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PureMathematics
n
Ø
ê
Æ
,2023,13(6),1758-1768
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136180
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OnGorensteinHomologicalDimensionof
Groups
YuxiangLuo
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:May21
st
,2023;accepted:Jun.22
nd
,2023;published:Jun.30
th
,2023
Abstract
Let
G
begroupand
R
commutativering.TheGorensteinhomologicaldimension
Ghd
R
G
ofthegroup
G
overthecoefficientring
R
isdefinedastheGorensteinflat
©
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[J].
n
Ø
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Æ
,2023,13(6):1758-1768.
DOI:10.12677/pm.2023.136180
Û
Œ
Œ
dimensionoftrivial
RG
-module
R
.Itisprovedthat
Ghd
S
G
=Ghd
R
G
foranyFrobe-
niusextensionofcommutativerings
R
→
S
.Inaddition,therelationshipbetween
theGorensteinhomologicaldimensionofagroupandtheGorensteinhomologicaldi-
mensionofitssubgroupsisstudied.Itisprovedthat
Ghd
R
G
≤
sup
λ<µ
Ghd
R
G
λ
for
anascendingfilltering
(
G
λ
)
λ<µ
ofgroup
G
;furthermore,if
[
G
:
G
λ
]
λ<µ
isfinite,then
Ghd
R
G
= sup
λ<µ
Ghd
R
G
λ
.
Keywords
GorensteinHomologicalDimension,GroupRing,GorensteinFlat,
FrobeniusExtension
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2023.1361801760
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,
n>
0
.
^
M
n
(
R
)
Ú
S
n
(
R
)
©
OL
«
n
-
Ý
‚
Ú
¥
%
é
¡
Ý
‚
.
K
S
n
(
R
)
→
M
n
(
R
)
´
Frobenius
*
Ü
.
„
(
[27]
½
n
3.1).
+
‚
þ
R
´
˜
‡
†‚
Ú
G
´
˜
‡
+
.
RG
´
G
ƒ
)
¤
g
d
R
-
,
Ï
d
RG
ƒ
Œ
±
•
˜
/
L
«
•
P
g
∈
G
r
(
g
)
g
,
Ù
¥
r
(
g
)
∈
R
,
…
é
A
¤
k
g
k
r
(
g
)= 0.
ù
¦
RG
¤
˜
‡
‚
,
¡
•
G
+
‚
.
+
‚
RG
þ
˜
‡
M
´
˜
‡
R
-
M
\
þ+
G
é
M
Š
^
.
é
?
¿
RG
-
M
,
G
²
…
Š
^
3
M
þ
•
Œ
f
•
M
G
:=
{
m
∈
M
|
gm
=
m,
∀
g
∈
G
}
.
a
q
/
,
G
²
…
Š
^
3
M
þ
•
Œ
û
d
M
†
/
X
{
gm
−
m
|∀
g
∈
G,m
∈
M
}
ƒ
)
¤
f
û
,
=
M
G
:=
M/
h
gm
−
m
i
.
du
R
´
†‚
,
+
G
k
g
‡
Ó
g
→
g
−
1
,
Œ
ò
?
Û
†
RG
-
M
À
•
m
RG
-
,
Ù
¥
é
?
¿
g
∈
G
Ú
m
∈
M
k
mg
=
g
−
1
m
.
ù
,
é
u
?
¿
ü
‡
†
RG
-
M
Ú
N
,
Ü
þ
È
M
⊗
RG
N
Ï
L'
X
g
−
1
m
⊗
n
=
mg
⊗
n
=
m
⊗
gn
DOI:10.12677/pm.2023.1361801761
n
Ø
ê
Æ
Û
Œ
Œ
Ú
\
C
k
¿Â
.
^
gm
O
†
m
,
l
k
m
⊗
n
=
g
−
1
(
gm
)
⊗
n
= (
gm
)
g
⊗
n
=
gm
⊗
gn
Ï
d
M
⊗
RG
N
= (
M
⊗
R
N
)
G
,
Ù
¥
G
“
é
”
Š
^
3
M
⊗
R
N
þ
,
=
g
(
m
⊗
n
)=
gm
⊗
gn
,
Ù
¥
m
∈
M,n
∈
N
,
g
∈
G
.
d
,
G
é
Hom
R
(
M,N
)
+
Š
^
,
d
“
é
”
Š
^
(
gu
)(
m
) =
g
·
u
(
g
−
1
m
)
‰
Ñ
,
Ù
¥
g
∈
G,u
∈
Hom
R
(
M,N
)
,m
∈
M.
d
d
Œ
Hom
RG
(
M,N
) = Hom
R
(
M,N
)
G
.
•
õ
[
!
ë
•
©
z
[4].
Ú
n
2.7.
R,S
´
†‚
,
G
´
+
.
X
J
R
→
S
´
‚
*
Ü
,
@
o
k
¼
f
g
,
d
:
(1)
S
⊗
R
−'
SG
⊗
RG
−
.
(2)
Hom
R
(
S,
−
)
'
Hom
RG
(
SG,
−
)
.
y
²
(1)
¼
f
S
:=
S
⊗
R
−
,
¼
f
T
:=
SG
⊗
RG
−
.
K
S
,
T
´
l
RG
-
‰
Æ
Mod(
RG
)
SG
-
‰
Æ
Mod(
SG
)
C
¼
f
.
-
τ
:
S
→
T
•
˜
‡
C
†
,
τ
=(
τ
M
:
S
M
→
T
M
)
M
∈
Mod(
RG
)
.
é
?
¿
f
:
M
→
M
0
,
Ù
¥
M,M
0
∈
Mod(
RG
).
k
e
†
ã
:
S
M
τ
M
/
/
S
f
T
M
T
f
S
M
0
τ
M
0
/
/
T
M
0
Ï
d
τ
´
˜
‡
g
,
C
†
.
d
é
?
¿
M
∈
Mod(
RG
),
k
S
⊗
R
M
∼
=
SG
⊗
SG
(
S
⊗
R
M
) = (
SG
⊗
S
S
⊗
R
M
)
G
∼
=
(
SG
⊗
R
M
)
G
=
SG
⊗
RG
M,
ù
Ò
¿
›
X
τ
´
˜
‡
g
,
Ó
.
Ï
d
S
⊗
R
−'
SG
⊗
RG
−
.
(2)
¼
f
S
0
:= Hom
R
(
S,
−
),
¼
f
T
0
:= Hom
RG
(
SG,
−
).
K
S
0
,
T
0
´
l
RG
-
‰
Æ
Mod(
RG
)
SG
-
‰
Æ
Mod(
SG
)
C
¼
f
.
-
η
:
S
0
→
T
0
•
˜
‡
C
†
,
η
= (
η
N
:
S
0
N
→
T
0
N
)
N
∈
Mod(
RG
)
.
é
?
¿
f
:
N
→
N
0
,
Ù
¥
N,N
0
∈
Mod(
RG
).
k
e
†
ã
:
S
0
N
η
N
/
/
S
0
f
T
0
N
T
0
f
S
0
N
0
η
N
0
/
/
T
0
N
0
DOI:10.12677/pm.2023.1361801762
n
Ø
ê
Æ
Û
Œ
Œ
Ï
d
η
´
˜
‡
g
,
C
†
.
d
é
?
¿
N
∈
Mod(
RG
),
k
Hom
RG
(
SG,N
)
∼
=
Hom
RG
(
RG
⊗
RG
SG,N
)
∼
=
Hom
RG
(
RG
⊗
R
S,N
)
∼
=
(Hom
R
(
RG
⊗
R
S,N
))
G
∼
=
(Hom
R
(
RG,
Hom
R
(
S,N
)))
G
∼
=
Hom
RG
(
RG,
Hom
R
(
S,N
))
∼
=
Hom
R
(
S,N
)
,
ù
Ò
¿
›
X
η
´
˜
‡
g
,
Ó
.
Ï
d
Hom
R
(
S,
−
)
'
Hom
RG
(
SG,
−
).
y
.
.
3.
+
Gorenstein
Ó
N
‘
ê
é
X
ê
‚
•
6
5
!
k
‰
Ñ
+
G
3
X
ê
‚
R
þ
Gorenstein
Ó
N
‘
ê
½
Â
,
,
?
Ø
Ù
3
Frobenius
*
Ü
e
5
Ÿ
.
é
?
¿
+
G
,
+
Gorenstein
Ó
N
‘
ê
½
Â
•
²
…
Z
G
Z
Gorenstein
²
"
‘
ê
([12]
½
Â
4.5).
a
q
/
,
3
©
z
[13]
¥
k
e
½
Â
½
Â
3.1.
[13]
R
´
†‚
Ú
G
´
?
¿
+
.
+
G
3
X
ê
‚
R
þ
Gorenstein
Ó
N
‘
ê
½
Â
•
²
…
RG
-
R
Gorenstein
²
"
‘
ê
,
P
Š
Ghd
R
G
.
5
3.2.
du
?
¿
²
"
´
Gorenstein
²
"
,
é
?
¿
†‚
R
Ú
+
G
k
Ghd
R
G
≤
hd
R
G
,
Ù
¥
hd
R
G
L
«
+
G
3
X
ê
‚
R
þ
Ó
N
‘
ê
,
=
²
…
RG
-
R
²
"
‘
ê
.
A
O
/
,
hd
R
G
k
•
ž
,
Ghd
R
G
= hd
R
G
.
Ú
n
3.3.
[13]
R
→
S
´
†‚
²
"
*
Ü
.
é
?
¿
+
G
,
k
Ghd
S
G
≤
Ghd
R
G.
Ú
n
3.4.
R,S
´
†‚
,
G
´
+
.
X
J
R
→
S
´
Frobenius
*
Ü
,
@
o
RG
→
SG
•
´
Frobenius
*
Ü
.
y
²
du
R
→
S
´
†‚
Frobenius
*
Ü
,
Ï
L
½
Â
2.5
Œ
•
,
•
3
¼
f
g
,
d
S
⊗
R
−'
Hom
R
(
S,
−
).
qd
Ú
n
2.7
Œ
•
S
⊗
R
−'
SG
⊗
RG
−
Ú
Hom
R
(
S,
−
)
'
Hom
RG
(
SG,
−
).
Ï
d
k
¼
f
g
,
d
SG
⊗
RG
−'
Hom
RG
(
SG,
−
),
=
RG
→
SG
´
˜
‡
Frobenius
*
Ü
.
y
.
.
e
¡
½
n
L
²
+
Gorenstein
Ó
N
‘
ê
Ghd
R
G
÷
X
‚
Frobenius
*
Ü
ä
k
ØC
5
.
½
n
3.5.
R
→
S
´
†‚
Frobenius
*
Ü
.
é
?
¿
+
G
,
k
Ghd
S
G
= Ghd
R
G.
y
²
du
R
→
S
´
†‚
Frobenius
*
Ü
,
g
,
•
´
‚
²
"
*
Ü
,
Ï
d
Š
â
½
n
3.3
á
=
k
Ghd
S
G
≤
Ghd
R
G.
e
y
Ghd
R
G
≤
Ghd
S
G
.
•
Ä
R
Ú
S
Š
•
RG
-
,
Ù
¥
G
²
…
Š
^
3
R
Ú
S
þ
,
@
o
R
´
S
†
Ú
‘
.
l
Š
â
Ú
n
2.3
Œ
•
Ghd
R
G
= Gfd
RG
R
≤
Gfd
RG
S
.
Ï
d
•
I
y
²
Ø
ª
Gfd
RG
S
≤
Ghd
S
G
=
Gfd
SG
S
¤
á
=
Œ
.
X
J
Ghd
S
G
=
∞
,
@
o
Gfd
RG
S
≤
Ghd
S
G
´
w
,
.
y
b
Ghd
S
G
=
n
´
k
•
.
Š
â
½
Â
•
DOI:10.12677/pm.2023.1361801763
n
Ø
ê
Æ
Û
Œ
Œ
3
SG
-
Ü
S
Q
= 0
→
Q
n
→
Q
n
−
1
→···→
Q
1
→
Q
0
→
S
→
0
Ù
¥
Q
i
∈GF
(
SG
)(0
≤
i
≤
n
).
M
´
?
¿
Gorenstein
²
"
SG
-
.
@
o
Š
â
½
Â
,
•
3
SG
-
²
"
©
)
F
=
···→
F
−
1
→
F
0
→
F
1
→···
¦
M
∼
=
Im(
F
0
→
F
1
).
du
R
→
S
´
†‚
Frobenius
*
Ü
,
Š
â
Ú
n
3.4
Œ
•
RG
→
SG
•
´
Frobenius
*
Ü
.
Ï
d
é
?
¿
S
m
RG
-
I
,
k
Ó
I
⊗
RG
SG
∼
=
Hom
RG
(
SG,I
)
Ú
I
⊗
RG
F
∼
=
(
I
⊗
RG
SG
)
⊗
SG
F
.
5
¿
p
I
⊗
RG
SG
´
S
SG
-
…
E
/
I
⊗
RG
F
´
Ü
.
Ï
d
SG
-
²
"
©
)
F
•
›
•
RG
-
S
E
•
²
"
©
)
,
l
M
•
›
•
Gorenstein
²
"
RG
-
.
Ï
d
Q
•
›
•
RG
-
S
E
•
Ü
…
z
‡
Q
i
´
Gorenstein
²
"
RG
-
,
=
Gfd
RG
S
≤
n
.
y
.
.
4.
+
C
z
e
+
Gorenstein
Ó
N
‘
ê
!
ò
ï
Ä
+
Gorenstein
Ó
N
‘
ê
†
Ù
f
+
Gorenstein
Ó
N
‘
ê
ƒ
m
é
X
.
R
´
†‚
,
H
´
+
G
f
+
.
w
,
RG
Š
•
RH
-
´
g
d
.
Ï
d
RH
→
RG
g
,
/
¤
˜
‡
²
"
*
Ü
,
?
l
RG
-
RH
-
•
›
¼
f
Res
G
H
´
Ü
…
±
S
Ú
²
"
.
P
l
RH
-
‰
Æ
Mod(
RH
)
RG
-
‰
Æ
Mod(
RG
)
p
¼
f
•
Ind
G
H
−
=
RG
⊗
RH
−
Ú
{p
¼
f
•
Coind
G
H
−
= Hom
RH
(
RG,
−
)
.
¡
+
G
˜
‡
f
+
S
(
G
λ
)
λ<µ
•
,
S
L
È
,
X
J
•
3
˜
‡
4
•
•
I
µ
Ú
S
{
1
}
=
G
0
⊆
G
1
⊆···⊆
G
λ
⊆···⊆
G,
¦
G
=
G
µ
= lim
−→
λ<µ
G
λ
.
Ú
n
4.1.
R
´
†‚
,
H
´
+
G
•
ê
k
•
f
+
.
é
?
¿
RH
-
M
,
p
Ind
G
H
M
´
Gorenstein
²
"
RG
-
.
y
²
du
M
´
Gorenstein
²
"
RH
-
,
Ï
d•
3
˜
‡
RH
-
²
"
©
)
F
=
···→
F
−
1
→
F
0
→
F
1
→···
5
¿
RG
Š
•
RH
-
´
²
"
,
Ï
d
k
²
"
RG
-
Ü
E
/
Ind
G
H
F
=
···→
Ind
G
H
F
−
1
→
Ind
G
H
F
0
→
Ind
G
H
F
1
→···
.
é
?
¿
S
m
RG
-
E
,
E
´
S
RH
-
…
k
Ó
E
⊗
RG
Ind
G
H
F
∼
=
E
⊗
RH
F
.
Ï
d
Ind
G
H
M
´
Gorenstein
²
"
RG
-
.
DOI:10.12677/pm.2023.1361801764
n
Ø
ê
Æ
Û
Œ
Œ
·
K
4.2.
R
´
†‚
,
G
´
+
.
•
Ä
+
G
˜
‡
,
S
L
È
(
G
λ
)
λ<µ
.
é
?
¿
RG
-
M
Ú
λ<µ
,
X
J
M
Š
•
RG
λ
-
´
Gorenstein
²
"
,
@
o
M
´
Gorenstein
²
"
RG
-
.
y
²
é
?
¿
λ<µ
,
Œ
À
M
•
˜
‡
RG
λ
-
.
y
•
Ä
p
M
λ
= Ind
G
G
λ
M
=
RG
⊗
RG
λ
M.
du
G
µ
=
G
,
Ï
d
M
µ
=
RG
⊗
RG
µ
M
=
M
.
qd
Ú
n
4.1
Œ
•
p
M
λ
•
´
Gorenstein
²
"
RG
-
.
é
?
¿
k
≤
λ<µ
,
d
i
\
G
k
→
G
λ
p
RG
-
÷
Ó
M
k
→
M
λ
.
l
,
·
‚
k
RG
-
S
M
0
→
M
1
→···→
M
λ
→···→
M
µ
=
M
5
¿
Š
â
([21]
½
n
3.7)
Ú
([28]
·
K
2.3)
Œ
•
GF
(
RG
)
é
?
¿
•
4
•
µ
4
.
Ï
d
M
=
M
µ
= lim
−→
λ<µ
M
λ
´
Gorenstein
²
"
RG
-
.
y
.
.
·
K
4.3.
R
´
†‚
,
G
´
+
.
•
Ä
+
G
˜
‡
,
S
L
È
(
G
λ
)
λ<µ
.
é
?
¿
RG
-
M
,
k
Gfd
RG
M
≤
sup
λ<µ
Gfd
RG
λ
M.
y
²
X
J
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