A Note on Projectively Coresolved Gorenstein Flat Complexes and Dimensions

Bowen DU; Bo LU

doi:10.1051/wujns/2025305447

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 30, Number 5, October 2025


Page(s)		447 - 452
DOI		https://doi.org/10.1051/wujns/2025305447
Published online		04 November 2025

Wuhan University Journal of Natural Sciences, 2025, Vol.30 No.5, 447-452

CLC number: O154.2

A Note on Projectively Coresolved Gorenstein Flat Complexes and Dimensions

关于PGF复形和维数的注记

Bowen DU (杜博文)¹ and Bo LU (卢博)¹^,2^†

¹ College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, Gansu, China
² Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou 730070, Gansu, China

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 10 April 2025

Abstract

In this article, we first establish a recollement related to projectively coresolved Gorenstein flat (PGF) complexes. Secondly, we define and study PGF dimension of complexes, we denote it PGF(X) for a complex X. It is shown that the PGF(X) is equal to the infimum of the set {supA | there exists a diagram of morphisms of complexes A←G→X, such that G→X is a special PGF precover of X and G→A is a PGF almost isomorphism}.

摘要

首先，建立了相对于projectively coresolved Gorenstein flat (PGF)复形的粘合。其次，定义并研究了PGF维数，对于任意的复形X，将其PGF维数记作PGF(X)。得到了PGF(X)是 ${s u p A | 存在复形的态射图 A \leftarrow G \to X, 使$ Mathematical equation $得 G \to X 是 X 的特殊 P G F 预覆盖$ ， $且 G \to A 是 P G F 几乎同构}$ 的下确界。

Key words: projectively coresolved Gorenstein flat (PGF) module / PGF complex / recollement / PGF dimension

关键字 : PGF模 / PGF复形 / 粘合 / PGF维数

Cite this article: DU Bowen, LU Bo. A Note on Projectively Coresolved Gorenstein Flat Complexes and Dimensions[J]. Wuhan Univ J of Nat Sci, 2025, 30(5): 447-452.

Biography: DU Bowen, male, Master candidate, research direction: homological algebra. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Foundation item: Supported by the National Natural Science Foundation of China (12061061), Young Talents Team Project of Gansu Province (2025QNTD49), Lanshan Talents Project of Northwest Minzu University (Xbmulsrc202412) and Longyuan Young Talents of Gansu Province

© Wuhan University 2025

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The famous "flat cover conjecture", that is, all modules have a flat cover over any ring, is addressed by Enochs^[1]. It has now been settled by Bican et al^[2]. It is natural to consider the Gorenstein version of "flat cover conjecture". Thus, the existence of Gorenstein flat covers of moduls is concerned by many scholars^[3-5]. In particular, Yang and Liu^[4] proved that all modules have Gorenstein flat covers over a left ring over which the class of Gorenstein flat modules is closed under extensions (GF-closed). Recently, Šaroch and Št'ovíček^[6] proved that every ring is always GF-closed in terms of the notion of projectively coresolved Gorenstein flat (PGF) modules. Thus, the Gorenstein version of flat cover conjecture holds as well. Recently, the researcheres extend the notion of PGF modules to the category of complexes, and call it PGF complexes^[7].

The definition of recollements of triangulated categories was first introduced by Beilinson et al^[8] to study the triangulated categories of perverse sheaves over singular spaces, and later was used by Scott et al^[9] to stratify the derived categories of quasi-hereditary algebras arising from the representation theory of semisimple Lie algebras and algebraic groups.

Homological dimension plays an important role in homological algebra, and has been extended to the setting of complexes by many researchers. Avramov and Foxby^[10] defined the notion of projective dimension for a complex X, pd(X), as the infimum of the set {sup P | P is a DG-projective resolution of X}. They showed that for any complexes X,

$p d (X) = i n f ｛ n \in Z | s u p H (P) \leq n, C o k e r δ_{n + 1}^{P} i s p r o j e c t i v e, w h e r e P \to X i s a D G - p r o j e c t i v e r e s o l u t i o n ｝ .$ Mathematical equation

Note that if $f : P \to X$ Mathematical equation is a quasi-isomorphism then $E x t^{1} (Q, K e r f) = 0$ for every DG-projective complex Q. Thus, the DG-projective resolution $P \to X$ is a special DG-projective precover. This means that for any complex X,

$p d (X) = i n f ｛ n \in Z | s u p H (P) \leq n, C o k e r δ_{n + 1}^{P} i s p r o j e c t i v e, w h e r e P \to X i s a s p e c i a l D G - p r o j e c t i v e p r o c o v e r ｝ .$ Mathematical equation

Gorenstein projective dimension of complexes, Gpd(X), was introduced by Veliche and it follows from Ref. [11] that

$G p d (X) = i n f ｛ n \in Z | s u p H (P) \leq n, C o k e r δ_{n + 1}^{P} i s G o r e n s t e i n p r o j e c t i v e, w h e r e P \to X i s a D G - p r o j e c t i v e r e s o l u t i o n ｝$ Mathematical equation

$= i n f ｛ n \in Z | s u p H (P) \leq n, C o k e r δ_{n + 1}^{P} i s G o r e n s t e i n p r o j e c t i v e, w h e r e P \to X i s a s p e c i a l D G - p r o j e c t i v e p r o c o v e r o f X ｝$ Mathematical equation

In Ref. [12], the authors introduced and studied the dimension of complexes, which is related to special Gorenstein projective precovers:

$G p p d (X) = i n f {n \in Z | s u p H (G) \leq n, a n d C o k e r δ_{n + 1}^{G} i s G o r e n s t e i n p r o j e c t i v e, w h e r e G \to X i s a s p e c i a l G o r e n s t e i n$ Mathematical equation

$p r o j e c t i v e p r e c o v e r o f X}$ Mathematical equation

If no such $G$ Mathematical equation exists, set $G p p d (X) = \infty$ .

Inspired by the above, in this article, we study the recollement and dimension related to PGF complexes.

1 Preliminaries

In this section we recall some necessary notations and definitions. Throughout the article, let R denote an associative ring with an identity and by the term "module" we always mean a left R-module.

A complex X of modules is often displayed as a sequence

$\dots \overset{}{\to} X_{n + 1} \overset{δ_{n + 1}^{X}}{\to} X_{n} \overset{δ_{n}^{X}}{\to} X_{n - 1} \overset{}{\to} \dots$ Mathematical equation

with $δ_{n}^{X} δ_{n + 1}^{X} = 0$ Mathematical equation for all $n \in Z$ . The n-th cycle (resp., homology) module is defined as $K e r δ_{n}^{X}$ (resp., $K e r δ_{n}^{X} / I m δ_{n + 1}^{X}$ ) and denoted by $Z_{n} (X)$ (resp., $H_{n} (X)$ ). The homology complex $H (X)$ is defined by setting $H {(X)}_{i} = H_{i} (X)$ Mathematical equation and $δ_{i}^{H (X)} = 0$ for all $i \in Z$ . A complex X is called exact if $H_{i} (X) = 0$ for all $i \in Z$ . For a complex X, we associate the numbers

$s u p X = s u p {i \in Z | X_{i} \neq 0} a n d i n f X = i n f {i \in Z | X_{i} \neq 0} .$ Mathematical equation

The complex X is called bounded above (resp., bounded below) if $s u p X < \infty$ Mathematical equation (resp., $i n f X > - \infty$ ). It is called bounded if it is both bounded above and bounded below.

In the following, we will use $C (R)$ Mathematical equation and $D (R)$ to denote the category of complexes of modules and the derived category. The symbol $≃$ is used to designate quasi-isomorphisms in $C (R)$ . For complexes $X$ and $Y$ , denote by $H o m (X, Y)$ the Abelian group of morphisms from $X$ to $Y$ in $C (R)$ Mathematical equation and $E x t^{i} (X, Y)$ for $i \geq 1$ will denote the groups we get from the right derived functor of $H o m (-, -)$ .

For two complexes $X$ Mathematical equation and $Y$ , denote by $H o m_{R} (X, Y)$ the complex with $H o m_{R} {(X, Y)}_{n} = \prod_{i \in Z} H o m_{R} (X_{i}, Y_{i + n})$ for all $n \in Z$ and with the boundary operator given by $δ_{n}^{H o m_{R} (X, Y)} (f) = (δ_{n + i}^{Y} f_{i} - {(- 1)}^{| f |} f_{i - 1} δ_{i}^{X})_{i \in Z}$ Mathematical equation for every $f \in H o m_{R} {(X, Y)}_{n}$ . We denote by $E x t_{R}^{n} (-, -)$ the right derived functors of $H o m_{R} (-, -)$ and call it the absolute cohomology functors. Two morphisms $α$ and $β$ in $H o m (X, Y) = K e r δ_{0}^{H o m_{R} (X, Y)}$ are called homotopic, denoted by $α ~ β$ Mathematical equation , if there exists a homomorphism $μ$ of degree 1 such that $δ_{1}^{H o m_{R} (X, Y)} (μ) = α - β$ .

Let X be a complex of right R-modules and Y a complex of left R-modules, denote the usual tensor product of X and Y, where $(X \otimes_{R} {Y)}_{n} = \underset{t \in Z}{\oplus} X_{t} \otimes_{R} Y_{n - t}$ Mathematical equation and $δ (x \otimes y) = δ_{t}^{X} (x) \otimes y + {(- 1)}^{t} x \otimes δ_{n - t}^{Y} (y)$ for $x \in X_{t}, y \in Y_{n - t}$ . We define $X \bar{\otimes} Y$ to be $\frac{(X \otimes_{R} Y)}{B (X \otimes_{R} Y)}$ with the maps

$\frac{(X \otimes_{R} {Y)}_{m}}{B_{m} (X \otimes_{R} Y)} \to \frac{(X \otimes_{R} {Y)}_{m - 1}}{B_{m - 1} (X \otimes_{R} Y)}, x \otimes y \to δ^{X} (x) \otimes y,$ Mathematical equation

where $x \otimes y$ Mathematical equation is used to denote the coset in $\frac{(X \otimes_{R} {Y)}_{m}}{B_{m} (X \otimes_{R} Y)}$ . Then we get a complex. Since $X \bar{\otimes} - : C (R) \to C (Z)$ is a right exact functor between Abelian categories with enough projectives, we can construct left derived functors which we denote by ${\bar{T o r}}_{i} (C, -)$ Mathematical equation .

Let $A$ Mathematical equation be an Abelian category and $X$ a subcategory of $A$ . For an object $M \in A$ write $M \in {}^{⊥}X$ , if $E x t_{A}^{1} (M, X) = 0$ for each object $X \in X$ . Dually, one can define $M \in X^{⊥}$ .

Recall that a pair $(X, Y)$ Mathematical equation of subcategories of $A$ is called a cotorsion pair or cotorsion theory provided that $X = {}^{⊥}Y$ and $Y = X^{⊥}$ . A morphism $ϕ : X \to M$ with $X \in X$ is called an $X$ -precover of M if for any morphism $f : X^{'} \to M$ with $X^{'} \in X$ , there is a morphism $g : X^{'} \to X$ Mathematical equation such that $ϕ g = f$ . An epimorphism $ψ : X \to M$ with $X \in X$ is said to be a special $X$ -precover of M if $ψ$ is an $X$ -precover of $M$ and $K e r ψ \in X^{⊥}$ .

A cotorsion pair $(X, Y)$ Mathematical equation is said to be complete provided that every object in $A$ has a special $Y$ -preenvelope and a special $X$ -precover. A cotorsion pair $(X, Y)$ cogenerated by a set $S$ of objects, i.e. such that $Y = S^{⊥}$ , is complete, see Ref. [13]. A cotorsion pair $(X, Y)$ is said to be hereditary provided that $E x t_{A}^{2} (X, Y) = 0$ Mathematical equation for each object $X \in X$ and each object $Y \in Y$ . Ref. [3] is a standard reference for cotorsion pairs^[3]. A complete cotorsion pair $(X, Y)$ is said to be a projective cotorsion pair^[14] if $Y$ is thick and $X ⋂ Y$ coincides with the class of projective objects. Recall from Ref. [14] that a class $W \subseteq A$ Mathematical equation is called thick provided that it is closed under direct summands, exte nsions, and taking kernels of epimorphisms and cokernels of monomorphisms.

2 A Recollement Related to PGF Complexes

We begin with the definition of PGF modules and PGF complexes. Note that PGF modules not only are Gorenstein flat, but also Gorenstein projective^[6].

Definition 1^[6-7] A module M is said to be PGF if there exists an exact sequence of projective modules

$\dots \to Q^{1} \to Q^{0} \to Q^{- 1} \to \dots$ Mathematical equation

with $M = K e r (Q^{0} \to Q^{- 1})$ Mathematical equation and which remains exact after applying $E \otimes_{R} -$ for any injective right $R$ -modules E.

A complex X is said to be PGF if there exists an exact sequence of projective complexes

$\dots \to P^{1} \to P^{0} \to P^{- 1} \to \dots$ Mathematical equation

with $X = K e r (P^{0} \to P^{- 1})$ Mathematical equation and which remains exact after applying $I \bar{\otimes} -$ for any injective complex I of right $R$ -modules.

In the following, we will use $P G F$ Mathematical equation and $P G ℱ$ to denote the category of PGF modules and the category of PGF complexes.

We give some basic results for PGF modules and PGF complexes which are obtained by Refs. [6-7, 14].

Lemma 1 (1) The class of PGF modules is projectively resolving, closed under arbitrary direct sums, under direct summands and transfinite extensions.

(2) $(P G F, P G F^{⊥})$ Mathematical equation is hereditary complete cotorsion pair.

(3) $(P G F, P G F^{⊥})$ Mathematical equation is a projective cotorsion pair which is cogenerated by a set.

(4) If M is a PGF module, then $T o r_{n}^{R} (E, M) = 0$ Mathematical equation for any injective right $R$ -modules E and any $n \geq 1$ .

(5) A complex G is a PGF complex if and only if $G_{n}$ Mathematical equation is a PGF module for each $n \in Z$ .

Remark 1 Gorenstein flat complexes are exactly the complexes with Gorenstein flat components over right coherent rings, independently^{[6, 15]}. Šaroch and Št'ovíček^[6] showed that the class of Gorenstein flat modules is always closed under extensions over any ring. Yang and Liu^[15] proved that Gorenstein flat complexes are exactly the complexes with Gorenstein flat components whenever the class of Gorenstein flat modules is closed under extensions. Thus, Gorenstein flat complexes are exactly the complexes with Gorenstein flat components.

We obtain the following result which gives equivalent characterizations of that Gorenstein projective modules are Gorenstein flat by Ref. [7].

Lemma 2 The following statements are equivalent:

(1) Every Gorenstein projective modules are Gorenstein flat modules.

(2) Every Gorenstein projective modules are PGF modules.

(3) Every Gorenstein projective complexes are Gorenstein flat complexes.

(4) Every Gorenstein projective complexes are PGF complexes.

Proof $(1) \Leftrightarrow (2)$ Mathematical equation follows from Ref. [6].

$(1) \Leftrightarrow (3)$ Mathematical equation follows by Ref. [16] and Remark 1.

$(2) \Leftrightarrow (4)$ Mathematical equation holds by Ref. [7].

Definition 2^[14] Let $D$ Mathematical equation , $D^{'}$ and $D^{″}$ be triangulated categories. We say that $D$ is a recollement of $D^{'}$ and $D^{″}$ if there are six triangle functors as in Fig. 1 such that

Thumbnail: Fig. 1 Refer to the following caption and surrounding text.

Fig. 1 The definition of recollement

(1) $(F_{λ}, F, F_{ρ})$ Mathematical equation and $(G_{λ}, G, G_{ρ})$ are adjoint triples;

(2) F, $G_{λ}$ Mathematical equation and $G_{ρ}$ are fully faithful;

(3) For any object $X \in D$ Mathematical equation , we have $G X = 0$ if and only if $X ≅ F X^{'}$ for some $X \in D^{'}$ .

Recall from Ref. [17] that a triple $ℋ = (Q, W, ℛ)$ Mathematical equation of classes of objects in an Abelian category is called a Hovey triple if $(Q ⋂ W, ℛ)$ and $(Q, W ⋂ ℛ)$ are complete cotorsion pairs and the class $W$ is thick. As a consequence of Refs. [6-7, 14], we obtain the following result.

Lemma 3 There exist two projective cotorsion pairs $(P G ℱ, P G ℱ^{⊥})$ Mathematical equation and $(ℰ P G ℱ, ℰ P G ℱ^{⊥})$ which are cogenerated by sets, where $ℰ P G ℱ$ denote the class of all exact PGF complexes. Moreover, $(P G ℱ, P G ℱ^{⊥}, C (R))$ and $(ℰ P G ℱ, ℰ P G ℱ^{⊥}, C (R))$ are Hovey triples. In particular, every complex has a special PGF precover and a special exact PGF precover.

Using Lemma 3 and Ref. [17], we also obtain two projective model structures on $C (R)$ Mathematical equation in which every object is fibrant, the objects in $P G ℱ$ (resp., $ℰ P G ℱ$ ) are cofibrant, and the objects in $P G ℱ^{⊥}$ (resp., $ℰ P G ℱ^{⊥}$ ) are trivial. We call these PGF model structures (resp., exact PGF model structures).

We use $K_{e x} (P G F)$ Mathematical equation and $K (P G F)$ to denote the homotopy category of all exact PGF complexes and the homotopy category of all PGF complexes, respectively. Then we obtain the following conclusion.

Theorem 1 Let R be a ring. There is a recollement in Fig. 2 where $E (E)$ Mathematical equation and $E (E P G F^{⊥})$ represent special exact preenvelopes and special $E P G F^{⊥}$ -preenvelopes, respectively, $C (D G)$ and $C (E P G F)$ represent special DG-projective precovers and special exact PGF precovers, respectively, $I$ denotes the inclusion functor.

Thumbnail: Fig. 2 Refer to the following caption and surrounding text.

Fig. 2 The recollement of PGF

Proof We can see that $(P G ℱ, P G ℱ^{⊥})$ Mathematical equation and $(ℰ P G ℱ, ℰ P G ℱ^{⊥})$ are projective cotorsion pairs in $C (R)$ by Lemma 3. We also notice that $(D G P, ℰ)$ is a projective cotorsion pair in $C (R)$ , $P G ℱ ⋂ ℰ = ℰ P G ℱ$ and $D G P \subseteq P G ℱ$ by Ref. [7]. As we know, the homotopy category of all DG-projective complexes $K (D G P)$ Mathematical equation is isomorphic to $D (R)$ . Thus, the result is obtain by Ref. [14].

3 PGF Dimension of Complexes

In this section, we will investigate the dimension with respect to PGF complexes.

Since $(P G F, P G F^{⊥})$ Mathematical equation is a projective cotorsion pair cogenerated by a set, thus by Ref. [14], we conclude that $(d w \tilde{P G F}, d w {\tilde{P G F}}^{⊥})$ , in which each degree component of the associated graded object is a PGF module, forms a projective cotorsion pair. By Lemma 1, the complexes in $d w \tilde{P G F}$ Mathematical equation are precisely the PGF complexes. Then by Ref. [18], we obtain the following definition.

Definition 3 Let X be a complex. The dimension of X related to special PGF precovers, PGF(X), is defined as

$P G F (X) = i n f ｛ n \in Z | s u p H (G) \leq n, a n d C o k e r δ_{n + 1}^{G} i s a P G F m o d u l e, w h e r e G \to X i s a s p e c i a l P G F p r e c o v e r o f X ｝ .$ Mathematical equation

If no such G exists, set $P G F (X) = \infty$ Mathematical equation .

Definition 4 Let A and G be two PGF complexes. An epimorphism of complexes $f : G \to A$ Mathematical equation is said to be PGF almost isomorphic if $K e r f$ is exact and bounded below. It is clear that $K e r f$ is a PGF complex.

Theorem 2 Suppose that G is a PGF complex. Then

$\begin{matrix} P G F (G) \end{matrix} \begin{matrix} = i n f {n \in Z | s u p H (G) \leq n, C o k e r δ_{n + 1}^{G} i s a P G F m o d u l e} \\ = i n f {s u p A | G \to A i s a P G F a l m o s t i s o m o r p h i s m} . \end{matrix}$ Mathematical equation

Proof Set $s = i n f {n \in Z | s u p H (G) \leq n, C o k e r δ_{n + 1}^{G} i s a P G F m o d u l e}$ Mathematical equation and $t = i n f {s u p A | G \to A i s a n P G F a l m o s t$

$i s o m o r p h i s m}$ Mathematical equation

Since G is a PGF complex, it can be seen that $P G F (G) \leq s$ Mathematical equation . We only need to concentrate on $P G F (G) \geq s$ . If $P G F (G) = \infty$ then the proof is evident. Suppose that $P G F (G) = n \in Z$ . Then there exists a special PGF precover $φ : F \to G$ such that $s u p H (F) \leq n$ , $C o k e r δ_{n + 1}^{F}$ is a PGF module and $E x t^{1} (G, K e r φ) = 0$ Mathematical equation . Hence the short exact sequence of complexes

$0 \to K e r φ \to F \to G \to 0$ Mathematical equation

is splits. Then we have $F ≅ G \oplus K e r φ$ Mathematical equation . This implies that $s u p H (G) \leq n$ . We can conclude that $s \leq n$ . If $P G F (G) = - \infty$ , then for any integer n, we can find a PGF precover $F \to G$ such that $s u p H (F) \leq n$ and $C o k e r δ_{n + 1}^{F}$ is a PGF module. Similar discussion yields that $s = - \infty$ Mathematical equation . This leads to the conclusion that $s u p H (G) \leq n$ , which is to say $P G F (G) \geq s$ . Thus $P G F (G) = s$ .

We show that $t \leq s$ Mathematical equation . If $s = \infty$ , then the proof is trivial. Assume s is an integer. Then $s u p H (G) \leq s$ , and $C o k e r δ_{n + 1}^{G}$ is a PGF module. Consider the commutative diagram (Fig. 3). Define K and A as the following sequences

$K = \dots \to G_{s + 2} \to G_{s + 1} \to I m δ_{s + 1}^{G} \to 0 \dots,$ Mathematical equation

$A = 0 \to C o k e r δ_{n + 1}^{G} \to G_{s - 1} \to G_{s - 2} \to \dots .$ Mathematical equation

Thumbnail: Fig. 3 Refer to the following caption and surrounding text.

Fig. 3 The commutative diagram about G

It is clealy that K is exact and $I m δ_{s + 1}^{G}$ Mathematical equation is PGF amodule. Then the map $G \to A$ is a PGF almost isomorphic and $s u p A \leq s$ . Consequently, $t \leq s$ . Suppose $s = - \infty$ , hence for any integer n, $s u p H (G) \leq n$ and $C o k e r δ_{n + 1}^{G}$ is a PGF module. Thus G is exact. Using a similar diagram, we can obtain $G \to A$ Mathematical equation is a PGF almost isomorphism such that $s u p A \leq n$ . This leads to the conclusion $t = - \infty$ . Thus $t \leq s$ .

Next, we demonstrate that $s \leq t$ Mathematical equation . If $t = - \infty$ , then the proof is trivial. Assume t is an integer. Then there exists a PGF almost isomorphism $G \to A$ with $s u p A = t$ . Let $E = K e r (G \to A)$ . Then E is exact and bounded below, which implies that $C o k e r δ_{n + 1}^{E}$ is a PGF module. Consider the commutative diagram (Fig. 4) with exact rows and the exact middle column, then it is evident that the final column of the diagram is exact, which implies that $C o k e r δ_{n + 1}^{G}$ Mathematical equation is a PGF module. Additionally, it is evident that $s u p H (G) \leq n$ . Consequently, we can conclude that $s \leq t$ .

Thumbnail: Fig. 4 Refer to the following caption and surrounding text.

Fig. 4 The commutative diagram about E

Now, consider the case where $t = - \infty$ Mathematical equation . For any integer n, there is a PGF almost isomorphism $G \to A$ such that $s u p A < n$ , hence G is exact at n+1 and $C o k e r δ_{n + 1}^{G}$ is a PGF module. This means that G is an exact complex, and $C o k e r δ_{n + 1}^{G}$ is a PGF complex for any integer n. This leads to the conclusion that $s = - \infty$ Mathematical equation .

The following result is an immediate consequence of Definition 3 and Theorem 2.

Theorem 3 Let X be a complex. Then

$P G F (X) = i n f {s u p A | t h e r e e x i s t s a d i a g r a m o f m o r p h i s m s o f c o m p l e x e s A \leftarrow G \to X s u c h t h a t G \to X i s a s p e c i a l P G F$ Mathematical equation

$p r e c o v e r o f X a n d G \to A i s a P G F a l m o s t i s o m o r p h i s m}$ Mathematical equation

Take $A = P G F$ Mathematical equation in Ref. [18], we get the following definition.

Definition 5 Let X be a complex. Then for any complex Y and any integer n, the n-th relative cohomology group $E x t_{P G ℱ}^{n} (X, Y)$ Mathematical equation is defined by the equality $E x t_{P G ℱ}^{n} (X, Y) = H^{n} (H o m_{R} (G, Y))$ where $G \to X$ is a special PGF precover of X.

As an immediate consequence of Ref. [18], we obtain the following result, which gives an equivalent characterization on PGF dimensions.

Corollary 1 Let X be a complex and n an integer. Then the following conditions are equivalent.

(1) $P G F (X) \leq n;$ Mathematical equation

(2) $E x t_{P G ℱ}^{i} (X, Y) = 0$ Mathematical equation for all $i > n - t$ and any bounded complex Y such that all $Y_{j}$ and $K e r (Y_{t} \to Y_{t - 1})$ are in $P G F^{⊥}$ , where $t = s u p H (Y)$ ;

(3) $E x t_{P G ℱ}^{i} (X, Y) = 0$ Mathematical equation for all $i > n$ and any module Y in $P G F^{⊥}$ ;

(4) $s u p H (X) \leq n$ Mathematical equation and for any special PGF precover $G \to X$ of X, the module $C o k e r (G_{n + 1} \to G_{n})$ is in $P G F$ .

References

Enochs E E. Injective and flat covers, envelopes and resolvents[J]. Israel Journal of Mathematics, 1981, 39(3): 189-209. [Google Scholar]
Bican L, El Bashir R, Enochs E. All modules have flat covers[J]. Bulletin of the London Mathematical Society, 2001, 33(4): 385-390. [Google Scholar]
Hochschild G. Relative homological algebra[J]. Transactions of the American Mathematical Society, 1956, 82(1): 246-269. [Google Scholar]
Yang G, Liu Z K. Gorenstein flat covers over GF-closed rings[J]. Communications in Algebra, 2012, 40(5): 1632-1640. [Google Scholar]
Yang G, Liang L. All modules have Gorenstein flat precovers[J]. Communications in Algebra, 2014, 42(7): 3078-3085. [Google Scholar]
Šaroch J, Št'ovíček J. Singular compactness and definability for ⅀-cotorsion and Gorenstein modules[J]. Selecta Mathematica, 2020, 26(2): 23. [Google Scholar]
Wu D J, Zhao Z H. Projectively coresolved Gorenstein flat complexes[J]. Journal of Lanzhou University of Technology, 2020, 46(6): 153-158(Ch). [Google Scholar]
Beilinson A, Bernstein J, Deligne P, et al. Faisceaux Pervers[M]. Paris: Société Mathématique de France, 2018. [Google Scholar]
Scott L, Parshall B, Cline E. Finite-dimensional algebras and highest weight categories[J]. Journal fur die Reine und Angewandte Mathematik, 1988, 391: 85-99. [Google Scholar]
Avramov L L, Foxby H B. Homological dimensions of unbounded complexes[J]. Journal of Pure and Applied Algebra, 1991, 71(2-3): 129-155. [Google Scholar]
Veliche O. Gorenstein projective dimension for complexes[J]. Transactions of the American Mathematical Society, 2006, 358(3): 1257-1283. [Google Scholar]
Liu Z K, Di Z X, Lu B. Relative cohomology of complexes II: Vanishing of relative cohomology[J]. Journal of Algebra, 2021, 565: 309-323. [Google Scholar]
Göbel R, Trlifaj J. Approximations and Andomorphism Algebras of Modules[M]. Berlin: Walter de Gruyter GmbH & Co. KG, 2006. [Google Scholar]
Gillespie J. Gorenstein complexes and recollements from cotorsion pairs[J]. Advances in Mathematics, 2016, 291: 859-911. [Google Scholar]
Yang G, Liu Z K. Stability of Gorenstein flat categories[J]. Glasgow Mathematical Journal, 2012, 54(1): 177-191. [Google Scholar]
Yang X Y, Liu Z K. Gorenstein projective, injective, and flat complexes[J]. Communications in Algebra, 2011, 39(5): 1705-1721. [Google Scholar]
Hovey M. Cotorsion pairs, model category structures, and representation theory[J]. Mathematische Zeitschrift, 2002, 241(3): 553-592. [Google Scholar]
Liu Z K, Li J L, Wang Z P. Relative cohomology dimensions of complexes based on degreewise cotorsion pairs[J]. Publicationes Mathematicae Debrecen, 2022, 101(1-2): 103-117. [Google Scholar]

All Figures

	Fig. 1 The definition of recollement
In the text

	Fig. 2 The recollement of PGF
In the text

	Fig. 3 The commutative diagram about G
In the text

	Fig. 4 The commutative diagram about E
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

[R1] Enochs E E. Injective and flat covers, envelopes and resolvents[J]. Israel Journal of Mathematics, 1981, 39(3): 189-209. [Google Scholar]

[R2] Bican L, El Bashir R, Enochs E. All modules have flat covers[J]. Bulletin of the London Mathematical Society, 2001, 33(4): 385-390. [Google Scholar]

[R3] Hochschild G. Relative homological algebra[J]. Transactions of the American Mathematical Society, 1956, 82(1): 246-269. [Google Scholar]

[R4] Yang G, Liu Z K. Gorenstein flat covers over GF-closed rings[J]. Communications in Algebra, 2012, 40(5): 1632-1640. [Google Scholar]

[R5] Yang G, Liang L. All modules have Gorenstein flat precovers[J]. Communications in Algebra, 2014, 42(7): 3078-3085. [Google Scholar]

[R6] Šaroch J, Št'ovíček J. Singular compactness and definability for ⅀-cotorsion and Gorenstein modules[J]. Selecta Mathematica, 2020, 26(2): 23. [Google Scholar]

[R7] Wu D J, Zhao Z H. Projectively coresolved Gorenstein flat complexes[J]. Journal of Lanzhou University of Technology, 2020, 46(6): 153-158(Ch). [Google Scholar]

[R8] Beilinson A, Bernstein J, Deligne P, et al. Faisceaux Pervers[M]. Paris: Société Mathématique de France, 2018. [Google Scholar]

[R9] Scott L, Parshall B, Cline E. Finite-dimensional algebras and highest weight categories[J]. Journal fur die Reine und Angewandte Mathematik, 1988, 391: 85-99. [Google Scholar]

[R10] Avramov L L, Foxby H B. Homological dimensions of unbounded complexes[J]. Journal of Pure and Applied Algebra, 1991, 71(2-3): 129-155. [Google Scholar]

[R11] Veliche O. Gorenstein projective dimension for complexes[J]. Transactions of the American Mathematical Society, 2006, 358(3): 1257-1283. [Google Scholar]

[R12] Liu Z K, Di Z X, Lu B. Relative cohomology of complexes II: Vanishing of relative cohomology[J]. Journal of Algebra, 2021, 565: 309-323. [Google Scholar]

[R13] Göbel R, Trlifaj J. Approximations and Andomorphism Algebras of Modules[M]. Berlin: Walter de Gruyter GmbH & Co. KG, 2006. [Google Scholar]

[R14] Gillespie J. Gorenstein complexes and recollements from cotorsion pairs[J]. Advances in Mathematics, 2016, 291: 859-911. [Google Scholar]

[R15] Yang G, Liu Z K. Stability of Gorenstein flat categories[J]. Glasgow Mathematical Journal, 2012, 54(1): 177-191. [Google Scholar]

[R16] Yang X Y, Liu Z K. Gorenstein projective, injective, and flat complexes[J]. Communications in Algebra, 2011, 39(5): 1705-1721. [Google Scholar]

[R17] Hovey M. Cotorsion pairs, model category structures, and representation theory[J]. Mathematische Zeitschrift, 2002, 241(3): 553-592. [Google Scholar]

[R18] Liu Z K, Li J L, Wang Z P. Relative cohomology dimensions of complexes based on degreewise cotorsion pairs[J]. Publicationes Mathematicae Debrecen, 2022, 101(1-2): 103-117. [Google Scholar]