightarrow G\rightarrow L\rightarrow 1$ is an extension of groups,
then ${\rm Ghd}_{R}G\leq{\rm Ghd}_{R}H+{\rm Ghd}_{R}L$ (see Proposition 4.3).

  3. (III)

${\rm Ghd}_{R}G={\rm Ghd}_{R}(G/H)$ for any finite normal subgroup $H$ of $G$
(see Proposition 4.4).

  4. (IV)

If $H$ is a finite subgroup of $G$, and $N_{G}(H)$ its normalizer in $G$, then
${\rm Ghd}_{R}W\leq{\rm Ghd}_{R}G$ for the Weyl group $W=N_{G}(H)/H$ (see
Corollary 4.5).

In particular, we generalize [1, Proposition 4.11] by (1) immediately: ${\rm
Ghd}_{\mathbb{Z}}H\leq{\rm Ghd}_{\mathbb{Z}}G$, where $H$ was assumed to be a
subgroup of $G$ of finite index in [1].

Moreover, we compare Gorenstein homological dimension of groups with the
generalized homological dimension of groups introduced by Ikenaga [19, III,
Definition]; see details in Proposition 4.6 and 4.7. In particular, we obtain
in Corollary 4.9 that ${\rm Ghd}_{\mathbb{Z}}G<\infty$ if and only if
$\underline{{\rm hd}}_{\mathbb{Z}}G<\infty$, and furthermore, ${\rm
Ghd}_{\mathbb{Z}}G=\underline{{\rm hd}}_{\mathbb{Z}}G$ in this case.

In Section 5, we are devoted to study the relation between ${\rm Ghd}_{R}G$
and some invariants such as ${\rm silp}RG$, ${\rm spli}RG$ and ${\rm sfli}RG$,
which are closely related to the theory of Gorenstein projective, Gorenstein
flat and PGF dimensions of modules; see for example [7, 8, 10]. First, we
observe that if ${\rm Ghd}_{R}G<\infty$, then ${\rm sfli}R<\infty$ if and only
if ${\rm sfli}RG<\infty$; moreover, if both ${\rm Ghd}_{R}G$ and ${\rm sfli}R$
are finite, then every Gorenstein projective $RG$-module is a PGF-module, and
furthermore is a Gorenstein flat module; see Proposition 5.1.

For any commutative Gorenstein ring $R$ and any group $G$, it follows from
[16, Theorem 2.4] that ${\rm spli}RG<\infty$ implies ${\rm silp}RG<\infty$.
For the finiteness of ${\rm sfli}RG$ and ${\rm silf}RG$, we show in
Proposition 5.2 that ${\rm silf}RG<\infty$ if and only if both ${\rm
sfli}RG<\infty$ and ${\rm splf}RG<\infty$, that is, the comparison between
${\rm sfli}RG$ and ${\rm silf}RG$ is essentially about the relation between
the projective and flat dimensions of modules. Then, we can show that if ${\rm
silf}RG<\infty$, then ${\rm Ghd}_{R}G<\infty$ and every Gorenstein projective
$RG$-module is Gorenstein flat; moreover, for any Gorenstein flat $RG$-module
$M$, ${\rm PGF}\text{-}{\rm dim}_{RG}M={\rm Gpd}_{RG}M<\infty$; see Corollary
5.3. In particular, if $R$ is commutative Gorenstein and $G$ is a group such
that $RG$ is a left noetherian ring, we show in Proposition 5.6 that ${\rm
sfli}RG<\infty$ if and only if ${\rm silf}RG<\infty$; in this case ${\rm
sfli}RG={\rm silf}RG$. It is trivially true that for any finite group, its
integral group is noetherian. Remark that for any virtually polycyclic group
$G$, also named a polycyclic-by-finite group, the integral group ring
$\mathbb{Z}G$ is noetherian; see details in [17]. Finally, we show in
Proposition 5.8 that there is an inequality ${\rm G.wgldim}RG\leq{\rm
PGF}\text{-}{\rm gldim}RG={\rm G.gldim}RG\leq{\rm G.wgldim}RG+{\rm splf}RG$
for any commutative ring $R$ and any group $G$.

##  2\. Preliminaries

In this section, we recall some notions and facts which will be needed in the
following.

### Gorenstein flat dimension of modules

Let $\Lambda$ be an associative ring with identity. Recall that
$\mathbf{F}=\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow
F_{-1}\rightarrow\cdots$ is called a totally acyclic complex of flat modules,
provided it is acyclic with each $F_{i}$ being a flat left $\Lambda$-module,
and for any injective right $\Lambda$-module $I$, the complex remains acyclic
after applying $I\otimes_{\Lambda}-$. A left $\Lambda$-module $M$ is called
Gorenstein flat [15], if there exists a totally acyclic complex of flat
modules $\mathbf{F}$, such that $M\cong\mathrm{Ker}(F_{0}\rightarrow F_{-1})$.

Let $M$ be any left $\Lambda$-module. The Gorenstein flat dimension of $M$,
denoted by $\mathrm{Gfd}_{\Lambda}M$, is defined by declaring that ${\rm
Gfd}_{\Lambda}M\leq n$ if and only if $M$ has a Gorenstein flat resolution
$0\rightarrow P_{n}\rightarrow\cdots\rightarrow P_{1}\rightarrow
P_{0}\rightarrow M\rightarrow 0$ of length $n$, where each $P_{i}$ is a
Gorenstein flat left $\Lambda$-module. The Gorenstein weak global dimension
[4] of $\Lambda$, denoted by ${\rm G.wgldim}\Lambda$, is defined as the
supremum of Gorenstein flat dimensions of all left $\Lambda$-modules.

It was proved in [18, Theorem 3.7] that for a right coherent ring, the class
of Gorenstein flat left modules is closed under extensions and direct
summands. This result was extended and generalized to any associative ring by
[25, Corollary 4.12]. This basic property is crucial for studying homology of
Gorenstein flat modules, however, it is not easy to prove. Thanks to [25,
Corollary 4.12], now we can remove the assumption of coherent rings in many
situations when dealing with Gorenstein flat modules and Gorenstein flat
dimension of modules, see for example [18, Theorem 3.14].

### Modules over group rings

Let $G$ be a group, and $R$ be a commutative ring. A module over the group
ring $RG$ is simply an $R$-module $M$ together with an action of $G$ on $M$.
In particular, $R$ is an $RG$-module with trivial $G$-action, that is $gr=r$
for any $g\in G$ and any $r\in R$.

For an $RG$-module $M$, the module of invariants of $M$, denoted by $M^{G}$,
is defined as the largest submodule of $M$ on which $G$ acts trivially, that
is, $M^{G}:=\\{m\in M|gm=m\text{ for all }g\in G\\}$. Analogously, the module
of coinvariants of $M$, denoted by $M_{G}$, is defined to be the largest
quotient of $M$ on which $G$ acts trivially.

Let $H$ be any subgroup of $G$. There exist simultaneously an induction
functor $\mathrm{Ind}_{H}^{G}=RG\otimes_{RH}-$ and a coinduction functor
$\mathrm{Coind}_{H}^{G}=\mathrm{Hom}_{RH}(RG,-)$ from the category of
$RH$-modules ${\rm Mod}(RH)$ to the category of $RG$-modules ${\rm Mod}(RG)$.
We denote the restriction functor from ${\rm Mod}(RG)$ to ${\rm Mod}(RH)$ by
${\rm Res}_{H}^{G}$. In particular, for any $RG$-module $M$, ${\rm
Res}_{H}^{G}M$ is exactly the underlying $R$-module if we consider the
subgroup $H$ formed only by the identity element of $G$. It is clear that
$({\rm Ind}_{H}^{G},{\rm Res}_{H}^{G})$ and $({\rm Res}_{H}^{G},{\rm
Coind}_{H}^{G})$ are adjoint pairs of functors. To simplify the notations, in
the following we will denote ${\rm Res}_{H}^{G}M$ by $M$ if there is no risk
of ambiguity.

### Gorenstein homological dimension of groups

Recall that for any group $G$, the Gorenstein homological dimension of $G$ is
defined to be the Gorenstein flat dimension of the trivial
$\mathbb{Z}G$-module $\mathbb{Z}$; see [1, Definition 4.5]. Analogously, we
may define the Gorenstein homological dimension of $G$ over any commutative
ring $R$, denoted by ${\rm Ghd}_{R}G$, to be the Gorenstein flat dimension of
the trivial $RG$-module $R$; see [21, Definition 2.5]. It follows from [21,
Corollary 2.8] that if the coefficient ring $R$ is $\mathbb{Z}$-torsion-free,
then ${\rm Ghd}_{R}G$ is a refinement of ${\rm Ghd}_{\mathbb{Z}}G$, that is,
${\rm Ghd}_{R}G\leq{\rm Ghd}_{\mathbb{Z}}G$.

### Invariants silp, spli, silf and sfli

The Gorenstein flat dimension is closely related to some homological
invariants. Recall that in connection with the existence of complete
cohomological functors in the category of left $\Lambda$-modules, Gedrich and
Gruenberg have defined in [16] the invariant ${\rm silp}\Lambda$ as the
supremum of the injective length (dimension) of projective left
$\Lambda$-modules, and the invariant ${\rm spli}\Lambda$ as the supremum of
the projective length (dimension) of injective left $\Lambda$-modules.
Analogously, we use ${\rm silf}\Lambda$ to denote the supremum of the
injective length (dimension) of flat left $\Lambda$-modules, and use ${\rm
sfli}\Lambda$ to denote the supremum of the flat length (dimension) of
injective left $\Lambda$-modules.

Note that if ${\rm sfli}\Lambda$ is finite, then any acyclic complex of flat
left $\Lambda$-modules is totally acyclic. It follows from [10, Theorem 5.3]
and [7, Corollary 1.5] that ${\rm G.wgldim}\Lambda={\rm G.wgldim}\Lambda^{op}$
is finite if and only if ${\rm sfli}\Lambda={\rm sfli}\Lambda^{op}$ is finite;
moreover, in this case one has ${\rm G.wgldim}\Lambda={\rm
sfli}\Lambda<\infty$, and $\Lambda$ is called a ring with finite Gorenstein
weak global dimension.

##  3\. Finiteness of Gorenstein homological dimension of groups

Throughout the paper, $R$ is assumed to be a commutative ring, $G$ is a group.
All modules over the group ring $RG$ are left $RG$-modules unless stated
otherwise.

In this section, we intend to characterize the finiteness of ${\rm Ghd}_{R}G$,
which is equivalent to that every $RG$-module has finite Gorenstein flat
dimension; see Theorem 3.12. We begin with the following observation.

######  Lemma 3.1.

If ${\rm sfli}R<\infty$, then any Gorenstein flat $RG$-module is also a
Gorenstein flat $R$-module.

###### Proof.

Let $M$ be a Gorenstein flat $RG$-module. There is a totally acyclic complex
of flat $RG$-modules $\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow
F_{-1}\rightarrow\cdots$ such that $M\cong\mathrm{Ker}(F_{0}\rightarrow
F_{-1})$. Since any flat $RG$-module is also $R$-flat, by restricting this
totally acyclic complex, we get an acyclic complex of flat $R$-modules. Noting
that ${\rm sfli}R<\infty$, every acyclic complex of flat $R$-modules is
totally acyclic. Hence, $M$ is also Gorenstein flat as an $R$-module. ∎

######  Proposition 3.2.

Assume ${\rm sfli}R<\infty$. Let $M$ be an $RG$-module with ${\rm
Gfd}_{RG}M<\infty$. If the flat dimension ${\rm fd}_{R}M$ of the underlying
$R$-module is finite, then there exists an $R$-pure $RG$-exact sequence
$0\rightarrow M\rightarrow N\rightarrow L\rightarrow 0$, for which $L$ is an
$R$-flat $RG$-module and ${\rm Gfd}_{RG}M={\rm fd}_{RG}N$.

###### Proof.

Let ${\rm Gfd}_{RG}M=n$. It follows from [18, Theorem 3.23] that there exists
an exact sequence $0\rightarrow K\rightarrow X\rightarrow M\rightarrow 0$,
where $X$ is a Gorenstein flat $RG$-module, and ${\rm fd}_{RG}K=n-1$. For $X$,
there is an exact sequence of $RG$-modules $0\rightarrow X\rightarrow
F\rightarrow L\rightarrow 0$, where $F$ is flat and $L$ is Gorenstein flat. We
consider the following pushout of $X\rightarrow M$ and $X\rightarrow F$:

---
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From the middle row we infer that ${\rm fd}_{RG}N={\rm fd}_{RG}K+1=n$, which
also implies the finiteness of ${\rm fd}_{R}N$. In view of the assumption
${\rm fd}_{R}M<\infty$, we infer from the right column that ${\rm
fd}_{R}L<\infty$. For the Gorenstein flat $RG$-module $L$, it follows from
Lemma 3.1 that $L$ is also Gorenstein flat as an $R$-module. Furthermore, we
imply that $L$ is $R$-flat since flat dimension of any Gorenstein flat module
is either zero or infinity; see [14, Corollary 10.3.4]. Hence, the exact
sequence of $RG$-modules $0\rightarrow M\rightarrow N\rightarrow L\rightarrow
0$ is $R$-pure exact. This completes the proof. ∎

######  Corollary 3.3.

Assume ${\rm sfli}R<\infty$. If ${\rm Ghd}_{R}G$ is finite, then there exists
an $R$-pure $RG$-exact sequence $0\rightarrow R\rightarrow A$, where $A$ is an
$R$-flat $RG$-module such that ${\rm Ghd}_{R}G={\rm fd}_{RG}A$.

Let $M$ and $N$ be left $RG$-modules. It is worth to note that there are two
different ways to define the $RG$-module structures for the tensor product
$M\otimes_{R}N$. On the one hand, $M\otimes_{R}N$ is an $RG$-module where $G$
acts diagonally, i.e. $g(m\otimes n)=gm\otimes gn$, but on the other hand,
$M\otimes_{R}N$ is an induced left $RG$-module by $g(m\otimes n)=gm\otimes n$.

######  Lemma 3.4.

For any left $RG$-modules $M$ and $N$, there is an isomorphism of induced left
$RG$-modules $M\otimes_{R}N\cong N\otimes_{R}M$.

###### Proof.

Since $R$ is assumed to be commutative, by using the anti-automorphism
$g\rightarrow g^{-1}$ of $G$, we can regard any left $RG$-module $X$ as a
right $RG$-module by setting $gx=xg^{-1}$ for any $g\in G$ and $x\in X$. Thus,
we may regard $M\otimes_{R}N$ as both a left $RG$-module with $G$-actions
induced from ${}_{RG}M\otimes_{R}-$ by setting $g(m\otimes n)=(gm)\otimes n$,
and as a right $RG$-module with $G$-actions induced from $-\otimes_{R}N_{RG}$
by setting $(m\otimes n)g^{-1}=m\otimes(ng^{-1})=m\otimes gn$.

Note that for any $g\in G$ and any $m\otimes n\in M\otimes_{R}N$, one has
$gm\otimes n=g(m\otimes n)=(m\otimes n)g^{-1}=m\otimes gn$. Similarly, for any
$n\otimes m\in N\otimes_{R}M$, $gn\otimes m=g(n\otimes m)=(n\otimes
m)g^{-1}=n\otimes gm$. It is direct to check that the map
$\varphi:M\otimes_{R}N\rightarrow N\otimes_{R}M$ given by $m\otimes n\mapsto
n\otimes m$ is an isomorphism of the induced left $RG$-modules. ∎

######  Remark 3.5.

Since $R$ is assumed to be commutative, for any $RG$-module, one can avoid
considering both left and right modules by using the anti-automorphism
$g\rightarrow g^{-1}$ of $G$. Thus, for any two left $RG$-modules $M$ and $N$,
the tensor product $M\otimes_{RG}N$ makes sense by introducing the relations
$gm\otimes n=mg^{-1}\otimes n=m\otimes g^{-1}n$.

For any $RG$-modules $M$ and $N$, By replacing $n$ in the above relations by
$gn$, we get that $gm\otimes gn=m\otimes n$. Hence,
$M\otimes_{RG}N=(M\otimes_{R}N)_{G}$, where $G$ acts diagonally on
$M\otimes_{R}N$. Moreover, there is an isomorphism $M\otimes_{RG}N\cong
N\otimes_{RG}M$, that is, the bifunctor $-\otimes_{RG}-$ is commutative.

######  Lemma 3.6.

Let $\iota:R\rightarrow A$ be an $R$-pure monomorphism of $RG$-modules, where
$A$ is $R$-flat. For any $RG$-module $N$, if the induced left $RG$-module
$N\otimes_{R}A$ is Gorenstein flat, then so is $N$.

###### Proof.

Assume that $N\otimes_{R}A$ is a Gorenstein flat $RG$-module. Then, there is
an exact sequence of $RG$-modules

$0\longrightarrow
N\otimes_{R}A\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}F_{0}\longrightarrow
L\longrightarrow 0,$

where $F_{0}$ is flat and $L$ is Gorenstein flat. By applying $N\otimes_{R}-$
to the $R$-pure monomorphism $\iota:R\rightarrow A$, we get a monomorphism of
induced left $RG$-modules ${\rm Id}_{N}\otimes\iota:N=N\otimes_{R}R\rightarrow
N\otimes_{R}A$. Let $\beta=\alpha({\rm Id}_{N}\otimes\iota):N\rightarrow
F_{0}$, and consider the following commutative diagram

---
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Id}_{N}\otimes\iota\quad}$$\textstyle{N\otimes_{R}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{N\otimes_{R}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

where $B={\rm Coker}\iota$, and $N^{\prime}={\rm Coker}\beta$.

Let $E$ be any injective right $RG$-module. It is clear that

${\rm Id}_{E}\otimes\alpha:E\otimes_{RG}(N\otimes_{R}A)\rightarrow
E\otimes_{RG}F_{0}$

is monic. Since $\iota:R\rightarrow A$ is an $R$-pure monomorphism, we infer
that

${\rm Id}_{E}\otimes({\rm
Id}_{N}\otimes\iota):E\otimes_{RG}N=(E\otimes_{RG}N)\otimes_{R}R\rightarrow(E\otimes_{RG}N)\otimes_{R}A\cong
E\otimes_{RG}(N\otimes_{R}A)$

is monic. Hence, ${\rm Id}_{E}\otimes\beta:E\otimes_{RG}N\rightarrow
E\otimes_{RG}F_{0}$ is a monomorphism, and moreover, we infer from the middle
row of the above diagram that ${\rm Tor}_{1}^{RG}(E,N^{\prime})=0$.

By applying $-\otimes_{R}A$ to the middle row, we get an exact sequence of
induced left $RG$-modules

$0\longrightarrow N\otimes_{R}A\longrightarrow
F_{0}\otimes_{R}A\longrightarrow N^{\prime}\otimes_{R}A\longrightarrow 0,$

where $F_{0}\otimes_{R}A$ is flat and $N\otimes_{R}A$ is assumed to be
Gorenstein flat. Since $A$ is a flat $R$-module, there is an isomorphism ${\rm
Tor}_{i}^{RG}(E,N^{\prime}\otimes_{R}A)\cong{\rm
Tor}_{i}^{RG}(E,N^{\prime})\otimes_{R}A$ for any $i>0$, and moreover, we get
${\rm Tor}_{1}^{RG}(E,N^{\prime}\otimes_{R}A)=0$ from ${\rm
Tor}_{1}^{RG}(E,N^{\prime})=0$. Note that by [25, Corollay 4.12], the class of
all Gorenstein flat modules is always closed under extensions. Then, we may
get rid of the coherent assumption on rings in [18, Proposition 3.8], and
obtain that the induced left $RG$-module $N^{\prime}\otimes_{R}A$ is also
Gorenstein flat.

Analogous to the above, for $N^{\prime}$ we obtain an exact sequence of left
$RG$-modules $0\rightarrow N^{\prime}\rightarrow F_{-1}\rightarrow
N^{\prime\prime}\rightarrow 0$ with $F_{-1}$ being flat, which remains exact
after applying $E\otimes_{RG}-$ for any injective right $RG$-module $E$, and
for $N^{\prime\prime}$ the induced left $RG$-module
$N^{\prime\prime}\otimes_{R}A$ is Gorenstein flat. Proceed in this manner, we
obtain an acyclic complex

$\textstyle{\mathbf{F}_{\leq
0}=0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

with each $F_{i}$ being a flat left $RG$-module, which remains acyclic after
applying $E\otimes_{RG}-$ for any injective right $RG$-module $E$.

Now consider a flat resolution of $N$:

$\mathbf{F}_{>0}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Since $A$ is $R$-flat, it is clear that the induced left $RG$-modules
$F_{i}\otimes_{R}A$ are flat, and
$\mathbf{F}_{>0}^{\bullet}\otimes_{R}A\rightarrow N\otimes_{R}A$ is a flat
resolution of the induced left $RG$-module $N\otimes_{R}A$. Invoking the
assumption that $N\otimes_{R}A$ is a Gorenstein flat left $RG$-module, we get
${\rm Tor}_{i}^{RG}(E,N\otimes_{R}A)=0$ for any injective right $RG$-module
$E$ and any $i\geq 1$, and this implies that the complex
$E\otimes_{RG}(\mathbf{F}_{>0}^{\bullet}\otimes_{R}A)\rightarrow
E\otimes_{RG}(N\otimes_{R}A)$ is acyclic. Thus, we infer from the isomorphisms

$(A\otimes_{R}E)\otimes_{RG}\mathbf{F}_{>0}\cong
A\otimes_{R}(E\otimes_{RG}\mathbf{F}_{>0})\cong(E\otimes_{RG}\mathbf{F}_{>0})\otimes_{R}A\cong
E\otimes_{RG}(\mathbf{F}_{>0}\otimes_{R}A)$

that the complex $(A\otimes_{R}E)\otimes_{RG}\mathbf{F}_{>0}$ is acyclic.

By applying $-\otimes_{R}E$ to the $R$-pure exact sequence $0\rightarrow
R\rightarrow A\rightarrow B\rightarrow 0$, we get an exact sequence of induced
right $RG$-modules

$0\longrightarrow E=R\otimes_{R}E\longrightarrow A\otimes_{R}E\longrightarrow
B\otimes_{R}E\longrightarrow 0.$

The sequence is split since $E$ is injective, and moreover, $E$ is a direct
summand of $A\otimes_{R}E$. Thus, as a direct summand of
$(A\otimes_{R}E)\otimes_{RG}\mathbf{F}_{>0}$, the complex
$E\otimes_{RG}\mathbf{F}_{>0}$ is acyclic.

Consequently, by pasting $\mathbf{F}_{\leq 0}$ and $\mathbf{F}_{>0}$ we will
get a totally acyclic complex of flat $RG$-modules such that
$N\cong\mathrm{Ker}(F_{0}\rightarrow F_{-1})$. This yields that $N$ is a
Gorenstein flat $RG$-module. This completes the proof. ∎

######  Lemma 3.7.

Let $H$ be any subgroup of $G$. For any Gorenstein flat $RH$-module $M$, the
induced $RG$-module ${\rm Ind}_{H}^{G}M$ is also Gorenstein flat.

###### Proof.

Let $M$ be a Gorenstein flat $RH$-module. There is a totally acyclic complex
of flat left $RH$-modules

$\mathbf{F}=\cdots\longrightarrow F_{1}\longrightarrow F_{0}\longrightarrow
F_{-1}\longrightarrow\cdots$

such that $M={\rm Ker}(F_{0}\rightarrow F_{-1})$. We infer that ${\rm
Ind}_{H}^{G}\mathbf{F}=RG\otimes_{RH}\mathbf{F}$ is an acyclic complex of flat
$RG$-modules such that ${\rm Ind}_{H}^{G}M={\rm Ker}({\rm
Ind}_{H}^{G}F_{0}\rightarrow{\rm Ind}_{H}^{G}F_{-1})$. For any injective right
$RG$-module $E$, it is restricted to be an injective $RH$-module. Then, we
infer from the isomorphism

$E\otimes_{RG}{\rm
Ind}_{H}^{G}\mathbf{F}=E\otimes_{RG}RG\otimes_{RH}\mathbf{F}\cong
E\otimes_{RH}\mathbf{F}$

that $E\otimes_{RG}{\rm Ind}_{H}^{G}\mathbf{F}$ is acyclic. Hence, ${\rm
Ind}_{H}^{G}\mathbf{F}$ is a totally acyclic complex of flat $RG$-modules, and
this implies that ${\rm Ind}_{H}^{G}M$ is a Gorenstein flat $RG$-module. ∎

######  Lemma 3.8.

Let $M$ be an $RG$-module. If $M$ is Gorenstein flat as an $R$-module, then
for any flat $RG$-module $F$, the induced left $RG$-module $F\otimes_{R}M$ is
also Gorenstein flat.

###### Proof.

Let $H=\\{1\\}$ be the subgroup formed by the identity element of $G$. Then,
the induction functor $\mathrm{Ind}_{H}^{G}=RG\otimes_{R}-$ sends every
$R$-module to be an $RG$-module. Let $M$ be an $RG$-module, and assume that
$M$ is Gorenstein flat as an $R$-module. Since any flat $RG$-module $F$ is
also restricted to be a flat $R$-module, we may express it as a colimit of
free $R$-modules. Hence, it follows that $F\otimes_{R}M$ is a Gorenstein flat
$R$-module, as the subcategory of Gorenstein flat $R$-modules is closed under
direct limits; see [25, Corollary 4.12] and [26, Lemma 3.1]. Then, by Lemma
3.7 we infer that the induced $RG$-module ${\rm Ind}_{H}^{G}(F\otimes_{R}M)$
is Gorenstein flat.

There is an isomorphism of $RG$-modules

$F\otimes_{R}{\rm Ind}_{H}^{G}M\cong{\rm Ind}_{H}^{G}(F\otimes_{R}M),$

where $G$ acts diagonally on the left tensor product; see for example [5,
Section III 5]. Note that the diagonal $RG$-module structure of
$F\otimes_{R}{\rm Ind}_{H}^{G}M$ coincides with the one induced by
${}_{RG}F\otimes_{R}-$. As $R$-modules, $M$ is a direct summand of ${\rm
Ind}_{H}^{G}M$. Then, the induced $RG$-module $F\otimes_{R}M$ is a direct
summand of the induced $RG$-module $F\otimes_{R}{\rm Ind}_{H}^{G}M$. Hence,
$F\otimes_{R}M$ is a Gorenstein flat $RG$-module. ∎

######  Proposition 3.9.

Let $M$ be any left $RG$-module which is Gorenstein flat as an $R$-module. If
there exists an $R$-pure monomorphism of $RG$-modules $\iota:R\rightarrow A$,
where $A$ is $R$-flat with $\mathrm{fd}_{RG}A<\infty$, then
$\mathrm{Gfd}_{RG}M\leq\mathrm{fd}_{RG}A$.

###### Proof.

Let $\mathrm{fd}_{RG}A=n$. There exists an exact sequence of left $RG$-modules

$0\longrightarrow F_{n}\longrightarrow\cdots\longrightarrow
F_{1}\longrightarrow F_{0}\longrightarrow A\longrightarrow 0,$

where each $F_{i}$ is flat. Since $A$ is a flat $R$-module and each $F_{i}$ is
restricted to be a flat $R$-module, the sequence is $R$-pure exact.

By applying $-\otimes_{R}M$ to the above $R$-pure exact sequence, we get an
exact sequence of induced left $RG$-modules

$0\longrightarrow F_{n}\otimes_{R}M\longrightarrow\cdots\longrightarrow
F_{1}\otimes_{R}M\longrightarrow F_{0}\otimes_{R}M\longrightarrow
A\otimes_{R}M\longrightarrow 0.$

Note that by Lemma 3.4, there is an isomorphism of induced left $RG$-modules
$M\otimes_{R}A\cong A\otimes_{R}M$. It follows from Lemma 3.8 that
$F_{i}\otimes_{R}M$ are Gorenstein flat $RG$-modules, and then ${\rm
Gfd}_{RG}(M\otimes_{R}A)={\rm Gfd}_{RG}(A\otimes_{R}M)\leq n$.

Now we consider a flat resolution of the $RG$-module $M$

$\cdots\longrightarrow F^{\prime}_{n}\longrightarrow\cdots\longrightarrow
F^{\prime}_{1}\longrightarrow F^{\prime}_{0}\longrightarrow M\longrightarrow
0,$

and let $K={\rm Ker}(F^{\prime}_{n-1}\rightarrow F^{\prime}_{n-2})$. By
applying $-\otimes_{R}A$, we get an exact sequence of induced left
$RG$-modules

$0\longrightarrow K\otimes_{R}A\longrightarrow
F^{\prime}_{n-1}\otimes_{R}A\rightarrow\cdots\longrightarrow
F^{\prime}_{0}\otimes_{R}A\longrightarrow M\otimes_{R}A\longrightarrow 0,$

where all $F^{\prime}_{i}\otimes_{R}A$ are flat $RG$-modules. We infer from
${\rm Gfd}_{RG}(M\otimes_{R}A)\leq n$ that $K\otimes_{R}A$ is a Gorenstein
flat left $RG$-module. Hence, it follows from Lemma 3.6 that $K$ is a
Gorenstein flat $RG$-module, and then the desired inequality
$\mathrm{Gfd}_{RG}M\leq n$ holds. ∎

######  Corollary 3.10.

If there exists an $R$-pure monomorphism of $RG$-modules $\iota:R\rightarrow
A$, where $A$ is $R$-flat with $\mathrm{fd}_{RG}A<\infty$, then for any
$RG$-module $M$, we have

$\mathrm{Gfd}_{RG}M\leq\mathrm{fd}_{RG}A+{\rm
Gfd}_{R}M\leq\mathrm{fd}_{RG}A+{\rm sfli}R.$

###### Proof.

Let $\mathrm{fd}_{RG}A=n$. The second inequality is obvious since ${\rm
G.wgldim}R={\rm sfli}R$; see [10, Theorem 5.3] and [7, Corollary 1.5].

For the first inequality, it suffices to assume that ${\rm Gfd}_{R}M=m$ is
finite. In this case, we may prove the result by induction on $m$. If $m=0$,
that is, $M$ is a Gorenstein flat $R$-module, then the assertion is immediate
from Proposition 3.9.

We now assume $m>0$, and consider a short exact sequence of $RG$-modules

$0\longrightarrow K\longrightarrow F\longrightarrow M\longrightarrow 0,$

where $F$ is flat. Since $F$ is restricted to be a flat $R$-module, we have
${\rm Gfd}_{R}K=m-1$. Invoking the induction hypothesis, we may conclude that
${\rm Gfd}_{RG}K\leq n+(m-1)$, and hence ${\rm Gfd}_{RG}M\leq n+m$. This
completes the proof. ∎

The following is immediate from Corollary 3.3 and 3.10.

######  Corollary 3.11.

Let $R$ be a commutative ring with ${\rm sfli}R<\infty$. Then ${\rm
Ghd}_{R}G<\infty$ if and only if ${\rm Gfd}_{RG}M<\infty$ for any $RG$-module
$M$. In this case, one has

$\mathrm{Gfd}_{RG}M\leq{\rm Ghd}_{R}G+{\rm Gfd}_{R}M.$

Now, we are in a position to state the main result of this section.

######  Theorem 3.12.

Let $G$ be a group, $R$ a commutative ring with ${\rm sfli}R<\infty$. The
following are equivalent:

  1. (1)

${\rm Ghd}_{R}G$ is finite.

  2. (2)

There exists an $R$-pure $RG$-exact sequence $0\rightarrow R\rightarrow A$,
where $A$ is an $R$-flat $RG$-module of finite flat dimension.

  3. (3)

Any $RG$-module has finite Gorenstein flat dimension.

  4. (4)

${\rm sfli}RG$ is finite.

In this case, there is an equality ${\rm Ghd}_{R}G={\rm fd}_{RG}A$, and
moreover, the following inequalities hold:

${\rm sfli}R\leq{\rm sfli}RG={\rm G.wgldim}RG\leq{\rm Ghd}_{R}G+{\rm sfli}R.$

###### Proof.

The implication (1)$\Longrightarrow$(2) follows by Corollary 3.3, and the
implication (2)$\Longrightarrow$(3) follows from Corollary 3.10. The
implication (3)$\Longrightarrow$(1) is trivial, and
(3)$\Longleftrightarrow$(4) follows from [10, Theorem 5.3].

For the first inequality ${\rm sfli}R\leq{\rm sfli}RG$, it suffices to assume
that ${\rm sfli}RG=n$ is finite. We consider any injective $R$-module $I$. For
the subgroup $H=\\{1\\}$ of $G$, note that ${\rm Coind}_{H}^{G}I$ is an
injective $RG$-module. Then ${\rm fd}_{RG}{\rm Coind}_{H}^{G}I\leq n$, which
yields ${\rm fd}_{R}{\rm Coind}_{H}^{G}I\leq n$. Moreover, as $R$-modules $I$
is a direct summand of ${\rm Coind}_{H}^{G}I$, and hence ${\rm fd}_{R}I\leq
n$. This implies that ${\rm sfli}R\leq n$.

By [10, Theorem 5.3] we have ${\rm sfli}R={\rm G.wgldim}R$ and ${\rm
sfli}RG={\rm G.wgldim}RG$. Then, the inequality ${\rm G.wgldim}RG\leq{\rm
Ghd}_{R}G+{\rm G.wgldim}R$ follows immediately from Corollary 3.11. ∎

##  4\. Properties of Gorenstein homological dimension of groups

Using the above characterization on the finiteness of ${\rm Ghd}_{R}G$, we
obtain the following properties of the Gorenstein homological dimensions of
groups.

######  Proposition 4.1.

Let $H$ be any subgroup of $G$. If ${\rm sfli}R<\infty$, then ${\rm
Ghd}_{R}H\leq{\rm Ghd}_{R}G$.

###### Proof.

There is nothing to prove if ${\rm Ghd}_{R}G=\infty$ and hence we may assume
that ${\rm Ghd}_{R}G=n$ is finite. It follows from Corollary 3.3 that there
exists an $R$-pure $RG$-exact sequence $0\rightarrow R\rightarrow A$ for which
$A$ is $R$-flat such that ${\rm fd}_{RG}A=n$. Note that every flat $RG$-module
is restricted to be a flat $RH$-module, and $0\rightarrow R\rightarrow A$ is
also an exact sequence of $RH$-modules. We infer that ${\rm fd}_{RH}A\leq{\rm
fd}_{RG}A$, and moreover, for the trivial $RH$-module $R$, by Proposition 3.9
we have ${\rm Ghd}_{R}H={\rm Gfd}_{RH}R\leq{\rm fd}_{RH}A\leq n$. This
completes the proof. ∎

The following generalizes [1, Proposition 4.11]: let $H$ be a subgroup of $G$
of finite index, then ${\rm Ghd}_{\mathbb{Z}}H\leq{\rm Ghd}_{\mathbb{Z}}G$. We
remove the assumption of finite index therein.

######  Corollary 4.2.

Let $H$ be any subgroup of $G$. Then ${\rm Ghd}_{\mathbb{Z}}H\leq{\rm
Ghd}_{\mathbb{Z}}G$, and ${\rm Ghd}_{\mathbb{Q}}H\leq{\rm Ghd}_{\mathbb{Q}}G$.

######  Proposition 4.3.

Let $1\rightarrow H\rightarrow G\rightarrow L\rightarrow 1$ be an extension of
groups. If ${\rm sfli}R<\infty$, then ${\rm Ghd}_{R}G\leq{\rm Ghd}_{R}H+{\rm
Ghd}_{R}L.$

###### Proof.

It suffices to assume that both ${\rm Ghd}_{R}H=m$ and ${\rm Ghd}_{R}L=n$ are
finite. Then, it follows from Corollary 3.3 that there exists an $R$-pure
$RL$-exact sequence $0\rightarrow
R\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}A$ for which $A$ is $R$-flat
such that ${\rm fd}_{RL}A={\rm Ghd}_{R}L=n$.

For the quotient group $L=G/H$, we may consider every $RL$-module as an
$RG$-module through the natural morphism of group rings $RG\rightarrow RL$. We
claim that for any flat $RL$-module $F$, one has ${\rm Gfd}_{RG}F\leq{\rm
Ghd}_{R}H$. Note that $RL=R[G/H]={\rm Ind}_{H}^{G}R$; see for example [5,
Proposition III 5.6]. Since ${\rm Ghd}_{R}H={\rm Gfd}_{RH}R=m$, there is a
Gorenstein flat $RH$-resolution of length $m$

$0\longrightarrow M_{m}\longrightarrow\cdots\longrightarrow
M_{1}\longrightarrow M_{0}\longrightarrow R\longrightarrow 0.$

By applying the induction functor, we get an exact sequence of $RG$-modules

$0\longrightarrow{\rm
Ind}_{H}^{G}M_{m}\longrightarrow\cdots\longrightarrow{\rm
Ind}_{H}^{G}M_{1}\longrightarrow{\rm Ind}_{H}^{G}M_{0}\longrightarrow{\rm
Ind}_{H}^{G}R=RL\longrightarrow 0.$

By Lemma 3.7 we infer that all ${\rm Ind}_{H}^{G}M_{i}$ are Gorenstein flat
$RG$-modules. Then, we get that ${\rm Gfd}_{RG}RL\leq m$, and moreover, by
[18, Proposition 3.13] we induce that ${\rm Gfd}_{RG}P\leq m$ for any free
$RL$-module $P$. For any flat $RL$-module $F$, it follows from Lazard-Govorov
Theorem that $F=\varinjlim P_{i}$ is a direct limit of finitely generated free
$RL$-modules $P_{i}$. Furthermore, since the subcategory of Gorenstein flat
modules is closed under direct limits (see [25, Corollary 4.12] and [26, Lemma
3.1]), we prove the claim that ${\rm Gfd}_{RG}F={\rm Gfd}_{RG}(\varinjlim
P_{i})\leq m={\rm Ghd}_{R}H$.

Recall that ${\rm fd}_{RL}A={\rm Ghd}_{R}L=n$. We consider an exact sequence
of $RL$-modules

$0\longrightarrow F_{n}\longrightarrow\cdots\longrightarrow
F_{1}\longrightarrow F_{0}\longrightarrow A\longrightarrow 0$

in which all $F_{i}$ are flat. It follows from the above argument that ${\rm
Gfd}_{RG}F_{i}\leq m$ for $0\leq i\leq n$. By a standard argument, we infer
from the above exact sequence that ${\rm Gfd}_{RG}A\leq m+n$.

Since $A$ is an $R$-flat module and ${\rm Gfd}_{RG}A\leq m+n$ is finite, it
follows from Proposition 3.2 that there is an $R$-pure $RG$-exact sequence
$0\rightarrow A\stackrel{{\scriptstyle\beta}}{{\rightarrow}}B$, where $B$ is
an $R$-flat $RG$-module such that ${\rm fd}_{RG}B={\rm Gfd}_{RG}A\leq m+n$.
Let $\iota=\beta\alpha$. Then, we obtain an $R$-pure $RG$-exact sequence
$0\rightarrow R\stackrel{{\scriptstyle\iota}}{{\rightarrow}}B$, and
consequently, it follows from Proposition 3.9 that

${\rm Ghd}_{R}G={\rm Gfd}_{RG}R\leq{\rm fd}_{RG}B\leq m+n.$

This completes the proof. ∎

######  Proposition 4.4.

Let $H$ be a finite normal subgroup of $G$, and $R$ be a commutative ring such
that ${\rm sfli}R<\infty$. Then ${\rm Ghd}_{R}G={\rm Ghd}_{R}(G/H)$.

###### Proof.

By Proposition 4.3, we have ${\rm Ghd}_{R}G\leq{\rm Ghd}_{R}H+{\rm
Ghd}_{R}(G/H)$. Since the subgroup $H$ is finite, we infer that ${\rm
Ghd}_{R}H=0$; see for example [21, Corollary 3.3] or [1, Proposition 4.12]. In
fact, $R$ is a Gorenstein flat $RH$-module whenever $H$ is a finite group.
This yields ${\rm Ghd}_{R}G\leq{\rm Ghd}_{R}(G/H)$.

Conversely, since the inequality ${\rm Ghd}_{R}(G/H)\leq{\rm Ghd}_{R}G$ is
obvious if ${\rm Ghd}_{R}G=\infty$, it suffices to assume ${\rm Ghd}_{R}G=n$
is finite. In this case, it follows immediately from Corollary 3.3 that there
exists an $R$-pure $RG$-exact sequence $0\rightarrow
R\stackrel{{\scriptstyle\iota}}{{\rightarrow}}A$ for which $A$ is $R$-flat
such that ${\rm fd}_{RG}A=n$. Since the group $G$ acts trivially on $R$, it
implies that ${\rm Im}\iota\subseteq A^{G}\subseteq A^{H}$. We may therefore
consider the $R[G/H]$-module $A^{H}$, and the $R$-pure $R[G/H]$-monomorphism
$R\rightarrow A^{H}$.

In the following, we will prove that $A^{H}$ is $R$-flat and ${\rm
fd}_{R[G/H]}A^{H}\leq n$. Then, the desired inequality ${\rm
Ghd}_{R}(G/H)={\rm Gfd}_{R[G/H]}R\leq n$ will hold by Proposition 3.9.

Since $A$ is $R$-flat with ${\rm fd}_{RG}A<\infty$ and $H$ is a finite group,
we infer that $A$ is a flat $RH$-module, and then $A=\varinjlim P_{i}$ for
some finitely generated free $RH$-modules. Since $(RH)^{H}\cong R$, we infer
that $P_{i}^{H}$ are finitely generated free $R$-modules. Note that for any
$RG$-module $M$, $M^{H}\cong{\rm Hom}_{RH}(R,M)$ as $R[G/H]$-modules; see for
example [5, pp.56]. Since $H$ is a finite group, $R$ is a finitely presented
$RH$-module. Hence, we induce from the isomorphisms

$A^{H}\cong{\rm Hom}_{RH}(R,\varinjlim P_{i})\cong\varinjlim{\rm
Hom}_{RH}(R,P_{i})\cong\varinjlim P_{i}^{H}$

that $A^{H}$ is a flat $R$-module.

For the finite group $H$, $R\rightarrow RH$ is a Frobenius extension of rings,
and then ${\rm RHom}_{R}(RH,R)\simeq RH$ in the derived category
$\mathbf{D}(RH)$. Hence, we have

$\begin{split}&{\rm RHom}_{RH}(R,RH)\simeq{\rm RHom}_{RH}(R,{\rm
RHom}_{R}(RH,R))\\\ &\simeq{\rm RHom}_{R}(R\otimes^{L}_{RH}RH,R)\simeq{\rm
RHom}_{R}(R,R)=R,\end{split}$

The isomorphisms yield that ${\rm Ext}^{1}_{RH}(R,RH)=0$. We remark that the
vanishing of ${\rm Ext}^{1}_{RH}(R,RH)$ can also be obtained by the fact that
$R$ is a Gorenstein projective $RH$-module, if one notices that $R\rightarrow
RH$ is a Frobenius extension of rings, and the restriction functor from the
category of $RH$-modules to the category of $R$-modules is a faithful
Frobenius functor; see [6, Theorem 3.2] and [23, Theorem 2.2]. Moreover, we
infer that ${\rm Ext}^{1}_{RH}(R,F)=0$ for any flat $RH$-module $F$ since $F$
is a direct limit of some finitely generated free $RH$-modules.

Since ${\rm fd}_{RG}A=n$, there is an exact sequence

$0\longrightarrow F_{n}\longrightarrow\cdots\longrightarrow
F_{1}\longrightarrow F_{0}\longrightarrow A\longrightarrow 0$

for which $F_{i}$ are flat $RG$-modules. Analogous to the above argument, we
get that $F_{i}^{H}\cong{\rm Hom}_{RH}(R,F)$ are flat modules over $R[G/H]$.
By restriction, $F_{i}$ are flat $RH$-modules, and then ${\rm
Ext}^{1}_{RH}(R,F_{i})=0$. By applying ${\rm Hom}_{RH}(R,-)$ to the above
sequence, we can obtain an exact sequence of $R[G/H]$-modules

$0\longrightarrow F_{n}^{H}\longrightarrow\cdots\longrightarrow
F_{1}^{H}\longrightarrow F_{0}^{H}\longrightarrow A^{H}\longrightarrow 0.$

Hence, ${\rm fd}_{R[G/H]}A^{H}\leq n$, as expected. This completes the proof.
∎

The Weyl groups of the subgroups of a given group $G$ are useful tools in the
study of actions of $G$ on topological spaces, as these Weyl groups act
naturally on the fixed point spaces of the actions. Let $H$ be a subgroup of
$G$. We denote by $N_{G}(H)$ the normalizer of $H$ in $G$.

######  Corollary 4.5.

Let $H$ be a finite subgroup of $G$, and $R$ be a commutative ring such that
${\rm sfli}R<\infty$. Then for the Weyl group $W=N_{G}(H)/H$, one has ${\rm
Ghd}_{R}W\leq{\rm Ghd}_{R}G$.

###### Proof.

In view of Proposition 4.1, we have ${\rm Ghd}_{R}N_{G}(H)\leq{\rm Ghd}_{R}G$.
It follows from Proposition 4.4 that ${\rm Ghd}_{R}W={\rm Ghd}_{R}N_{G}(H)$.
Then, we get the inequality ${\rm Ghd}_{R}W\leq{\rm Ghd}_{R}G$. ∎

For a group $G$, recall that Ikenaga introduced the generalized homological
dimension of $G$ over the ring of integers $\mathbb{Z}$; see [19, III,
Definition]. Analogously, we may define the generalized homological dimension
of $G$ over any coefficient ring $R$ as follows:

$\underline{{\rm hd}}_{R}G={\rm sup}\\{i\in\mathbb{N}~{}~{}|~{}~{}{\rm
Tor}^{RG}_{i}(I,M)\neq 0,~{}~{}\exists
M~{}~{}R\text{-flat},~{}~{}\exists~{}~{}I~{}~{}RG\text{-injective}\\}.$

We conclude this section by comparing the Gorenstein homological dimension and
generalized homological dimension of groups.

######  Proposition 4.6.

Let $R$ be a commutative ring of coefficients. If ${\rm Ghd}_{R}G<\infty$,
then ${\rm Ghd}_{R}G\leq\underline{{\rm hd}}_{R}G$. Moreover, if ${\rm
sfli}R<\infty$, then ${\rm Ghd}_{R}G=\underline{{\rm hd}}_{R}G$.

###### Proof.

Assume ${\rm Ghd}_{R}G={\rm Gfd}_{RG}R$ is finite. It follows from [18,
Theorem 3.14] that

${\rm Ghd}_{R}G={\rm sup}\\{i\in\mathbb{N}~{}~{}|~{}~{}{\rm
Tor}^{RG}_{i}(I,R)\neq 0,~{}~{}\exists~{}~{}I~{}~{}RG\text{-injective}\\}.$

Then, we infer from the definition of $\underline{{\rm hd}}_{R}G$ that ${\rm
Ghd}_{R}G\leq\underline{{\rm hd}}_{R}G$.

Moreover, if we assume ${\rm sfli}R<\infty$, then every $RG$-module has finite
Gorenstein flat dimension; see Theorem 3.12. Since both ${\rm Ghd}_{R}G$ and
${\rm sfli}R$ are assumed to be finite, for any $RG$-module $M$, by Corollary
3.11 we have

$\mathrm{Gfd}_{RG}M\leq\mathrm{Ghd}_{R}G+{\rm Gfd}_{R}M.$

Then, $\underline{{\rm hd}}_{R}G={\rm sup}\\{{\rm
Gfd}_{RG}M~{}~{}|~{}~{}M~{}~{}R\text{-flat}\\}\leq{\rm Ghd}_{R}G$.
Consequently, we obtain the desired equality ${\rm Ghd}_{R}G=\underline{{\rm
hd}}_{R}G$. ∎

In the following, we denote by ${\rm wgldim}R$ the weak global dimension of
$R$.

######  Proposition 4.7.

For any group $G$ and any commutative ring $R$, there are inequalities

$\underline{{\rm hd}}_{R}G\leq{\rm sfli}RG\leq\underline{{\rm hd}}_{R}G+{\rm
wgldim}R.$

###### Proof.

If we assume ${\rm sfli}RG<\infty$, it follows from [10, Theorem 5.3] that
${\rm sfli}RG={\rm G.wgldim}RG$, and then the first inequality is clear.

In order to prove the second inequality, it suffices to assume that both
$\underline{{\rm hd}}_{R}G=m$ and ${\rm wgldim}R=n$ are finite. If $n=0$, then
every $R$-module is flat, and moreover, $\underline{{\rm hd}}_{R}G={\rm
G.wgldim}RG$ holds immediately from the definitions. Note that
$\underline{{\rm hd}}_{R}G$ is assumed to be finite, and then we have ${\rm
sfli}RG=\underline{{\rm hd}}_{R}G$.

Now assume $n>0$. In this case, the argument goes by induction on the flat
dimension of any $RG$-module $M$. We consider the exact sequence of
$RG$-modules

$0\longrightarrow K\longrightarrow P_{n-1}\longrightarrow\cdots
P_{1}\longrightarrow P_{0}\longrightarrow M\longrightarrow 0$

for which each $P_{i}$ is projective for $0\leq i<n$. Since $P_{i}$ is
restricted to be a projective $R$-module, we get that $K$ is a flat $R$-module
as ${\rm wgldim}R=n$. For any injective right $RG$-module $I$ and any $k>0$,
there are isomorphisms

${\rm Tor}^{RG}_{k}(I,M)\cong{\rm Tor}^{RG}_{k-n}(I,K).$

We infer from the assumption $\underline{{\rm hd}}_{R}G=m$ that ${\rm
Tor}^{RG}_{k-n}(I,K)=0$ for any $k-n>m$, and then ${\rm Tor}^{RG}_{k}(I,M)=0$
for any $k>m+n$. Hence, we have ${\rm fd}_{(RG)^{op}}I\leq m+n$, and moreover,
${\rm sfli}(RG)^{op}\leq m+n$. Since Gorenstein weak global dimension is
symmetric, it follows from [10, Theorem 5.12] or [7, Corollary 2.5] that ${\rm
sfli}RG={\rm sfli}(RG)^{op}$, and consequently, ${\rm sfli}RG\leq m+n$. This
completes the proof. ∎

######  Corollary 4.8.

Let $R$ be a commutative ring with finite weak global dimension. If
$\underline{{\rm hd}}_{R}G$ is finite, then ${\rm Ghd}_{R}G=\underline{{\rm
hd}}_{R}G$.

###### Proof.

Following the assumptions, we infer from Proposition 4.7 that ${\rm
sfli}RG\leq\underline{{\rm hd}}_{R}G+{\rm wgldim}R<\infty$. By Theorem 3.12,
this induces ${\rm Ghd}_{R}G<\infty$. Then, the equality ${\rm
Ghd}_{R}G=\underline{{\rm hd}}_{R}G$ holds by Proposition 4.6 immediately. ∎

######  Corollary 4.9.

${\rm Ghd}_{\mathbb{Z}}G<\infty$ if and only if $\underline{{\rm
hd}}_{\mathbb{Z}}G<\infty$. In this case, ${\rm
Ghd}_{\mathbb{Z}}G=\underline{{\rm hd}}_{\mathbb{Z}}G$.

##  5\. Some relative homological dimensions and invariants

Let $\Lambda$ be any associative ring with identity. We consider an acyclic
complex of projective left $\Lambda$-modules

$\mathbf{P}=\cdots\longrightarrow P_{n+1}\longrightarrow P_{n}\longrightarrow
P_{n-1}\longrightarrow\cdots$

and a $\Lambda$-module $M\cong{\rm Ker}(P_{0}\rightarrow P_{-1})$. Recall that
the module $M$ is _Gorenstein projective_ [14] if the complex $\mathbf{P}$
remains acyclic after applying $\mathrm{Hom}_{\Lambda}(-,Q)$ for any
projective module $Q$; $M$ is called a projectively coresolved Gorenstein flat
module, or a PGF-module for short [25], provided that the complex $\mathbf{P}$
remains acyclic after applying $I\otimes_{\Lambda}-$ for any injective right
$\Lambda$-module $I$. It is clear that PGF-modules are Gorenstein flat. As
shown in [25, Theorem 4.4], PGF-modules are also Gorenstein projective.

Every projective module is flat, however, it is not at all clear from the
definitions that Gorenstein projective modules are Gorenstein flat. It was
shown in [18, Proposition 3.4] that every left Gorenstein projective module is
Gorenstein flat if the base ring is right coherent and has finite left
finitistic dimension. In general, the relation between Gorenstein projective
and Gorenstein flat modules remains somehow mysterious, and it is still an
open problem that if every Gorenstein projective module is Gorenstein flat.

We have the following observation, which gives a sufficient condition for
affirmative answer of Gorenstein projective-flat problem.

######  Proposition 5.1.

Let $G$ be a group and $R$ be a commutative ring. If ${\rm Ghd}_{R}G<\infty$,
then ${\rm sfli}R<\infty$ if and only if ${\rm sfli}RG<\infty$. Moreover, if
both ${\rm Ghd}_{R}G$ and ${\rm sfli}R$ are finite, then every Gorenstein
projective $RG$-module is a PGF-module, and furthermore is a Gorenstein flat
module.

###### Proof.

The equivalence of finiteness of ${\rm sfli}R$ and ${\rm sfli}RG$ is easy to
see. The “if” part follows from the inequality ${\rm sfli}R\leq{\rm sfli}RG$.
The “only if” part follows from Theorem 3.12.

Note that the finiteness of both ${\rm Ghd}_{R}G$ and ${\rm sfli}R$ implies
${\rm sfli}(RG)^{op}={\rm sfli}RG<\infty$. In this case, by induction on the
flat dimension of any injective right $RG$-module $I$, we infer that any
acyclic complex of projective $RG$-modules remains acyclic after applying
$I\otimes_{RG}-$. Hence, invoking the definitions it is immediate that every
Gorenstein projective $RG$-module is a PGF-module, and moreover, is a
Gorenstein flat module; see also [10, Remark 2.3 (ii)]. ∎

Let $\Lambda$ be a ring. It follows from [16, 1.6] that the finiteness of both
${\rm spli}\Lambda$ and ${\rm silp}\Lambda$ implies that ${\rm
spli}\Lambda={\rm silp}\Lambda$. If $\Lambda$ is a commutative noetherian
ring, by [20, 5.9] one has the equality ${\rm spli}\Lambda={\rm silp}\Lambda$.
Let $R$ be a commutative Gorenstein ring (noetherian ring with finite self-
injective dimension), and $G$ be any group. It follows from [9, Theorem 4.4]
that ${\rm spli}RG={\rm silp}RG$.

We will consider ${\rm sfli}RG$ and ${\rm silf}RG$, and show that the
comparison between ${\rm sfli}RG$ and ${\rm silf}RG$ is essentially about a
problem on the relation between the projective and flat dimensions of modules.
Let $\overline{\mathcal{P}}(RG)$ and $\overline{\mathcal{F}}(RG)$ denote the
classes of $RG$-modules with finite projective dimension and finite flat
dimension, respectively. The invariant ${\rm splf}RG$ is defined as the
supremum of projective length (dimension) of flat $RG$-modules, which is due
to [22, Section II.3.3], while the notation “splf” is introduced in [11]. See
also [7, Proposition 3.2] for a characterization for “splf”.

######  Proposition 5.2.

Let $R$ be a commutative Gorenstein ring, $G$ be a group. The following are
equivalent:

  1. (1)

${\rm silf}RG<\infty$.

  2. (2)

${\rm sfli}RG<\infty$ and ${\rm splf}RG<\infty$.

  3. (3)

${\rm sfli}RG<\infty$ and
$\overline{\mathcal{P}}(RG)=\overline{\mathcal{F}}(RG)$.

###### Proof.

(1)$\Longrightarrow$(2) Since the coefficient ring $R$ is commutative
Gorenstein, combining with [9, Theorem 4.4] and [11, Proposition 2.1], we have
${\rm spli}RG={\rm silp}RG={\rm silf}RG$. Hence, we infer from ${\rm
silf}RG<\infty$ that ${\rm sfli}RG\leq{\rm spli}RG<\infty$. Moreover, the
finiteness of both ${\rm silf}RG$ and ${\rm spli}RG$ yields that the
projective dimension of any flat $RG$-module is finite, and hence ${\rm
splf}RG<\infty$.

(2)$\Longrightarrow$(1) The finiteness of both ${\rm sfli}RG$ and ${\rm
splf}RG$ implies that the projective dimension of any injective $RG$-module is
finite, that is, ${\rm spli}RG<\infty$. Then, it follows from [16, Theorem
2.4] that ${\rm silp}RG$ is also finite. Moreover, it follows from [11,
Proposition 2.1] that ${\rm silf}RG={\rm silp}RG<\infty$.

(2)$\Longleftrightarrow$(3) Since projective modules are flat, it is clear
that $\overline{\mathcal{P}}(RG)\subseteq\overline{\mathcal{F}}(RG)$. It is
easy to see that the projective dimension of any flat $RG$-module is finite if
and only if $\overline{\mathcal{F}}(RG)\subseteq\overline{\mathcal{P}}(RG)$.
Hence, the assertion follows. ∎

For any $\Lambda$-module $M$, the _Gorenstein projective dimension_ , denoted
by $\mathrm{Gpd}_{\Lambda}M$, is defined in the standard way via resolutions
by Gorenstein projective modules; see [14, 18]. The Gorenstein global
dimension of $\Lambda$ [4], denoted by ${\rm G.gldim}\Lambda$, is defined as
the supremum of Gorenstein projective dimension of all $\Lambda$-modules.
Analogously, the PGF-dimension ${\rm PGF}\text{-}{\rm dim}_{\Lambda}M$ of any
$\Lambda$-module $M$, and the PGF-global dimension ${\rm PGF}\text{-}{\rm
gldim}\Lambda$ are defined. The Gorenstein projective dimension is a
refinement of the PGF-dimension in the sense that ${\rm
Gpd}_{\Lambda}M\leq{\rm PGF}\text{-}{\rm dim}_{\Lambda}M$ for any module $M$,
and ${\rm Gpd}_{\Lambda}M={\rm PGF}\text{-}{\rm dim}_{\Lambda}M$ if ${\rm
PGF}\text{-}{\rm dim}_{\Lambda}M<\infty$; see [8, Corollary 3.7].

######  Corollary 5.3.

Let $R$ be a commutative Gorenstein ring, $G$ be any group. If ${\rm silf}RG$
is finite, then the following hold:

  1. (1)

${\rm Ghd}_{R}G<\infty$, and every Gorenstein projective $RG$-module is
Gorenstein flat.

  2. (2)

For any Gorenstein flat $RG$-module $M$, ${\rm PGF}\text{-}{\rm
dim}_{RG}M={\rm Gpd}_{RG}M<\infty$.

###### Proof.

By Proposition 5.2, the finiteness of ${\rm silf}RG$ implies both ${\rm
sfli}RG<\infty$ and ${\rm splf}RG<\infty$. Then the assertion (1) follows by
Theorem 3.12 and Proposition 5.1, and the assertion (2) is immediate from [8,
Proposition 3.9] together with [8, Corollary 3.7 (2)]. ∎

######  Corollary 5.4.

Let $R$ be a commutative Gorenstein ring. For any group $G$, the following are
equivalent:

  1. (1)

${\rm silf}RG<\infty$.

  2. (2)

${\rm Ghd}_{R}G<\infty$, every Gorenstein flat $RG$-module has finite
Gorenstein projective dimension and every flat Gorenstein projective
$RG$-module is projective.

###### Proof.

Note that for any commutative Gorenstein ring $R$, ${\rm spli}R={\rm
silp}R<\infty$. Then, ${\rm sfli}R\leq{\rm spli}R<\infty$, and by Theorem 3.12
we have ${\rm Ghd}_{R}G<\infty$ if and only if ${\rm sfli}RG<\infty$. It
follows from [7, Proposition 3.2] that ${\rm splf}RG<\infty$ if and only if
every Gorenstein flat $RG$-module has finite Gorenstein projective dimension
and every flat Gorenstein projective $RG$-module is projective. Hence, the
assertion follows immediately from Proposition 5.2. ∎

It follows from [14, Proposition 10.2.3] that any Gorenstein projective module
of finite projective dimension is necessarily projective. For modules over
group rings, the hypothesis on the finiteness of the projective dimension may
be relaxed as shown below.

######  Corollary 5.5.

Let $R$ be a commutative Gorenstein ring, and $G$ be a group. If ${\rm
silf}RG<\infty$, then any Gorenstein projective $RG$-module of finite flat
dimension is necessarily projective.

It is trivially true that for any finite group $G$, the integral group ring
$\mathbb{Z}G$ is noetherian, while it is quite subtle for infinite $G$. It
follows from [17, Theorem 1.1] that for any virtually polycyclic group $G$,
the group ring $\mathbb{Z}G$ is noetherian. Recall that a group $G$ is
polycyclic if there exists a subnormal sequence

$G=G_{1}\rhd G_{2}\rhd\cdots\rhd G_{n}=1$

such that $G_{k}/G_{k+1}$ is cyclic for all $1\leq k<n$. The minimal length of
such a subnormal sequence is the Hirsch length of $G$. A group is virtually
polycyclic, also named a polycyclic-by-finite group, if it has a polycyclic
subgroup of finite index.

######  Proposition 5.6.

Let $R$ be a commutative Gorenstein ring. Let $G$ be a group such that $RG$ is
a left noetherian ring. Then ${\rm sfli}RG<\infty$ if and only if ${\rm
silf}RG<\infty$. In this case, one has ${\rm sfli}RG={\rm silf}RG$.

###### Proof.

The “if” part follows from Proposition 5.2, that is, ${\rm silf}RG<\infty$
implies ${\rm sfli}RG<\infty$. By [9, Proposition 4.2] we have an inequality
${\rm sfli}RG\leq{\rm silp}RG$. Moreover, it follows from [11, Proposition
2.1] that ${\rm silf}RG={\rm silp}RG$; see also [1, Theorem 3.3]. Hence, we
have ${\rm sfli}RG\leq{\rm silf}RG$. Here, it is not necessary to assume that
the group ring $RG$ is noetherian.

For the “only if” part, we assume $RG$ is a left noetherian ring, and ${\rm
sfli}RG=n<\infty$. Let $F$ be any flat left $RG$-module. Consider the exact
sequence of left $RG$-modules

$0\longrightarrow F\longrightarrow I_{0}\longrightarrow\cdots\longrightarrow
I_{n-1}\longrightarrow L\longrightarrow 0,$

for which $I_{i}$ ($0\leq i\leq n-1$) are injective modules. Denote by
$(-)^{+}$ the Pontryagin dual ${\rm
Hom}_{\mathbb{Z}}(-,\mathbb{Q}/\mathbb{Z})$. Then we get an exact sequence of
right $RG$-modules

$0\longrightarrow L^{+}\longrightarrow
I_{n-1}^{+}\longrightarrow\cdots\longrightarrow I_{1}^{+}\longrightarrow
I_{0}^{+}\longrightarrow F^{+}\longrightarrow 0.$

It is clear that $F^{+}$ is an injective $RG$-module. Since $RG$ is left
noetherian, it follows that $I_{i}^{+}$ are flat right $RG$-modules for $0\leq
i\leq n-1$; see for example [14, Corollary 3.2.17]. Then, the assumption ${\rm
sfli}RG=n$ implies that ${\rm sfli}(RG)^{op}=n$, and furthermore $L^{+}$ is a
flat right $RG$-module. Hence, $L$ is an injective left $RG$-module by [14,
Corollary 3.2.17]. Consequently, we get the desired inequality ${\rm
silf}RG\leq n$. This completes the proof. ∎

It is well known that ${\rm G.gldim}\Lambda<\infty$ if and only if ${\rm
spli}\Lambda={\rm silp}\Lambda<\infty$; in this case, one has ${\rm
G.gldim}\Lambda={\rm spli}\Lambda={\rm silp}\Lambda$; see for example [10,
Theorem 4.1]. For PGF-global dimension, it follows from [8, Theorem 5.1] that
${\rm PGF}\text{-}{\rm gldim}\Lambda<\infty$ if and only if ${\rm
spli}\Lambda={\rm silp}\Lambda<\infty$ and ${\rm sfli}\Lambda={\rm
sfli}\Lambda^{op}<\infty$; in this case one has

${\rm PGF}\text{-}{\rm gldim}\Lambda={\rm spli}\Lambda={\rm silp}\Lambda={\rm
G.gldim}\Lambda.$

Moreover, we have the following.

######  Proposition 5.7.

Let $R$ be a commutative ring, and $G$ be a group. Then ${\rm spli}RG<\infty$
if and only if ${\rm PGF}\text{-}{\rm gldim}RG={\rm G.gldim}RG<\infty$; in
this case, ${\rm PGF}\text{-}{\rm gldim}RG={\rm spli}RG$.

###### Proof.

It suffices to prove the “only if” part. It follows from [8, Corollary 5.4]
and [16, 1.6] that ${\rm silp}RG\leq{\rm spli}RG$ with the equality if ${\rm
spli}RG<\infty$. Invoking the finiteness of ${\rm spli}RG$, we infer that
${\rm sfli}(RG)^{op}={\rm sfli}RG<\infty$. Then the assertion and the equality
${\rm PGF}\text{-}{\rm gldim}RG={\rm spli}RG$ follow immediately from [8,
Theorem 5.1]. ∎

It is well known that flat dimension of any module is a refinement of its
projective dimension, while it is not clear if the inequality ${\rm
Gfd}M\leq{\rm Gpd}M$ holds “locally” for any module $M$. However, for group
rings the inequality follows “globally” as shown below.

######  Proposition 5.8.

Let $R$ be an commutative ring and $G$ a group. Then there is an inequality

${\rm G.wgldim}RG\leq{\rm PGF}\text{-}{\rm gldim}RG={\rm G.gldim}RG\leq{\rm
G.wgldim}RG+{\rm splf}RG$.

###### Proof.

The inequality ${\rm G.wgldim}RG\leq{\rm G.gldim}RG$ follows from [4,
Corollary 1.2 (1)]. For completeness, we briefly include an argument in [10,
Remark 5.4 (ii)], which is different from that of [4]. The inequality is
obvious if ${\rm G.gldim}RG=\infty$ and hence we may assume that ${\rm
G.gldim}RG=n<\infty$. Then it follows from [10, Theorem 4.1] that ${\rm
silp}RG={\rm spli}RG=n$. By the definition, we have ${\rm sfli}RG\leq{\rm
spli}RG$. Hence, it follows from [10, Theorem 5.3] that ${\rm G.wgldim}RG={\rm
sfli}RG\leq n$.

The inequality ${\rm G.gldim}RG\leq{\rm G.wgldim}RG+{\rm splf}RG$ is immediate
from [7, Theorem 3.3]. It remains to prove the equality ${\rm PGF}\text{-}{\rm
gldim}RG={\rm G.gldim}RG$. It follows from [25, Theorem 4.4] that every PGF-
module is Gorenstein projective, and then the inequality ${\rm
G.gldim}RG\leq{\rm PGF}\text{-}{\rm gldim}RG$ is clear. Conversely, in proving
the inequality ${\rm G.gldim}RG\geq{\rm PGF}\text{-}{\rm gldim}RG$, it
suffices to assume that ${\rm G.gldim}RG=n$ is finite. Then ${\rm
sfli}RG\leq{\rm spli}RG={\rm silp}RG=n$. Hence, we infer from [8, Theorem 5.1]
that ${\rm PGF}\text{-}{\rm gldim}RG<\infty$, and moreover ${\rm
G.gldim}RG={\rm PGF}\text{-}{\rm gldim}RG$. This completes the proof. ∎

Acknowledgements. The first author is grateful to Rui-Peng Zhu for his help in
proving Proposition 4.4. The authors also thank Prof. Emmanouil and Talelli
for sharing their preprint [13]. W. Ren is supported by National Natural
Science Foundation of China (No. 11871125), and G. Yang is supported by
National Natural Science Foundation of China (No. 12161049) and the Natural
Science Foundation of Gansu Province of China (No. 21JR7RA295).

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Wei Ren,
School of Mathematical Sciences, Chongqing Normal University, Chongqing
401331, PR China

Gang Yang,
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.
China