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PureMathematics
n
Ø
ê
Æ
,2021,11(5),767-775
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115091
Y
-Gorenstein
SSS
ÚÚÚ
Frobenius
VVV
~~~
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c
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µ
2021
c
5
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Á
‡
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‡
ï
Ä
Y
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S
Ú
Frobenius
V
ƒ
m
'
X
.
‚
R
Ú
S
Ñ
´
k
ü
(
Ü
‚
,
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
.
y
²
(1)
R
op
-
X
´
Y
-Gorenstein
S
…
=
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
;(2)
R
-
Y
´
Y
-Gorenstein
S
…
=
M
⊗
R
Y
´
Y
-Gorenstein
S
S
-
"
'
…
c
Y
-Gorenstein
S
§
Frobenius
V
§
)
¤
f
Y
-GorensteinInjectiveModuleandFrobeniusBimodules
XiaomeiWang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.10
th
,2021;accepted:May11
th
,2021;published:May18
th
,2021
Abstract
Inthispaper,wemainlystudytherelationshipbetween
Y
-Gorensteininjectivemodule
andFrobeniusbimodules.Let
R
and
S
beassociativeringswithanidentity,
S
M
R
be
©
Ù
Ú
^
:
~
.
Y
-Gorenstein
S
Ú
Frobenius
V
[J].
n
Ø
ê
Æ
,2021,11(5):767-775.
DOI:10.12677/pm.2021.115091
~
Frobeniusbimodulewith
M
R
agenerator.Weprovedthat(1)
R
op
-module
X
is
Y
-
Gorensteininjectivemoduleifandonlyif
Hom
R
op
(
M,X
)
is
Y
-Gorensteininjective
S
op
-module;(2)
R
-module
Y
is
Y
-Gorensteininjectivemoduleifandonlyif
M
⊗
R
Y
is
Y
-Gorensteininjective
S
-module.
Keywords
Y
-GorensteinInjectiveModule,FrobeniusBimodules,Generator
Copyright
c
2021byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
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[1]
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Noether
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k
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.1995
c
,Enochs
<
3
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z
[2]
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\
Gorenstein
S
Ú
Ý
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g
.2008
c
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<
3
©
z
[3]
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Â
GorensteinFP-
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¿
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Ä
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Ó
N
5
Ÿ
.2009
c
,Ding
<
3
[4]
¥
½
Â
r
Gorenstein
²
"
.2010
c
,Gillespie
3
[5]
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ò
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Gorenstein
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<
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[6]
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<
3
[7]
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-Gorenstein
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Ä
Ù
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5
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3
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ê
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:
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Ú
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Frobenius
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V
g
.
3
[9,10]
¥
,Nakayama,Tsuzuku
Ú
Morita
Š
?
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ï
Ä
.1999
c
,Kadison
3
[11]
¥
•
g
/
ï
Ä
Frobenius
*
Ü
,
¿
…
J
Ñ
Frobenius
V
V
g
.2018-2019
c
,Ren
3
[12–14]
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ï
Ä
Frobenius
*
Ü
þ
Gorenstein
Ý
(
S
,
²
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,
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DOI:10.12677/pm.2021.115091768
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¡
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V
M
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J
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M
R
Ñ
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•
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;
(2)
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3
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:=
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S
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w
R
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op
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)
S
=:
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R
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E
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M
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•
3
Ý
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-
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P
=
···→
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−
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→
P
−
1
→
P
0
→
P
1
→···
¦
M
∼
=
Ker
(
P
0
→
P
1
),
…
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?
¿
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∈X
,Hom
R
(
P
,F
)
Ü
.
·
‚
^
X−GP
(
R
)
L
«
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-Gorenstein
Ý
a
.
5
P
1
.
5(1)
d
½
Â
•
,
X
-Gorenstein
Ý
´
Gorenstein
Ý
.
(2)
X
J
X
´
Ý
R
-
,
@
o
X
-Gorenstein
Ý
R
-
Ò
´
Gorenstein
Ý
R
-
.
(3)
X
J
X
´
²
"
R
-
,
@
o
X
-Gorenstein
Ý
R
-
Ò
´
Ding
Ý
R
-
.
(4)
X
J
X
´
Gorenstein
Ý
R
-
,
@
o
X
-Gorenstein
Ý
R
-
Ò
´
Ý
.
3.
Y
-Gorenstein
S
Ú
Frobenius
V
!
Ì
‡
ï
Ä
Frobenius
V
Ú
Y
-Gorenstein
S
5
Ÿƒ
m
'
X
.
Ú
n
2
.
1[11,
1
2
Ù
][16,
1
2.1
!
]
R
Ú
S
´
‚
,
S
M
R
´
Frobenius
V
,
-
N
:=
∗
M
.
K
±
e
Q
ã
¤
á
:
(1)
R
N
S
´
Frobenius
V
.
DOI:10.12677/pm.2021.115091769
n
Ø
ê
Æ
~
(2)
M
⊗
R
−
∼
=
Hom
R
(
N,
−
):
R
M→
S
M
,
N
⊗
S
−
∼
=
Hom
S
(
M,
−
):
S
M→
R
M
.
(3)Hom
R
op
(
M,
−
)
∼
=
−⊗
R
N
:
M
R
→M
S
,Hom
S
op
(
N,
−
)
∼
=
−⊗
S
M
:
M
S
→M
R
.
(4)
e
X
´
Ý
(
S
,
²
"
)
R
-
,
K
M
⊗
R
X
´
Ý
(
S
,
²
"
)
S
-
.
(5)
e
Y
´
Ý
(
S
,
²
"
)
S
-
,
K
Hom
S
(
M,Y
)
´
Ý
(
S
,
²
"
)
R
-
.
(6)
é
?
¿
i
≥
0,
9
?
¿
S
-
X
Ú
R
-
Y
,
Ext
i
S
(
X,M
⊗
R
Y
)
∼
=
Ext
i
R
(Hom
S
(
M,X
)
,Y
)
,
Ext
i
R
(
Y,N
⊗
S
X
)
∼
=
Ext
i
S
(Hom
R
(
N,Y
)
,X
)
(7)
é
?
¿
i
≥
0,
?
¿
R
op
-
X
Ú
R
-
Y
,
Tor
R
i
(Hom
R
op
(
M,X
)
⊗
S
M,Y
)
∼
=
Tor
S
i
(Hom
R
op
(
M,X
)
,M
⊗
R
Y
)
Ú
n
2
.
2[15,
½
n
2.2]
S
M
R
´
Frobenius
V
,
-
N
:=
∗
M
…
F
:=
M
⊗
R
−
:
R
M→
S
M
´
Frobenius
¼
f
.
K
±
e
Q
ã
¤
á
:
(1)
F
´
§
¢
¼
f
;
(2)
R
N
´)
¤
f
;
(3)
M
R
´)
¤
f
;
(4)
M
R
´
§
¢
;
(5)
é
?
¿
R
-
X
,
N
ϕ
X
:
X
→
Hom
S
(
M,M
⊗
R
X
)(
ϕ
X
(
x
)(
m
)=
m
⊗
R
x
))
´
ü
Ó
,
Ù
¥
x
∈
X
…
m
∈
M
;
(6)
é
?
¿
R
op
-
Y
,
N
ψ
Y
:Hom
R
op
(
M,Y
)
⊗
S
M
→
Y
(
ψ
Y
(
f
⊗
S
m
)=
f
(
m
))
´
÷
Ó
,
Ù
¥
f
∈
Hom
R
op
(
M,Y
)
…
m
∈
M
;
(7)
é
?
¿
R
-
P
∈P
(
R
),
N
φ
P
:
N
⊗
S
Hom
R
(
M,P
)
→
P
(
φ
P
(
n
⊗
S
f
)=
f
(
n
))
´
÷
Ó
,
Ù
¥
f
∈
Hom
R
(
M,P
)
…
n
∈
N
;
(8)
R
op
-
E
∈I
(
R
op
),
N
θ
E
:
E
→
Hom
S
op
(
N,E
⊗
R
N
)(
θ
E
(
x
)(
n
)=
x
⊗
R
n
)
´
ü
Ó
,
Ù
¥
x
∈
E
…
n
∈
N
.
Ú
n
2
.
3[15,
í
Ø
2.3]
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
.
(1)
?
¿
Ý
R
-
P
´
Hom
S
(
M,M
⊗
R
P
)
†
Ú
‘
;
(2)
?
¿
R
op
-
E
´
Hom
R
op
(
M,E
)
⊗
S
M
†
Ú
‘
.
Ú
n
2
.
4
S
M
R
´
Frobenius
V
.
(1)
e
X
´
Y
-Gorenstein
S
R
op
-
,
K
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
.
(2)
e
Y
´
Y
-Gorenstein
S
S
op
-
,
K
Y
⊗
S
M
´
Y
-Gorenstein
S
R
op
-
.
y
²
(1)
Ï
•
X
´
Y
-Gorenstein
S
R
op
-
,
¤
±
•
3
S
R
op
-
Ü
E
=
···→
E
1
→
E
0
→
E
−
1
→
E
−
2
→···
,
DOI:10.12677/pm.2021.115091770
n
Ø
ê
Æ
~
¦
X
∼
=
Ker(
E
−
1
→
E
−
2
),
¿
…
é
?
¿
I
∈Y
,Hom
R
op
(
I,
E
)
Ü
.
Ï
E
i
´
S
R
op
-
.
¤
±
Hom
R
op
(
M,E
i
)
´
S
S
op
-
.
Ï
d
Hom
R
op
(
M,
E
)
´
S
S
op
-
Ü
,
…
Hom
R
op
(
M,X
)
∼
=
Ker
(Hom
R
op
(
M,E
−
1
)
→
Hom
R
op
(
M,E
−
2
)).
Q
´
S
S
op
-
,
K
Q
⊗
S
M
´
S
R
op
-
,
Hom
R
op
(
Q
⊗
S
M,
E
)
Ü
.
d
Ó
Hom
R
op
(
Q
⊗
S
M,
E
)
∼
=
Hom
S
op
(
Q,
Hom
R
op
(
M,
E
))
•
,Hom
S
op
(
Q,
Hom
R
op
(
M,
E
))
Ü
.
Ï
d
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
.
(2)
Y
´
Y
-Gorenstein
S
S
op
-
,
-
N
:=
∗
M
.
Ï
•
S
M
R
´
Frobenius
V
,
¤
±
R
N
S
´
Frobenius
V
.
Ï
d
d
(1)
•
,Hom
S
op
(
N,Y
)
´
Y
-Gorenstein
S
R
op
-
.
q
Ï
•
Hom
S
op
(
N,Y
)
∼
=
Y
⊗
S
M
,
¤
±
Y
⊗
S
M
´
Y
-Gorenstein
S
R
op
-
.
Ú
n
2
.
5[7,
Ú
n
2.8]
±
e
Q
ã
d
:
(1)
M
´
Y
-Gorenstein
S
R
op
-
;
(2)
é
?
¿
R
op
-
H
∈Y
,Ext
≥
1
R
(
H,M
)=0,
…
•
3
S
R
op
-
Ü
µ
···→
E
2
→
E
1
→
E
0
→
X
→
0
,
¦
é
?
¿
H
∈Y
,Hom
R
(
H,
−
)
Ü
;
(3)
•
3
á
Ü
0
→
G
→
I
→
M
→
0,
Ù
¥
I
´
S
,
G
´
Y
-Gorenstein
S
.
Ú
n
2
.
6[7,
Ú
n
2.10]
Y
-Gorenstein
S
'
u
*
Ü
,
†
Ú
‘
Ú
÷
Ó
Ø
µ
4
.
½
n
2
.
7
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
.
(1)
R
op
-
X
´
Y
-Gorenstein
S
…
=
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
.
(2)
R
-
Y
´
Y
-Gorenstein
S
…
=
M
⊗
R
Y
´
Y
-Gorenstein
S
S
-
.
y
²
(1)
⇒
)
d
·
K
2.4
Œ
.
⇐
)
X
´
R
op
-
¦
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
.
e
y
X
´
Y
-Gorenstein
S
R
op
-
.
E
´
S
R
op
-
,
¤
±
Hom
R
op
(
M,E
)
´
S
S
op
-
.
Ï
•
Hom
R
op
(
M,X
)
´
Y
-Gorenstein
S
S
op
-
,
¤
±
é
?
¿
i
≥
1,Ext
i
S
op
(Hom
R
op
(
M,E
)
,
Hom
R
op
(
M,X
))=0.
d
Ó
Ext
i
S
op
(Hom
R
op
(
M,E
)
,
Hom
R
op
(
M,X
))
∼
=
Ext
i
R
op
(Hom
R
op
(
M,E
)
⊗
S
M,X
)
•
,Ext
i
R
op
(Hom
R
op
(
M,E
)
⊗
S
M,X
)=0.
qd
Ú
n
2.3(2),
S
R
op
-
E
´
Hom
R
op
(
M,E
)
⊗
S
M
†
Ú
‘
,
¤
±
é
?
¿
i
≥
1,Ext
i
R
op
(
E,X
)=0.
e
¡
E
X
Hom
R
op
(
I
(
R
op
)
,
−
)
Ü
†
I
(
R
op
)-
©
)
.
d
·
K
2.4
•
,Hom
R
op
(
M,E
)
⊗
S
M
´
Y
-Gorenstein
S
R
op
-
,
¤
±
d
Ú
n
2.5
•
,
k
Ü
0
→
L
1
→
I
0
→
Hom
R
op
(
M,E
)
⊗
S
M
→
0,
Ù
¥
I
0
´
S
R
op
-
,
L
1
´
Y
-Gorenstein
S
R
op
-
.
Ï
•
M
R
´)
¤
f
,
¤
±
•
3
R
op
-
Ü
0
→
K
→
Hom
R
op
(
M,E
)
⊗
S
M
→
X
→
0.
•
Ä
e
í
Ñ
ã
DOI:10.12677/pm.2021.115091771
n
Ø
ê
Æ
~
0
0
L
1
L
1
0
/
/
H
1
/
/
I
0
/
/
X
/
/
0
0
/
/
K
/
/
Hom
R
op
(
M,E
)
⊗
S
M
/
/
X
/
/
0
00
^
Hom
R
op
(
M,
−
)
Š
^u
,
0
0
Hom
R
op
(
M,L
1
)
Hom
R
op
(
M,L
1
)
0
/
/
Hom
R
op
(
M,H
1
)
/
/
Hom
R
op
(
M,I
0
)
/
/
Hom
R
op
(
M,X
)
/
/
0
0
/
/
Hom
R
op
(
M,K
)
/
/
Hom
R
op
(
M,
Hom
R
op
(
M,E
))
⊗
S
M
/
/
Hom
R
op
(
M,X
)
/
/
0
00
N
´
Hom
R
op
(
M,
Hom
R
op
(
M,E
))
⊗
S
M
→
Hom
R
op
(
M,X
)
´
Œ
÷
Ó
,
¤
±
d
Ú
n
2.6
•
Hom
R
op
(
M,K
)
´
Y
-Gorenstein
S
S
op
-
…
Hom
R
op
(
M,H
1
)
´
Y
-Gorenstein
S
S
op
-
.
k
Ü
0
→
H
1
→
I
0
→
X
→
0,
Ù
¥
I
0
´
S
R
op
-
,Hom
R
op
(
M,H
1
)
´
Y
-Gorenstein
S
S
op
-
.
Ï
d
é
?
¿
i
≥
1,Ext
i
R
(
E,H
1
)=0.
-
E
þ
ã
L
§
,
R
op
-
Ü
···→
I
2
→
I
1
→
I
0
→
X
→
0,
¦
é
?
¿
R
op
-
E
∈Y
,Hom
R
op
(
E,
−
)
Ü
.
¤
±
X
´
Y
-Gorenstein
S
R
op
-
.
(2)
Ï
•
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
,
¤
±
R
N
S
´
Frobenius
V
…
R
N
´)
¤
f
.
Ï
d
,
R
-
Y
´
Y
-Gorenstein
S
…
=
Hom
R
(
N,Y
)
´
Y
-Gorenstein
S
S
-
.
qd
Ó
ª
Hom
R
(
N,Y
)
∼
=
M
⊗
R
Y
•
,
M
⊗
R
Y
´
Y
-Gorenstein
S
S
-
.
4.
X
-Gorenstein
Ý
Ú
Frobenius
V
!
Ì
‡
ï
Ä
Frobenius
V
Ú
X
-Gorenstein
Ý
5
Ÿƒ
m
'
X
.
·
K
3
.
1
S
M
R
´
Frobenius
V
.
(1)
e
X
´
X
-Gorenstein
Ý
R
-
,
K
M
⊗
R
X
´
X
-Gorenstein
Ý
S
-
.
DOI:10.12677/pm.2021.115091772
n
Ø
ê
Æ
~
(2)
e
Y
´
X
-Gorenstein
Ý
S
-
,
K
Hom
S
(
M,Y
)
´
X
-Gorenstein
Ý
R
-
.
y
²
(1)
Ï
•
X
´
X
-Gorenstein
Ý
R
-
,
¤
±
•
3
Ý
m
R
-
Ü
P
:
···→
P
1
→
P
0
→
P
−
1
→
P
−
2
→···
,
¦
X
∼
=
Ker(
P
−
1
→
P
−
2
),
…
é
?
¿
Q
∈X
,Hom
R
(
P
,Q
)
Ü
.
Ï
•
P
i
´
Ý
R
-
,
¤
±
M
⊗
R
P
i
´
Ý
S
-
.
M
⊗
R
P
´
Ý
S
-
Ü
,
…
M
⊗
R
X
∼
=
Ker(
M
⊗
R
P
−
1
→
M
⊗
R
P
−
2
).
A
´
Ý
S
-
,
K
Hom
S
(
M,A
)
´
Ý
R
-
.
Hom
R
(
P
,
Hom
S
(
M,A
))
Ü
.
d
Ó
Hom
R
(
P
,
Hom
S
(
M,A
))
∼
=
Hom
S
(
M
⊗
R
P
,A
)
•
,Hom
S
(
M
⊗
R
P
,A
)
Ü
.
Ï
d
M
⊗
R
X
´
X
-Gorenstein
Ý
S
-
.
(2)
Ï
•
Y
´
X
-Gorenstein
Ý
S
-
,
-
N
:=
∗
M
,
Ï
•
S
M
R
´
Frobenius
V
,
¤
±
R
N
S
´
Frobenius
V
.
Ï
d
d
(1)
•
,
N
⊗
S
Y
´
X
-Gorenstein
Ý
R
-
,
qd
Ó
ª
N
⊗
S
Y
∼
=
Hom
S
(
M,Y
)
•
,
¤
±
Hom
S
(
M,Y
)
´
X
-Gorenstein
Ý
R
-
.
Ú
n
3
.
2[6,
·
K
2.2]
X
´
R
-
,
K
±
e
Q
ã
d
:
(1)
X
´
X
-Gorenstein
Ý
;
(2)
é
?
¿
F
∈X
,
i>
0,Ext
i
R
(
X,F
)=0,
…
•
3
Ý
R
-
Ü
0
→
X
→
P
0
→
P
1
→
P
2
→···
,
¦
é
?
¿
F
∈X
,Hom
R
(
−
,F
)
Ü
;
(3)
•
3
á
Ü
0
→
X
→
P
→
L
→
0,
Ù
¥
P
´
Ý
,
L
´
X
-Gorenstein
Ý
.
Ú
n
3
.
3[6,
½
n
2.3(1)]
á
Ü
0
→
A
→
B
→
C
→
0,
Ù
¥
C
´
X
-Gorenstein
Ý
.
K
A
´
X
-Gorenstein
Ý
…
=
B
´
X
-Gorenstein
Ý
.
½
n
3
.
4
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
.
(1)
R
-
X
´
X
-Gorenstein
Ý
…
=
M
⊗
R
X
´
X
-Gorenstein
Ý
S
-
.
(2)
R
op
-
Y
´
X
-Gorenstein
Ý
…
=
Hom
R
op
(
M,Y
)
´
X
-Gorenstein
Ý
S
op
-
.
y
²
(1)
⇒
)
d
·
K
3.1
Œ
.
⇐
)
X
´
R
-
¦
M
⊗
R
X
´
X
-Gorenstein
Ý
S
-
.
e
y
X
´
X
-Gorenstein
Ý
R
-
.
P
´
Ý
R
-
,
¤
±
M
⊗
R
P
´
Ý
S
-
.
Ï
•
M
⊗
R
X
´
X
-Gorenstein
Ý
S
-
,
¤
±
é
?
¿
i
≥
1,Ext
i
S
(
M
⊗
R
X,M
⊗
R
P
)=0.
d
Ó
Ext
i
S
(
M
⊗
R
X,M
⊗
R
P
)
∼
=
Ext
i
R
(
X,
Hom
S
(
M,M
⊗
R
P
))
•
,Ext
i
R
(
X,
Hom
S
(
M,M
⊗
R
P
))=0.
qd
Ú
n
2.3(1),
Ý
R
-
P
´
Hom
S
(
M,M
⊗
R
P
)
†
Ú
‘
,
¤
±
é
?
¿
i
≥
1,Ext
i
R
(
X,P
)=0.
e
¡
E
X
Hom
R
(
−
,
P
(
R
))
Ü
m
P
(
R
)-
©
)
.
d
·
K
3.1
•
,Hom
S
(
M,M
⊗
R
X
)
´
X
-Gorenstein
Ý
R
-
,
¤
±
d
Ú
n
3.2
•
,
•
3
á
Ü
0
→
Hom
S
(
M,M
⊗
R
X
)
→
P
0
→
L
1
→
0,
Ù
¥
P
0
´
Ý
R
-
,
L
1
´
X
-Gorenstein
Ý
R
-
.
Ï
•
M
R
´)
¤
f
,
¤
±
•
3
R
-
á
Ü
0
→
X
→
Hom
S
(
M,M
⊗
R
X
)
→
K
→
0.
•
Ä
í
Ñ
ã
DOI:10.12677/pm.2021.115091773
n
Ø
ê
Æ
~
0
0
0
/
/
/
/
X
/
/
Hom
S
(
M,M
⊗
R
X
)
/
/
/
/
K
/
/
0
0
/
/
X
/
/
P
0
/
/
H
1
/
/
0
L
1
L
1
00
^
M
⊗
R
−
Š
^u
í
Ñ
ã
0
0
0
/
/
/
/
M
⊗
R
X
/
/
M
⊗
R
Hom
S
(
M,M
⊗
R
X
)
/
/
/
/
M
⊗
R
K
/
/
0
0
/
/
M
⊗
R
X
/
/
M
⊗
R
P
0
/
/
M
⊗
R
H
1
/
/
0
M
⊗
R
L
1
M
⊗
R
L
1
00
N
´
M
⊗
R
X
→
M
⊗
R
Hom
S
(
M,M
⊗
R
X
)
´
Œ
ü
Ó
.
d
Ú
n
3.3
•
M
⊗
R
K
´
X
-Gorenstein
Ý
S
-
,
…
M
⊗
R
H
1
´
X
-Gorenstein
Ý
S
-
.
Ï
d
k
Ü
0
→
X
→
P
0
→
H
1
→
0,
Ù
¥
P
0
´
Ý
R
-
,
M
⊗
R
H
1
´
X
-Gorenstein
Ý
S
-
.
Ï
d
é
?
¿
i
≥
1,Ext
i
R
(
H
1
,P
)=0.
-
E
þ
ã
L
§
,
R
-
Ü
0
→
X
→
P
0
→
P
1
→
P
2
→···
Ù
¥
P
i
∈P
(
R
)
¦
é
?
¿
Q
∈X
,Hom
R
(
−
,Q
)
Ü
.
¤
±
X
´
X
-Gorenstein
Ý
R
-
.
(2)
Ï
•
S
M
R
´
Frobenius
V
…
M
R
´)
¤
f
,
-
N
:=
∗
M
.
¤
±
R
N
S
´
Frobenius
V
…
R
N
´)
¤
f
.
Ï
d
R
op
-
Y
´
X
-Gorenstein
Ý
…
=
Y
⊗
R
N
´
X
-Gorenstein
Ý
S
op
-
.
qd
Ó
ª
Y
⊗
R
N
∼
=
Hom
R
op
(
M,Y
)
•
,Hom
R
op
(
M,Y
)
´
X
-Gorenstein
Ý
S
op
-
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(11561061).
DOI:10.12677/pm.2021.115091774
n
Ø
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•
©
z
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