On the Restricted Projective Dimension of Complexes
We study the restricted projective dimension of complexes and give some new characterizations of the restricted projective dimension. In particular, it is shown that the restricted projective dimension can be computed in terms of the so-called restricted projective resolutions. As applications, we g...
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| author | Li, Liang Wu, Dejun Лі, Ліанг Ву, Дюн |
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| description | We study the restricted projective dimension of complexes and give some new characterizations of the restricted projective dimension. In particular, it is shown that the restricted projective dimension can be computed in terms of the so-called restricted projective resolutions. As applications, we get some results on the behavior of the restricted projective dimension under the change of rings. |
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| fulltext |
UDC 517.91
Li Liang (Lanzhou Jiaotong Univ., China),
Dejun Wu (Lanzhou Univ. Technology, China)
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES*
ПРО ОБМЕЖЕНУ ПРОЕКТИВНУ РОЗМIРНIСТЬ КОМПЛЕКСIВ
We study the restricted projective dimension of complexes. We give some new characterizations of the restricted projective
dimension. In particular, we show that the restricted projective dimension can be computed in terms of the so-called
restricted projective resolutions. As applications, we get some results on the behavior of the restricted projective dimension
under change of rings.
Вивчається обмежена проективна розмiрнiсть комплексiв. Наведено деякi новi властивостi обмеженої проективної
розмiрностi. Зокрема, показано, що обмежену проективну розмiрнiсть можна обчислити через так званi обмеженi
проективнi резольвенти. Як застосування отримано деякi результати про поведiнку обмеженої проективної розмiр-
ностi при змiнi кiлець.
Introduction. As is well known, the classical homological dimensions — projective, flat and injective
dimensions are defined in terms of resolutions, but they can also be computed in terms of vanishing
of appropriate derived functors. For example, the flat dimension of R-module M can be computed
as follows:
fdR(M) = sup
{
i ∈ N0 | TorRi (T,M) 6= 0 for some module T
}
.
The restricted flat dimension was defined solely in terms of the vanishing of the derived functor Tor
over some classes of test modules that are restricted to assure automatic finiteness over commutative
Noetherian rings of finite Krull dimension (see [3]). Accurately, the restricted flat dimension, denoted
RfdRM, of an R-module M is defined as
RfdR(M) = sup
{
i ∈ N0 | TorRi (T,M) 6= 0 for some module T with fdR(T ) <∞
}
.
Christensen, Foxby and Frankild [3] further studied the restricted flat dimension of complexes,
and they gave a number of interesting properties. For example, they showed the restricted flat dimen-
sion is finite for any homologically bounded complex over commutative Noetherian rings of finite
Krull dimension, and it is a refinement of both flat and Gorenstein flat dimensions. Sharif and Yassemi
[4] studied the behavior of the restricted flat dimension under change of rings, and generalized some
classical results.
Let X be a homologically bounded below complex of R-modules. The restricted projective di-
mension, denoted RpdRX, of X was defined by Christensen, Foxby and Frankild in [3]. They
showed that this dimension is also finite for any homologically bounded complex over commuta-
tive Noetherian rings of finite Krull dimension. In this paper, we give some new characterizations
of the restricted projective dimension of complexes as follows, which show that the restricted pro-
jective dimension can be computed in terms of the so-called restricted projective resolutions (see
Theorem 2.1).
* This work was partially supported by NSF of China (Grant No. 11226059) and the Young Scholars Science Foundation
of Lanzhou Jiaotong University (Grant No. 2012020).
c© LI LIANG, DEJUN WU, 2013
936 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES 937
Theorem A. Let X be a homologically bounded below complex and n ∈ Z. Consider the
following conditions:
(1) RpdRX ≤ n.
(2) X is equivalent to a bounded complex P of restricted projective R-modules with sup{i ∈
∈ Z | Pi 6= 0} ≤ n; and P can be chosen such that Pl = 0 for l < inf X.
(3) Hi(RHomR(X,T )) = 0 for any i < −n and any R-module T with idR(T ) <∞.
(4) supX ≤ n and Cn(P ) is a restricted projective R-module whenever P is a bounded below
complex of restricted projective R-modules which is equivalent to X.
(5) −inf(RHom(X,U)) + inf U ≤ n for any non-exact complex U with idR U <∞.
Then we have (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇐ (5). If, furthermore, X is homologically degree-wise finite,
then all the above statements are equivalent.
As applications of the above theorem, we get the following result on the behavior of the restricted
projective dimension under change of rings (see Propositions 2.4 and 2.5).
Theorem B. Let ϕ : R → S be a homomorphism of rings and X a homologically bounded
below and degree-wise finite complex of S-modules. Then the following statements hold:
(1) If Y is a homologically bounded below complex of R-modules with fdR Y < ∞, then we
have
RpdR(X ⊗L
R Y ) ≤ RpdS X + RpdR Y + RpdR S
and
RpdS(X ⊗L
R Y ) ≤ RpdS X + RfdR S + supY + dimS.
(2) If Y is a homologically bounded below complex of S-modules with fdS Y < ∞, then we
have
RpdR(X ⊗L
S Y ) ≤ RpdS X + RpdR Y.
1. Preliminaries. We begin with some notations and terminology for use throughout this paper,
which can be found in [2].
1.1. A complex . . . −→ X1
δX1−→ X0
δX0−→ X−1 −→ . . . of R-modules will be denoted by
(X, δX) or simply X. We frequently (and without warning) identify R-modules with complexes con-
centrated in degree 0. A complex X is bounded above (resp., bounded below, bounded) if Xn = 0
for n � 0 (resp., n � 0, |n| � 0). The nth boundary (resp., cycle, homology) of X is defined
as Im δXn+1 (resp., Ker δXn , Ker δXn / Im δXn+1) and it is denoted by Bn(X) (resp., Zn(X), Hn(X)).
A complex X is homologically bounded above (resp., homologically bounded below, homologically
bounded) if the homology complex H(X) is bounded above (resp., bounded below, bounded). We
use the notation Cn(X) for the cokernel of the differential δXn+1. The soft truncations of X at n are
the complexes
X⊂n ≡ 0 −→ Cn(X)
δXn−→ Xn−1
δXn−1−→ Xn−2 −→ . . .
and
X⊃n ≡ . . . −→ Xn+2
δXn+2−→ Xn+1
δXn+1−→ Zn(X) −→ 0.
The hard truncations of X at n are the complexes
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
938 LI LIANG, DEJUN WU
X≤n ≡ 0 −→ Xn −→ Xn−1 −→ Xn−2 −→ . . .
and
X≥n ≡ . . . −→ Xn+2 −→ Xn+1 −→ Xn −→ 0.
The supremum and infimum of X are given by the following formulas:
sup(X) = sup
{
i ∈ Z | Hi(X) 6= 0
}
and inf(X) = inf
{
i ∈ Z | Hi(X) 6= 0
}
.
For any m ∈ Z, ΣmX denotes the complex with the degree-n term (ΣmX)n = Xn−m and whose
boundary operators are (−1)mδXn−m.
1.2. If X and Y are both complexes, then by a morphism α : X −→ Y we mean a sequence
αn : Xn −→ Yn such that αn−1δXn = δYn αn for each n ∈ Z. A quasiisomorphism, indicated by
the symbol “'” next to their arrows, is a morphism of complexes that induces an isomorphism in
homology. The mapping cone Cone(α) of α is defined as Cone(α)n = Yn⊕Xn−1 with nth boundary
operator δCone(α)
n =
(
δNn αn−1
0− δMn−1
)
. It is well known that a morphism α is a quasiisomorphism if and
only if its mapping cone Cone(α) is exact. Two complexes X and Y are equivalent, we write X ' Y,
if there is the third complex Z and two quasiisomorphisms: X
'−→ Z
'←− Y .
1.3. Throughout this paper, all rings are assumed to be commutative Noetherian. The category of
complexes of R-modules is denoted C(R), and we use subscripts =, <, and � to denote boundedness
conditions, and use subscripts (=), (<), and (�) to denote homologically boundedness conditions.
For example, C=(R) and C(=)(R) are the full subcategories of C(R) of bounded below and homo-
logically bounded below complexes, respectively. Superscript “(f)” signifies that the homology is
degree-wise finitely generated. Thus, C(f)(=) denotes the full subcategory of C(R) of homologically
bounded below complexes with finitely generated homology modules.
1.4. A projective (resp., flat) resolution of X ∈ C(=)(R) is a bounded below complex P of
projective (resp., flat) R-modules such that P ' X, and an injective resolution of a complex Y ∈
C(<)(R) is a bounded above complex I of injective R-modules such that Y ' I. The projective, flat
and injective dimensions are defined as follows:
pdRX = inf
{
sup{l ∈ Z | Pl 6= 0} | P is a projective resolution of X
}
,
fdRX = inf
{
sup{l ∈ Z | Fl 6= 0} | F is a flat resolution of X
}
,
and
idR Y = inf
{
− inf{l ∈ Z | Il 6= 0} | I is an injective resolution of X
}
.
We use P(R) (resp., F(R), I(R)) to denote the full subcategory of C(�)(R) of complexes of finite
projective (resp., flat, injective) dimension, and use P0(R) (resp., F0(R), I0(R)) to denote the full
subcategory of R-modules of finite projective (resp., flat, injective) dimension.
We use the standard notations RHomR(−,−) and − ⊗L
R − for the derived Hom and derived
tensor product of complexes; they are computed by way of the resolutions defined above.
The next two results can be found in [1] ((4.1)) and [2] ((A.4.21) and (A.4.24)).
Lemma 1.1. Let ϕ : R −→ S be a homomorphism of rings, and let Z ∈ C(S) and X ∈ C(R).
Then
idS(RHomR(Z, Y )) ≤ fdS(Y ) + idR(Y ).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES 939
Lemma 1.2. Assume that ϕ : R −→ S is a homomorphism of rings. Then the following state-
ments hold:
(1) Let Z ∈ C(=)(S), Y ∈ C(S) and X ∈ C(=)(R). Then
RHomR(Z ⊗L
R Y,X) = RHomS(Z,RHomR(Y,X)).
(2) Let Z ∈ C(f)(=)(S), Y ∈ C(�)(S) and X ∈ C(<)(R) with idRX <∞. Then
Z ⊗L
R RHomR(Y,X) = RHomR(RHomS(Z, Y ), X).
1.5. Following from [3], the restricted flat dimension, RfdRX, of X ∈ C(=)(R) is defined as
RfdRX = sup
{
sup(T ⊗L
R X) | T ∈ F0(R)
}
.
There are inequalities supX ≤ RfdRX ≤ supX+dimR. In particular, RfdRX = −∞ if and only
if X ' 0, and if dimR <∞ then RfdRX <∞ if and only if X ∈ C(�)(R).
2. Restricted projective modules and restricted projective dimension. We say that an R-
module P is restricted projective if ExtiR(P, T ) = 0 for any R-module T of finite injective dimension
and any i > 0.
Lemma 2.1. If P ∈ C=(R) is an exact complex of restricted projective R-modules and I ∈
∈ C�(R) is a complex of R-modules in I0(R), then HomR(P, I) is a exact complex.
Proof. We may assume that I is non-zero, and let s = sup{i ∈ Z | Ii 6= 0}. We proceed by
induction on s. Without loss of generality, we assume that Pl = 0 and Il = 0 for l < 0.
If s = 0 then I ∈ I0(R). Note that P ∈ C=(R) is exact and ExtiR(Pl, I) = 0 for all i > 0 and
l ∈ Z. One can check, by “Dimension Shift”, that HomR(P, I) is exact.
Let s > 0 and assume that Hom(P, I) is exact for any complex I ∈ C�(R) of R-modules in
I0(R) with sup{i ∈ Z | Ii 6= 0} ≤ s− 1. Consider the degree-wise split exact sequence
0 −→ I≤s−1 −→ I −→ ΣsIs −→ 0
of complexes, then it stays exact after application of HomR(P,−). The complex HomR(P, Is) and
HomR(P, I≤s−1) are exact by the induction base and hypothesis, respectively. Thus HomR(P, I) is
exact.
Lemma 2.2. If X ' P and U ' I, where P ∈ C=(R) is a complex of restricted projective
R-modules and I ∈ C�(R) is a complex of R-modules in I0(R), then RHomR(X,U) is represented
by HomR(P, I).
Proof. Take a projective resolution X
'←− Q ∈ C=(R), then RHomR(X,U) is represented
by HomR(Q,U). Since Q ' P, there exists a quasiisomorphism α : Q −→ P by [1, (1.4.P)], and
hence we have a morphism
HomR(α, I) : HomR(P, I) −→ HomR(Q, I).
Since Cone(α) ∈ C=(R) is an exact complex of restricted projective R-modules, we have
Cone(HomR(α, I)) ∼= Σ1 HomR(Cone(α), I) is exact by Lemma 2.1, and hence HomR(α, I) is
a quasiisomorphism. Thus HomR(Q,U) ' HomR(P, I). This implies that RHomR(X,U) is repre-
sented by HomR(P, I).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
940 LI LIANG, DEJUN WU
Lemma 2.3. Let P ∈ C=(R) be a complex of restricted projective R-modules and T ∈ I0(R),
and let X be a complex of R-modules such that supX ≤ n <∞ and X ' P. Then, for any i > 0,
we have
ExtiR(Cn(P ), T ) = H−(i+n)(RHomR(X,T )).
Proof. Since supP = supX ≤ n, we have P≥n ' ΣnCn(P ), and hence Cn(P ) ' Σ−nP≥n.
Thus by Lemma 2.2, for each i > 0, we have
ExtiR(Cn(P ), T ) = H−i(RHomR(Cn(P ), T )) =
= H−i(HomR(Σ−nP≥n, T )) = H−i(Σ
n HomR(P≥n, T )) =
= H−(i+n)(HomR(P≥n, T )) = H−(i+n)(HomR(P, T )) =
= H−(i+n)(RHomR(X,T )).
Following from [3], the restricted projective dimension, RpdRX, of X ∈ C(=)(R) is defined as
RpdRX = sup
{
−inf(RHomR(X,T )) | T ∈ I0(R)
}
.
It can be checked easily that, for X ∈ C(=)(R), RfdRX ≤ RpdRX, and there are inequalities
supX ≤ RpdRX ≤ supX + dimR.
In particular, RpdRX = −∞ if and only if X ' 0, and if dimR < ∞ then RpdRX < ∞ if and
only if X ∈ C(�)(R).
The next theorem gives some new characterizations of the restricted projective dimension of
complexes.
Theorem 2.1. Let X ∈ C(=)(R) and n ∈ Z. Consider the following conditions:
(1) RpdRX ≤ n.
(2) X is equivalent to a bounded complex P of restricted projective R-modules with sup{i ∈
∈ Z | Pi 6= 0} ≤ n; and P can be chosen such that Pl = 0 for l < inf X.
(3) Hi(RHomR(X,T )) = 0 for any i < −n and any T ∈ I0(R).
(4) supX ≤ n and Cn(P ) is a restricted projective R-module whenever P is a bounded below
complex of restricted projective R-modules which is equivalent to X.
(5) −inf(RHom(X,U)) + inf U ≤ n for any nonexact complex U ∈ I(R).
Then we have (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇐ (5). If, furthermore, X ∈ C(f)(=)(R), then all the above
statements are equivalent.
Proof. (1) ⇒ (4). Obviously, supX ≤ RpdRX ≤ n. Let P ∈ C=(R) be a complex of
restricted projective R-modules such that P ' X, and let T ∈ I0(R). Then, by Lemma 2.3,
ExtiR(Cn(P ), T ) = H−(i+n)(RHomR(X,T )) = 0 for any i > 0 since RpdRX ≤ n, and so
Cn(P ) is an restricted projective R-module.
(4) ⇒ (2). Take a projective resolution X
'←− P ∈ C=(R) of X with Pl = 0 for l < inf X.
Since supP = supX ≤ n, we have X ' P ' P⊂n. Obviously, P⊂n is a bounded complex of
restricted projective R-modules.
(2) ⇒ (3). Let T ∈ I0(R). By Lemma 2.2, we have Hi(RHomR(X,T )) = Hi(HomR(P, T )).
For i < −n, HomR(P, T )i =
∏
t∈Z HomR(Pt, Tt+i) = 0 since Pt = 0 for t > n, and so
Hi(RHomR(X,T )) = 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES 941
(3) ⇒ (1) and (5)⇒ (3) are trivial.
Finally, we let X ∈ C(f)(=)(R), and let E be a faithfully injective R-module. Then, for any non-
exact complex U ∈ I(R), we have
− inf(RHomR(X,U)) = sup(RHomR(RHomR(X,U), E)) =
= sup(X ⊗L
R RHomR(U,E)) ≤ RfdRX + sup(RHomR(U,E)) ≤
≤ RpdRX − inf U,
where the second equality holds by Lemma 1.2(2), the third inequality by [3] (Theorem 2.4(a)) and
the last by [3] (Lemma 5.6). Thus the implication (1)⇒ (5) holds.
Recall, from [5], that an R-module M is strongly torsion free if TorR1 (T,M) = 0 for any
T ∈ F0(R). One can check easily that M is strongly torsion free if and only if TorRi (T,M) = 0 for
any T ∈ F0(R) and any i > 0. Using a similar method as proved in Theorem 2.1, we get the next
result.
Proposition 2.1. Let X ∈ C(=)(R) and n ∈ Z. Then the following statements are equivalent:
(1) RfdRX ≤ n.
(2) X is equivalent to a bounded complex F of strongly torsion free R-modules with sup{i ∈
∈ Z | Fi 6= 0} ≤ n; and F can be chosen such that Fl = 0 for l < inf X.
(3) Hi(T ⊗L
R X) = 0 for any i > n and any T ∈ F0(R).
(4) supX ≤ n and Cn(F ) is a strongly torsion free R-module whenever F is a bounded below
complex of strongly torsion free R-modules which is equivalent to X.
Let X ∈ C(=)(R). We say that P is a restricted projective resolution of X if P is a bounded
below complex of restricted projective R-modules such that P ' X. A restricted projective resolution
of an R-module M is a sequence
. . . −→ Pl −→ . . . −→ P1 −→ P0 −→ 0
of restricted projective R-modules which is exact at Pi for i > 0 and satisfies
P0/ Im(P1 −→ P0) ∼= M.
That is, the sequence
. . . −→ Pl −→ . . . −→ P1 −→ P0 −→ M −→ 0
is exact.
The next two corollaries are immediate by Theorem 2.1.
Corollary 2.1. If X ∈ C(=)(R), then
RpdRX = inf
{
sup{l ∈ Z | Pl 6= 0} | P is a restricted projective resolution of X
}
.
Corollary 2.2. If X ∈ C(f)(=)(R), then
RpdRX = sup
{
inf U − inf(RHomR(X,U)) | U ∈ I(R) ∧ U 6' 0
}
.
In particular, − inf(RHomR(X,U)) ≤ RpdRX − inf U for any X ∈ C(f)(=)(R) and any U ∈ I(R).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
942 LI LIANG, DEJUN WU
The next lemma can be proved easily.
Lemma 2.4. Let 0 −→ M ′ −→ M −→ M ′′ −→ 0 be an exact sequence of R-modules.
Then the following statements hold:
(1) If M ′′ is restricted projective, then M is restricted projective if and only if M ′ is so.
(2) If the sequence splits, then M is restricted projective if and only if both M ′ and M ′′ are so.
Lemma 2.5. Let M be an R-module, and let P ∈ C=(R) be a complex of restricted projec-
tive R-modules such that P ' M. Then the soft truncated complex P⊃0 is a restricted projective
resolution of M.
Proof. Since P ' M, we have inf P = 0, and hence P⊃0 ' P ' M. Thus we have an exact
sequence
. . . −→ P2 −→ P1 −→ Z0(P ) −→ M −→ 0
of R-modules. In the following we show that Z0(P ) is restricted projective. Let i = inf{l ∈ Z | Pl 6=
6= 0}, then the sequence
0 −→ Z0(P ) −→ P0 −→ . . . −→ Pi+1 −→ Pi −→ 0
of R-modules is exact, and so Z0(P ) is restricted projective by Lemma 2.4.
Corollary 2.3. Let M 6= 0 be an R-module. Then M is restricted projective if and only if
RpdRM = 0.
Proof. Immediately by Corollary 2.1 and Lemma 2.5.
Corollary 2.4. Let M be an R-module and n ∈ N0. Then the following statements are equiva-
lent:
(1) RpdRM ≤ n.
(2) There is an exact sequence 0 −→ Pn −→ . . . −→ P1 −→ P0 −→ M −→ 0 of R-modules
with Pi restricted projective for each 0 ≤ i ≤ n.
(3) ExtiR(M,T ) = 0 for any i > n and any T ∈ I0(R).
(4) For any restricted projective resolution . . . −→ P1 −→ P0 −→ M −→ 0 of M, Kn =
= Ker(Pn−1 −→ Pn−2) is a restricted projective R-module, where K0 = M and K1 =
= Ker(P0 −→ M).
Proof. We notice that if the sequence . . . −→ P1 −→ P0 −→ M −→ 0 is exact, then M is
equivalent to the complex P = . . . −→ P1 −→ P0 −→ 0, and C0(P ) ∼= M, C1(P ) ∼=
∼= Ker(P0 −→ M) and Cl(P ) ∼= Zl−1(P ) = Ker(Pl−1 −→ Pl−2) for l ≥ 2. In view of Lemma 2.5,
the equivalence of the four conditions now follows from Theorem 2.1.
Similarly, by Proposition 2.1, we get the following result.
Corollary 2.5. Let M be an R-module and n ∈ N0. Then the following statements are equiva-
lent:
(1) RfdRM ≤ n.
(2) There is an exact sequence 0 −→ Fn −→ . . . −→ F1 −→ F0 −→ M −→ 0 of R-modules
with Fi strongly torsion free for each 0 ≤ i ≤ n.
(3) TorRi (T,M) = 0 for all i > n and all T ∈ F0(R).
(4) For strongly torsion free resolution . . . −→ Fl −→ . . . −→ F0 −→ M −→ 0 ofM, Kn =
= Ker(Fn−1 −→ Fn−2) is a strongly torsion free R-module, where K0 = M and K1 =
= Ker(F0 −→ M).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES 943
Recall that a finite R-module M belongs to the G-class G(R) if ExtiR(M,R) = 0 =
= ExtiR(HomR(M,R), R) for i > 0 and the biduality map
δM : M −→ HomR(HomR(M,R), R),
defined by δM (x)(ψ) = ψ(x) for ψ ∈ HomR(M,R) and x ∈ M, is an isomorphism. A complex
G is said to be a G-resolution of X ∈ C(f)(=)(R) if G is a bounded below complex of R-modules in
G(R) such that G ' X. The G-dimension, G-dimRX, of X is defined as
G-dimRX = inf
{
sup{l ∈ Z | Gl 6= 0} | G is a G-resolution of X
}
.
The next lemma shows that the restricted projective dimension is a refinement of the G-dimension.
Lemma 2.6. If X ∈ C(f)(=)(R), then RpdRX ≤ G-dimRX, and the equality hold if
G-dimRX <∞.
Proof. If G-dimRX =∞, then the inequality is trivial. If G-dimRX <∞, then, by [2] ((2.4.7)),
G-dimRX = sup{− inf(RHomR(X,T )) | T ∈ I0(R)} = RpdRX.
By [3] ((5.17)), RpdRX ≤ pdRX for any X ∈ C(=)(R), and if R is local, X ∈ C(f)(=)(R) and
pdRX < ∞, then RpdRX = pdRX. In the following we see that the condition “R is local” is
superfluous.
Proposition 2.2. If X ∈ C(=)(R), then RpdRX ≤ pdRX, and the equality hold if X ∈
∈ C(f)(=)(R) and pdRX <∞.
Proof. Note that G-dimRX ≤ pdRX for X ∈ C(f)(=)(R) and the equality holds if pdRX < ∞
(see [2] (2.3.10)), then we get the desired result by Lemma 2.6.
A complex P is said to be a Gorenstein projective resolution of X ∈ C(=)(R), if P is a bounded
below complex of Gorenstein projective R-modules such that P ' X. The Gorenstein projective
dimension, GpdRX, of X is defined as
GpdRX = inf
{
sup{l ∈ Z | Pl 6= 0} | P is a Gorenstein projective resolution of X
}
.
Proposition 2.3. If R is a Gorenstein local ring, then RpdRX = GpdRX for any X ∈
∈ C(=)(R).
Proof. We first prove RpdRX ≤ GpdRX. If GpdRX =∞ then the inequality is trivial. Now
we assume that GpdRX <∞, then we have
GpdRX = sup{−inf(RHomR(X,T )) | T ∈ F0(R)} =
= sup{−inf(RHomR(X,T )) | T ∈ I0(R)} = RpdRX,
where the first equality holds by [2] ((4.4.5)), and the second by [2] ((3.3.4)).
Next we show that GpdRX ≤ RpdRX. If RpdRX = ∞ then the inequality is trivial. Now
we assume that RpdRX < ∞, then X ∈ C(�)(R). Thus GpdRX < ∞ by [2] ((4.4.8)), and so
GpdRX = RpdRX as proved above.
Proposition 2.4. Let ϕ : R −→ S be a homomorphism of rings, X ∈ C(f)(=)(S) and Y ∈ F(R).
Then we have the following inequalities:
(1) RpdR(X ⊗L
R Y ) ≤ RpdS X + RpdR Y + RpdR S.
(2) RpdS(X ⊗L
R Y ) ≤ RpdS X + RfdR S + supY + dimS.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
944 LI LIANG, DEJUN WU
Proof. (1) Choose T ∈ I0(R) such that
RpdR(X ⊗L
R Y ) = − inf(RHomR(X ⊗L
R Y, T )) =
= − inf(RHomR((X ⊗L
S S)⊗L
R Y, T )) =
= − inf(RHomR(X ⊗L
S (S ⊗L
R Y ), T )) =
= − inf(RHomS(X,RHomR(S ⊗L
R Y, T ))) ≤
≤ RpdS X − inf(RHomR(S ⊗L
R Y, T )) =
= RpdS X − inf(RHomR(S,RHomR(Y, T ))) ≤
≤ RpdS X + RpdR S − inf(RHomR(Y, T )) ≤
≤ RpdS X + RpdR S + RpdR Y.
Where the fourth equality holds by Lemma 1.2(1). Since
idS
(
RHomR(S ⊗L
R Y, T )
)
≤ fdS(S ⊗L
R Y ) + idR T ≤ fdR Y + idR T <∞
by [1] ((4.1)), the fifth inequality follows from Corollary 2.2. The sixth equality comes from
Lemma 1.2(1), and the seventh inequality holds by Corollary 2.2 since idR(RHomR(Y, T )) ≤
≤ fdR Y + idR T <∞ by Lemma 1.1.
(2) Choose T ∈ I0(S) such that
RpdS(X ⊗L
R Y ) = − inf(RHomS(X ⊗L
R Y, T )) =
= − inf(RHomS((X ⊗L
S (S ⊗L
R Y ), T )) =
= − inf(RHomS(X,RHomS(S ⊗L
R Y, T ))) ≤
≤ RpdS X − inf(RHomS(S ⊗L
R Y, T )) ≤
≤ RpdS X + sup(Y ⊗L
R S) + idS T ≤
≤ RpdS X + RfdR S + supY + dimS,
where the third equality holds by Lemma 1.2(1), the fourth inequality by Corollary 2.2 since
idS(RHomS(S ⊗L
R Y, T )) ≤ fdS(S ⊗L
R Y ) + idS T ≤ fdR Y + idS T < ∞ by Lemma 1.1, the
fifth by [2] ((A.5.2)), and the last by [3] ((2.4(1))).
Proposition 2.5. Let ϕ : R −→ S be a homomorphism of rings, X ∈ C(f)(=)(S) and Y ∈ F(S).
Then
RpdR(X ⊗L
S Y ) ≤ RpdS X + RpdR Y.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON RESTRICTED PROJECTIVE DIMENSION OF COMPLEXES 945
Proof. Choose T ∈ I0(R) such that
RpdR(X ⊗L
S Y ) = − inf(RHomR(X ⊗L
S Y, T )) =
= − inf(RHomS(X,RHomR(Y, T ))) ≤
≤ RpdS X − inf(RHomR(Y, T )) ≤ RpdS X + RpdR Y,
where the second equality holds by Lemma 1.2(1), and the third inequality by Corollary 2.2 since
idS(RHomR(Y, T )) ≤ fdS Y + idR T <∞ by Lemma 1.1.
Corollary 2.6. Let ϕ : R −→ S be a homomorphism of rings and X ∈ C(f)(=)(S). Then
RpdRX ≤ RpdS X + RpdR S.
Proof. Immediately by Proposition 2.4(1) or 2.5.
1. Avramov L. L., Foxby H.-B. Homological dimensions of unbounded complexes // J. Pure and Appl. Algebra. – 1991. –
71. – P. 129 – 155.
2. Christensen L. W. Gorenstein dimensions // Lect. Notes Math. – 2000. – 1747.
3. Christensen L. W., Foxby H.-B., Frankild A. Restricted homological dimensions and Cohen – Macaulayness // J.
Algebra. – 2002. – 251. – P. 479 – 502.
4. Sharif T., Yassemi S. Depth formulas, restricted Tor-dimension under base change // Rocky Mountain J. Math. –
2004. – 34. – P. 1131 – 1146.
5. Xu J. Flat covers of modules // Lect. Notes Math. – 1996. – 1634.
Received 28.05.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
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| id | umjimathkievua-article-2479 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:14Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7c/13c3d3de8aec91d0fd9aa846fcd87b7c.pdf |
| spelling | umjimathkievua-article-24792020-03-18T19:16:28Z On the Restricted Projective Dimension of Complexes Про обмежену проективну розмірність комплексів Li, Liang Wu, Dejun Лі, Ліанг Ву, Дюн We study the restricted projective dimension of complexes and give some new characterizations of the restricted projective dimension. In particular, it is shown that the restricted projective dimension can be computed in terms of the so-called restricted projective resolutions. As applications, we get some results on the behavior of the restricted projective dimension under the change of rings. Вивчається обмежена проективна розмірність комплексів. Наведено дєякі нові властивості обмеженої проективної розмірності. Зокрема, показано, що обмежену проективну розмірність можна обчислити через так звані обмежені проективні резольвенти. Як застосування отримано деякі результати про поведінку обмеженої проективної розмірності при зміні кілець. Institute of Mathematics, NAS of Ukraine 2013-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2479 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 7 (2013); 936–945 Український математичний журнал; Том 65 № 7 (2013); 936–945 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2479/1724 https://umj.imath.kiev.ua/index.php/umj/article/view/2479/1725 Copyright (c) 2013 Li Liang; Wu Dejun |
| spellingShingle | Li, Liang Wu, Dejun Лі, Ліанг Ву, Дюн On the Restricted Projective Dimension of Complexes |
| title | On the Restricted Projective Dimension of Complexes |
| title_alt | Про обмежену проективну розмірність комплексів |
| title_full | On the Restricted Projective Dimension of Complexes |
| title_fullStr | On the Restricted Projective Dimension of Complexes |
| title_full_unstemmed | On the Restricted Projective Dimension of Complexes |
| title_short | On the Restricted Projective Dimension of Complexes |
| title_sort | on the restricted projective dimension of complexes |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2479 |
| work_keys_str_mv | AT liliang ontherestrictedprojectivedimensionofcomplexes AT wudejun ontherestrictedprojectivedimensionofcomplexes AT lílíang ontherestrictedprojectivedimensionofcomplexes AT vudûn ontherestrictedprojectivedimensionofcomplexes AT liliang proobmeženuproektivnurozmírnístʹkompleksív AT wudejun proobmeženuproektivnurozmírnístʹkompleksív AT lílíang proobmeženuproektivnurozmírnístʹkompleksív AT vudûn proobmeženuproektivnurozmírnístʹkompleksív |