Research article Special Issues

Italian landscape macrosystem (ILM) from urban pressure to a National Wildway

  • Thirty years after the widespread establishment of the concepts of environmental continuity in Europe and Italy, linked to the EU Habitat Directive and the Natura 2000 project, these have almost completely lost their impetus. After having fostered a very intense scientific and political/administrative/managerial period, these concepts have produced only a few virtuous examples of implementation. The study presented in this paper shows that today national environmental quality is almost entirely based on land use categories that are scarcely protected and not subject to sufficiently binding planning. Moreover, their functional structure and extension/quality depend on how the local and global market impact spatially complementary categories mostly affected by anthropic action, which instead undergo efficiently planned changes. From the fundamental paradigm of "ecological networks" the interests of the scientific world and local authorities have shifted to other fronts, losing that physical and conceptual cohesion inherent in "environmental continuity" as a value in itself. The availability of recent, very advanced data helps take stock of this issue and verify critical points on the national territory that have been unresolved for several decades, but still rife with potential for the application of models of international value.

    Citation: Bernardino Romano, Andrea Agapito Ludovici, Francesco Zullo, Alessandro Marucci, Lorena Fiorini. Italian landscape macrosystem (ILM) from urban pressure to a National Wildway[J]. AIMS Environmental Science, 2020, 7(6): 505-525. doi: 10.3934/environsci.2020032

    Related Papers:

    [1] Dingguo Wang, Chenyang Liu, Xuerong Fu . Weakly Gorenstein comodules over triangular matrix coalgebras. AIMS Mathematics, 2022, 7(8): 15471-15483. doi: 10.3934/math.2022847
    [2] Qianqian Guo . Gorenstein projective modules over Milnor squares of rings. AIMS Mathematics, 2024, 9(10): 28526-28541. doi: 10.3934/math.20241384
    [3] Juxiang Sun, Guoqiang Zhao . Homological conjectures and stable equivalences of Morita type. AIMS Mathematics, 2025, 10(2): 2589-2601. doi: 10.3934/math.2025120
    [4] Nasr Anwer Zeyada, Makkiah S. Makki . Some aspects in Noetherian modules and rings. AIMS Mathematics, 2022, 7(9): 17019-17025. doi: 10.3934/math.2022935
    [5] Xuanyi Zhao, Jinggai Li, Shiqi He, Chungang Zhu . Geometric conditions for injectivity of 3D Bézier volumes. AIMS Mathematics, 2021, 6(11): 11974-11988. doi: 10.3934/math.2021694
    [6] Senrong Xu, Wei Wang, Jia Zhao . Twisted Rota-Baxter operators on Hom-Lie algebras. AIMS Mathematics, 2024, 9(2): 2619-2640. doi: 10.3934/math.2024129
    [7] Hongyu Chen, Li Zhang . A smaller upper bound for the list injective chromatic number of planar graphs. AIMS Mathematics, 2025, 10(1): 289-310. doi: 10.3934/math.2025014
    [8] Yanyi Li, Lily Chen . Injective edge coloring of generalized Petersen graphs. AIMS Mathematics, 2021, 6(8): 7929-7943. doi: 10.3934/math.2021460
    [9] Pengcheng Ji, Jialei Chen, Fengxia Gao . Projective class ring of a restricted quantum group ¯Uq(sl2). AIMS Mathematics, 2023, 8(9): 19933-19949. doi: 10.3934/math.20231016
    [10] Samah Gaber, Abeer Al Elaiw . Evolution of null Cartan and pseudo null curves via the Bishop frame in Minkowski space R2,1. AIMS Mathematics, 2025, 10(2): 3691-3709. doi: 10.3934/math.2025171
  • Thirty years after the widespread establishment of the concepts of environmental continuity in Europe and Italy, linked to the EU Habitat Directive and the Natura 2000 project, these have almost completely lost their impetus. After having fostered a very intense scientific and political/administrative/managerial period, these concepts have produced only a few virtuous examples of implementation. The study presented in this paper shows that today national environmental quality is almost entirely based on land use categories that are scarcely protected and not subject to sufficiently binding planning. Moreover, their functional structure and extension/quality depend on how the local and global market impact spatially complementary categories mostly affected by anthropic action, which instead undergo efficiently planned changes. From the fundamental paradigm of "ecological networks" the interests of the scientific world and local authorities have shifted to other fronts, losing that physical and conceptual cohesion inherent in "environmental continuity" as a value in itself. The availability of recent, very advanced data helps take stock of this issue and verify critical points on the national territory that have been unresolved for several decades, but still rife with potential for the application of models of international value.


    Gorenstein homological algebra, roughly speaking, which is a relative version of homological algebra with roots on one hand in commutative algebra and on the other hand in modular representation theory of finite groups, has been developed to a high level. We refer the reader to [6] for some basic knowledge. It is well known that a very natural and important way to study homological algebra is by extending the homological theory on the category of modules to one on the category of complexes. Based on this idea, Gorenstein homological theory of complexes concerned many scholars, see [5,8,12,20,22,27] and so on. In particular, Gorensein injective complexes were introduced and studied by Enochs and García Rozas in [5]. It was shown that for each nZ, a complex C is Gorenstein-injective if and only if Cn is a Gorenstein-injective module over an n-Gorenstein ring. Liu and Zhang proved that this result holds over left Noetherian rings [12]. These have been further developed by Yang [22], Yang and Liu [27], independently.

    Cartan-Eilenberg projective and injective complexes play important roles not only in the category of complexes but also in homotopy category of complexes, see [2,Sections 12.4 and 12.5]. In 2011, Enochs further studied Cartan-Eilenberg projective, injective and flat complexes, and meanwhile, introduced and investigated Cartan-Eilenberg Gorenstein-injective complexes in [4]. In particular, he proved that a complex X is Cartan-Eilenberg Gorenstein-injective if and only if Bn(X) and Hn(X) are Gorenstein-injective modules for all nZ. Recently, Cartan-Eilenberg complexes have attracted a lot of attention. For instance, Yang and Liang in [23,24] studied the notion of Cartan-Eilenberg Gorenstein projective and flat complexes. For a self-orthogonal class W of modules, Lu et al. studied Cartan-Eilenberg W-Gorenstein complexes and stability [17].

    The notion of m-complexes has important applications in theoretical physics, quantum theory and representation theory of algebras, and has concerned many authors, see [1,7,9,11,14,15,16,19,25,26,28] and so on. For instance, Bahiraei, Gillespie, Iyama, Yang and their collaborators [1,9,11,19,26,28] investigate the homotopy category and the derived category of m-complexes. Yang and Wang in [28] introduce the notions of dg-projective and dg-injective m-complexes, and study the existence of homotopy resolutions for m-complexes. Estrada investigates the notion of Gorenstein-injective m-complexes in [7]. Recently, Lu et al. in [13,15,16] investigate the notions of Cartan-Eilenberg m-complexes and Gorenstein m-complexes. For a self-orthogonal subcategory W of an abelian category A, it is shown that a Cartan-Eilenberg injective m-complex can be divided into direct sums of an injective m-complex and an injetive grade module, a m-complex G is Gorenstein-injective if and only if G is a m-complex consisting of Gorenstein modules which improves a result of Estrada in [7].

    The motivation of this paper comes from the following incomplete diagram of related notions:

    The main purpose of the present paper is to define and investigate Cartan-Eilenberg Gorenstein injective m-complexes such that the above diagram will be completed. We establish the following result which gives a relationship between a Cartan-Eilenberg Gorenstein-injective m-complex and the corresponding level modules, cycle modules, boundary modules and homology modules.

    Theorem 1.1. (= Theorem 3.8) Let G be a m-complex. Then the following conditions are equivalent:

    (1) G is a Cartan-Eilenberg Gorenstein-injective m-complex.

    (2) Gn, Ztn(G), Btn(G) and Htn(G) are Gorenstein-injective modules for each nZ and t=1,2,,m.

    The stability of Gorenstein categories, initiated by Sather-Wagstaff et al. in [18], is an important research subject, and has also been considered by Ding, Liu, Lu, Xu and so on, see [17,21]. In this paper, as an application of Theorem 1.1 we prove a stability result for Cartan-Eilenberg Gorenstein-injective m-complexes, see Theorem 4.2.

    We conclude this section by summarizing the contents of this paper. Section 2 contains necessary notation and definitions for use throughout this paper. In section 3, we mainly give an equivalent characterization of Cartan-Eilenberg Gorenstein-injective m-complexes. An application of Theorem 1.1 is given in Section 4.

    In what follows, we will use the abbreviation 'CE' for Cartan-Eilenberg.

    In this section we recall some necessary notation and definitions. Throughout the paper, R denotes an associative ring with an identity and by the term "module" we always mean a left R-module. For two modules M and N, we will let HomR(M,N) denote the group of morphisms from M to N. ExtiR for i0 denotes the groups we get from the right derived functor of Hom.

    A m-complex X (m2) is a sequence of left R-modules

    dXn+2Xn+1dXn+1XndXnXn1dXn1

    satisfying dm=dXn+1dXn+2dXn+m=0 for any nZ. That is, composing any m-consecutive morphisms gives 0. So a 2-complex is a chain complex in the usual sense. We use dlnX to denote dXnl+1dXn1dXn. A chain map or simply map f:XY of m-complexes is a collection of morphisms fn:XnYn making all the rectangles commute. In this way we get a category of m-complexes of left R-modules, denoted by Cm(R), whose objects are m-complexes and whose morphisms are chain maps. This is an abelian category having enough projectives and injectives. Let C and D be m-complexes. We use HomCm(R)(C,D) to denote the abelian group of morphisms from C to D and ExtiCm(R)(C,D) for i0 to denote the groups we get from the right derived functor of Hom. In particular, we use Hom(C,D) to denote the abelian group of morphisms from C to D whenever m=2.

    Unless stated otherwise, m-complexes will always the m-complexes of left R-modules. For a m-complex X, there are m1 choices for homology. Indeed for t=1,,m, we define Ztn(X)=Ker(dnt+1dn1dn), Btn(X)=Im(dn+1dn+2dn+t), and Htn(X)=Ztn(X)/Bmtn(X) the amplitude homology modules of X. In particular, we have

    Z1n(X)=Kerdn,Zmn(X)=Xn

    and

    B1n(X)=Imdn+1,Bmn(X)=0.

    We say X is m-exact, or just exact, if Htn(X)=0 for all n and t. Given a left R-module A, we define m-complexes Dtn(A) for t=1,,m as follows: Dtn(A) consists of A in degrees n,n1,,nt+1, all joined by identity morphisms, and 0 in every other degree.

    Two chain maps f,g:XY are called chain homotopic, or simply homotopic if there exists a collection of morphisms {sn:XnYn+m1} such that

    gnfn=dm1sn+dm2sn1d++sn(m1)dm1=m1i=0d(m1)isnidi,  n.

    If f and g are homotopic, then we write fg. We call a chain map f null homotopic if f0. There exists an additive category Km(R), called the homotopy category of m-complexes, whose objects are the same as those of Cm(R) and whose Hom sets are the equivalence classes of Hom sets in Cm(R). An isomorphism in Km(R) is called a homotopy equivalence. We define the shift functor Θ:Cm(R)Cm(R) by

    Θ(X)i=Xi1anddΘ(X)i=dXi1

    for X=(Xi,dXi)Cm(R). The m-complex Θ(ΘX) is denoted Θ2X and inductively we define ΘnX for all nZ. This induces the shift functor Θ:Km(R)Km(R) which is a triangle functor.

    Recall from [6,Definition 10.1.1] that a left R-module M is called Gorenstein-injective if there exists an exact sequence of injective left R-modules

    E1E0E1

    with MKer(E0E1) and which remains exact after applying HomR(I,) for any injective left R-module I.

    A complex I is injective if and only if I is exact with Zn(I) injective in R-Mod for each nZ. A complex X is said to be Gorenstein-injective if there exists an exact sequence of injective complexes

    E1E0E1

    with XKer(E0E1) and which remains exact after applying Hom(I,) for any injective complex I [8,Definition 3.2.1].

    A complex I is said to be CE injective if I,Z(I),B(I) and H(I) are complexes consisting of injective modules [4,Definition 3.1]. A sequence of complexes

    C1C0C1

    is CE exact if C1C0C1 and Z(C1)Z(C0)Z(C1) are exact by [4,Lemma 5.2 and Definition 5.3]. Eonchs also introduced and studied the concept of CE Gorenstein-injective complexes, see [4,Definition 8.4]. A complex G is said to be CE Gorenstein-injective, if there exists a CE exact sequence

    I=I1I0I1

    of CE injective complexes such that G=Ker(I0I1) and the sequence remains exact when Hom(E,) is applied to it for any CE injective complex E.

    Remark 2.1. (1) For any injective module E, E is a Gorenstein-injective module and 0E1E0 is an injective complex.

    (2) For any Gorenstein-injective module E, 0E0 is a Gorenstein-injective complex by [27,Proposition 2.8].

    (3) According to [4,Theorem 8.5] and [27,Proposition 2.8], any injective complex is CE Gorenstein-injective, and any CE Gorenstein-injective complex is Gorenstein-injective.

    (4) Note that strongly Gorenstein-injective modules are Gorenstein-injective by [3,Definition 2.1]. The Gorenstein injective modules are not necessarily injective by [3,Example 2.5]. Thus Gorenstein-injective complexes and CE Gorenstein-injective complexes are not necessarily injective by [4,Theorem 8.5] and [27,Proposition 2.8].

    In this section, we will give an equivalent characterization on Cartan-Eilenberg Gorenstein-injective m-complexes. To this end, we first give some preparations.

    Definition 3.1. ([16,Definition 1]) A sequence of m-complexes

    C1C0C1

    is said to be CE exact if

    (1) C1nC0nC1n,

    (2) Ztn(C1)Ztn(C0)Ztn(C1),

    (3) Btn(C1)Btn(C0)Btn(C1),

    (4) C1/Ztn(C1)C0/Ztn(C0)C1/Ztn(C1),

    (5) C1/Btn(C1)C0/Btn(C0)C1/Btn(C1),

    (6) Htn(C1)Htn(C0)Htn(C1) are all exact for t=1,,m and nZ. $

    Remark 3.2. ([16,Remark 1]) Obviously, in the above definition, exactness of (1) and (2)(or (1) and (3), or (1) and (4), or (2) and (3), or (2) and (5), or (3) and (4), or (3) and (5), or (4) and (5)) implies exactness of all (1)(6).

    Definition 3.3. ([16,Remark 2]) A m-complex I is called CE CE if In,Ztn(I),Btn(I),Htn(I) are injective left R-modules for nZ and t=1,,m.

    The following lemma palys an important role in the proof of Theorem 3.8.

    Lemma 3.4. ([16,Proposition 3] and [15,Proposition 4.1]) Let I be a m-complex. Then

    (1) I is injective if and only if I=nZDmn+m1(En), where En is an injective module for each nZ.

    (2) I is CE injective if and only if I can be divided into direct sums I=II, where I is an injective m-complex and I is a graded module with In injective modules. Specifically,

    I=(nZDmn+m1(Bm1n(I)))(nZD1n+m1(H1n(I))).

    Here, we give the notion of CE Gorenstein-injective objects in the category of m-complexes.

    Definition 3.5. A m-complex G is said to be CE Gorenstein-injective, if there exists a CE exact sequence

    I=I1I0I1

    of CE injective m-complexes such that

    (1) G=Ker(I0I1);

    (2) the sequence remains exact when HomCm(R)(E,) is applied to it for any CE injective m-complex V.

    In this case, I is called a complete CE injective resolution of G.

    Remark 3.6. (1) It is clear that any CE injective m-complex is CE Gorenstein-injective.

    (2) Any CE Gorenstein-injective m-complex is Gorenstein-injective.

    (3) Take m=2. Then CE Gorenstein-injective m-complexes are CE Gorenstein-injective complexes.

    The following lemma is used to prove Theorem 3.8.

    Lemma 3.7. ([25,Lemma 2.2]) For any module M, X,YCm(R) and nZ,i=1,2,,m, we have the following natural isomorphisms.

    (1) HomCm(R)(Dmn(M),Y)HomR(M,Yn).

    (2) HomCm(R)(X,Dmn+m1(M))HomR(Xn,M).

    (3) HomCm(R)(Din(M),Y)HomR(M,Zin(Y)).

    (4) HomCm(R)(X,Din(M))HomR(Xn(i1)/Bin(i1)(Y),M).

    (5) Ext1Cm(R)(Dmn(M),Y)Ext1R(M,Yn).

    (6) Ext1Cm(R)(X,Dmn+m1(M))Ext1R(Xn,M).

    (7) If Y is m-exact, then Ext1Cm(R)(Din(M),Y)Ext1R(M,Zin(Y)).

    (8) If X is m-exact, then Ext1Cm(R)(X,Din(M))Ext1R(Xn(i1)/Bin(i1)(X),M).

    Now, we give the following result which gives an equivalent characterization of m-complexes which extends the corresponding result [4,Theorem 8.5] to the setting of m-complexes.

    Theorem 3.8. Let G be a m-complex. Then the following conditions are equivalent:

    (1) G is a CE Gorenstein-injective m-complex.

    (2) Gn, Ztn(G), Btn(G) and Htn(G) are Gorenstein-injective modules for each nZ and t=1,2,,m.

    In this case, Gn/Ztn(G) and Gn/Btn(G) are Gorenstein-injective modules.

    Proof. (1) (2). Suppose that

    I=I1I0I1

    is a complete CE injective resolution of G such that G=Ker(I0I1). Then, there is an exact sequence of injective modules

    I1nI0nI1n

    such that Gn=Ker(I0nI1n) for all nZ by Definition 3.3. For any injective module E, Dmn(E) is a CE injective m-complex for each nZ by Definition 3.3 again. It follows from Lemma 3.7 that there is a natural isomorphism

    HomCm(R)(Dmn(E),Ii)HomR(E,Iin)

    for all i,nZ. Then the following commutative diagram with the first row exact

    yields that the lower row is exact. Hence, Gn is a Gorenstein-injective module.

    We notice that Ztn(Ii)=Iin for t=m. So we only to show that Ztn(G) is a Gorenstein-injective module for t=1,2,,m1. Since I is CE exact, then, for any nZ and t=1,2,,m1, the sequence

    Ztn(I)=Ztn(I1)Ztn(I0)Ztn(I1)

    is exact with Ztn(Ii) injective module since Ii is CE injective for each iZ.

    For any injective module E, Dtn(E) is a CE injective m-complex for each nZ and t=1,2,,m, and there is a natural isomorphism HomCm(R)(Dtn(E),Ii)HomR(E,Ztn(Ii)) by Lemma 3.7 for each iZ. Applying HomCm(R)(Dtn(E),) to the sequence I, we obtain that Ztn(I) is HomR(E,) exact. Note that Ztn(G)=Ker(Ztn(I0)Ztn(I1)). Thus Ztn(G) is a Gorenstein-injective module.

    Meanwhile, there are exact sequences of modules

    0Ztn(G)GnBtnt(G)0

    and

    0Bmtn(G)Ztn(G)Htn(G)0

    for all nZ and t=1,2,,m. Then we get that Btnt(G) and Htn(G) are Gorenstein-injective by [10,Theorem 2.6].

    (2) (1). For a fixed nZ, there exist the following exact sequences of modules

    0Zm1n(G)GnBm1nm+1(G)00Zm2n(G)Zm1n(G)Bm1nm+2(G)00Z1n(G)Z2n(G)Bm1n1(G)00Bm1n(G)Z1n(G)H1n(G)0

    with Bm1ni(G) and H1n(G) Gorenstein-injective modules for i=0,1,2,,m1.

    Suppose Ei is a complete injective resolution of Bm1n(G), and Fn is a complete injective resolution of H1n(G) respectively, i=0,1,2,,m1, nZ. In an iterative way, we can construct a complete injective resolution of Gn+m

    EnEn1Enm+1Fn

    by the Horseshoe Lemma.

    Put

    Iln=(EnEn1Enm+1Fn)l=EnlEn1lEnm+1lFnl

    and define

    dn:IlnIln1
    (xn,xn1,,xnm+2,xnm+1,yn)(xn1,xn2,,xnm+1,0,0),

    for any (xn,xn1,,xnm+2,xnm+1,yn)Iln. Then (Il,dn) is a m-complex and GnKer(IlnIl1n). It is easily seen that Il is a CE injective m-complex for all lZ and G=Ker(IlIl1). For any nZ and t=1,2,,m,

    Ztn(I)=Ztn(Il+1)Ztn(Il)Ztn(Il1)

    is a complete injective resolution of Ztn(G), and

    In=Il+1nIlnIl1n

    is a complete injective resolution of Gn, so they both are exact. Hence, we can get that

    I=Il+1IlIl1

    is CE exact by Remark 3.2.

    It remains to prove that, for any CE injective m-complex I, I is still exact when HomCm(R)(I,) applied to it. However, it suffices to prove that the assertion holds when we pick I particularly as I=Dmn(E) and I=D1n(E) for any injective module E and all nZ by Lemma 3.4. Note that In and Ztn(I) are complete injective resolutions, hence from Lemma 3.7, the desired result follows.

    Moreover, if G is a CE Gorenstein-injective m-complex. then Gn/Ztn(G) and Gn/Btn(G) are also Gorenstein-injective modules by [10,Theorem 2.6].

    Take m=2 in Theorem 3.8. We obtain the following corollary which is a main result of [4].

    Corollary 3.9. [4,Theorem 8.5] Let G be a complex. Then the following conditions are equivalent:

    (1) G is a CE Gorenstein-injective complex.

    (2) Gn, Zn(G), Bn(G) and Hn(G) are Gorenstein-injective modules for each nZ.

    Take m=3 in Theorem 3.8. We obtain the following result.

    Corollary 3.10. Let G be a 3-complex. Then the following conditions are equivalent:

    (1) G is a CE Gorenstein-injective 3-complex.

    (2) Gn, Z1n(G), B1n(G) , H1n(G), Z2n(G), B2n(G) and H2n(G) are Gorenstein-injective modules for each nZ.

    The following result establishes a relationship between an m-exact m-complex and its cycle modules under certain hypotheses.

    Proposition 3.11. Let C be a m-exact m-complex with HomR(D1n(E),C) exact for any injective module E. Then C is a Gorenstein-injective complex if and only if Ztn(C) is a Gorenstein-injective module for each nZ and t=1,2,,m.

    Proof. () Let C be a Gorenstein-injective m-complex. There exists an exact sequence of injective m-complexes

    I=I1f1I0f0I1f1 (★)

    with C=Kerf0 such that I is HomCm(R)(E,) exact for any injective m-complex E. We also have Ker(fi) is m-exact for all iZ since Ker(f0)=C and Ii are m-exact.

    Applying HomCm(R)(Dtn(R),) to the sequence (), there is a commutative diagram:

    with the upper exact since Ext1Cm(R)(Dtn(R),Kerfi)=0 by Lemma 3.7. Then the second row is exact. Thus we obtain an exact sequence of injective modules

    Ztn(I1)Ztn(I0)Ztn(I1) (★★)

    with Ztn(C)Ker(Ztn(I0)Zn(I1)). So we only need to show that HomR(E,) leave the sequence () exact for any injective module E.

    Let E be an injective module and g:EZtn(C) be a morphism of modules. Since HomR(D1n(E),C) is m-exact, there exists a morphism f:ECn+mt such that the following diagram:

    commutes.

    Note that I is HomCm(R)(Dmn+mt(E),) exact since Dmn+mt(E) is an injective m-complex by [15,Proporition 4.1 and Corollary 4.4]. For the exact sequence

    0Kerf1I1C0,

    there is an exact sequence

    0HomCm(R)(Dmn+mt(E),Kerf1)HomCm(R)(Dmn+mt(E),I1)HomCm(R)(Dmn+mt(E),C)0.

    It follows from Lemma 3.7 that

    0HomR(E,(Kerf1)n+mt)HomR(E,I1n+mt)HomR(E,Cn+mt)0

    is exact. Thus, for fHomR(E,Cn+mt), there exists a morphism h:EI1n+mt such that f=hρ. Therefore, we obtain the following commutative diagram:

    where

    σ=dI1n+1dI1n+mt1dI1n+mt:I1n+mtZtn(I1),
    π=dCn+1dCn+mt1dCn+mt:Cn+mtZtn(C),

    and

    ϕ:Ztn(I1)Ztn(C)

    is induced by the morphism I1C, which means g=ϕσh. We notice that

    σhHomR(E,Ztn(I1)).

    Then

    0HomR(E,Ztn(Kerf1))HomR(E,Ztn(I1))HomR(E,Ztn(C))0

    is exact. Similarly, we can prove that

    0HomR(E,Ztn(Kerfi))HomR(E,Ztn(Ii))HomR(E,Ztn(Kerfi+1))0

    is exact. Hence, HomR(E,) leaves the sequence () exact. This completes the proof.

    () Note that there is an exact sequence

    0Zmtn+mt(C)CnZtn(C)0

    and Zmtn+mt(C) and Ztn(C) are Gorenstein-injective modules. Then Cn is a Gorenstein-injective module for each nZ by [10,Theorem 2.6], as desired.

    As an immediately consequence of Proposition 3.11, we establish the following Corollary which appears in [22,Theorem 4].

    Corollary 3.12. Let C be an exact complex with HomR(E,C) exact for any injective module E. Then C is a Gorenstein-injective complex if and only if Zn(C) is a Gorenstein-injective module for each nZ.

    In this section, as an application of Theorem 3.8, we investigate the stability of CE Gorenstein-injective categories of CE Gorenstein-injective m-complexes. That is, we show that an iteration of the procedure used to define the CE Gorenstein-injective m-complexes yields exactly the CE Gorenstein-injective m-complexes. We first introduce the notion of two-degree CE Gorenstein-injective m-complexes.

    Definition 4.1. A m-complex C is said to be two-degree CE Gorenstein-injective if there is a CE exact sequence of CE Gorenstein-injective m-complexes

    G=G1G0G1

    such that CKer(G0G1) and the functors HomCm(R)(H,) leave G exact whenever H is a CE Gorenstein-injective m-complex. In this case, G is called a complete CE Gorenstein-injective resolution of C.

    It is obvious that any CE Gorenstein-injective m-complex is two-degree CE Gorenstein-injective. In the following, we prove that two-degree CE Gorenstein-injective m-complexes are CE Gorenstein-injective.

    Theorem 4.2. Let C be a m-complex. Then the following statements are equivalent:

    (1) C is two-degree CE Gorenstein-injective.

    (2) C is a CE Gorenstein-injective.

    Proof. (2)(1) is trivial.

    (1)(2). We need to show that Gn, Ztn(G), Btn(G) and Htn(G) are Gorenstein-injective modules for each nZ and t=1,2,,m by Theorem 3.8. By (1), there exists a CE exact sequence of CE Gorenstein-injective m-complexes

    G=G1G0G1 (†)

    with C=Ker(G0G1) and such that the functor HomCm(R)(H,) leave G exact whenever H is a CE Gorenstein-injective m-complex. Then there is an exact sequence of Gorenstein-injective modules

    Gn=G1nG0nG1n

    with Cn=Ker(G0nG1n) for all nZ by Theorem 3.8.

    Let E be an injective module. Then Dmn(E) is a CE Gorenstein-injective m-complex for each nZ by Theorem 3.8. We apply the functor HomCm(R)(Dmn(E),) to the sequence (), we get the following exact sequence

    HomR(E,G1n)HomR(E,G0n)HomR(E,G1n)

    by Lemma 3.7. Thus, Cn is a Gorenstein-injective module for each nZ by Corollary 3.12.

    There also exists an exact sequence

    Ztn(G)=Ztn(G1)Ztn(G0)Ztn(G1)

    of Gorenstein-injective modules with Ztn(C)=Ker(Ztn(G0)Ztn(G1)) for all nZ and t=1,2,,m.

    We notice that, for any injective module E, Dtn(E) is CE Gorenstein-injective m-complexes for each nZ and t=1,2,,m by Theorem 3.8. Applying the functor HomCm(R)(Dtn(E),) to the sequence (), we get the following exact sequence

    HomR(E,Ztn(G1))HomR(E,Ztn(G0))HomR(E,Ztn(G1))

    by Lemma 3.7. Hence, Ztn(C) is a Gorenstein-injective module for each nZ by Corollary 3.12 again.

    Meanwhile, there are exact sequences of modules

    0Ztn(G)GnBtnt(G)0

    and

    0Bmtn(G)Ztn(G)Htn(G)0

    for all nZ and t=1,2,,m. Then we get that Btnt(G) and Htn(G) are Gorenstein-injective by [10,Theorem 2.6]. Therefore, C is a CE Gorenstein-injective m-complexes using Theorem 3.8 again.

    Take m=2, as a consequence of Theorem 4.2, we obtain the following corollary which establishes the stability for CE Gorenstein-injective complexes.

    Corollary 4.3. Let C be a complex. Then the following statements are equivalent:

    (1) C is a two-degree CE Gorenstein-injective complex.

    (2) C is a CE Gorenstein-injective complex.

    Take m=3, as a consequence of Theorem 4.2, we obtain the following corollary.

    Corollary 4.4. Let C be a 3-complex. Then the following statements are equivalent:

    (1) C is two-degree CE Gorenstein-injective 3-complex.

    (2) C is a CE Gorenstein-injective 3-complex.

    We can construct the following example by Corollary 3.10 and Corollary 4.4.

    Example 4.5. The 3-complex

    H=:000G1G1G000

    is CE Gorenstein-injective if and only if G is a Gorenstein-injective module if and only if H is two-degree CE Gorenstein-injective.

    This work was partially supported by National Natural Science Foundation of China (No. 12061061), Innovation Team Projec of Northwest Minzu University (No. 1110130131), and first-rate discipline of Northwest Minzu University. We would like to thank the referee for a careful reading of the paper and for many useful comments and suggestions.

    All authors declare no conflicts of interest.



    [1] Hudson WE (1991) Landscape Linkages and Biodiversity, Island Press USA, 195.
    [2] Noss RF (1991) Landscape Connectivity, Different functions at different scales, Landscape linkages and biodiversity, Island Press, 27-40.
    [3] IUCN (1993) Parks for life, Workshop Ⅲ.9, Corridors, transition zones and buffers: tools for enhancing the effectiveness of protected areas, IUCN, Gland Switzerland.
    [4] Ahern J (1994) Greenways as ecological networks in rural areas, Landscape planning and ecological networks, Elsevier, Amsterdam, 159-177.
    [5] Bennet G (1994) Conserving Europes Natural Heritage: Towards a European Ecological Network, Graham Bennett, Graham & Trotman, Great Britain, 254.
    [6] Forman RTT (1995) Some general principles of landscape and regional ecology. Landscape Ecol 10: 133-142.
    [7] Jongman RHG (1995) Nature conservation planning in Europe, developing ecological networks. Landscape Urban Plan 32: 169-183.
    [8] Collinge SK (1996) Ecological consequences of habitat fragmentation, implications for landscape architecture and planning. Landscape Urban Plan 36: 59-77.
    [9] Romano B (1999) Planning and Environmental Continuity. Urbanistica 112: 156-160.
    [10] Jongman RHG, Bouwma, I.M, Griffioen, et al. (2011) The Pan European Ecological Network: PEEN. Landscape Ecol 26: 311-326.
    [11] Battisti C (2011) Ecological network planning - from paradigms to design and back: a cautionary note. J Land Use Sci 8: 1-9.
    [12] EEA (2011) Landscape fragmentation in Europe. EEA-FOEN, 92.
    [13] Ragni B (2009) RERU, Rete Ecologica Regionale dell'Umbria. Petruzzi, 241.
    [14] Montanari I, Carati M, Costantino, R, et al. (2010) Qualità ecologica, l'approccio emiliano-romagnolo. Ecoscienza 3: 56-59.
    [15] Lombardi M, Giunti M, Castelli C (2014) La rete ecologica toscana: aspetti metodologici e applicativi. Ri-Vista XXII(1), 90-101.
    [16] Malcevschi S, Lazzarini M (2013) Tecniche e metodi per la realizzazione della Rete Ecologica Regionale. Regione Lombardia, 240.
    [17] Frontoni E, Mancino A, Zingaretti P, et al. (2014) SIT-REM: an interoperable and interactive web geographic information system for fauna, flora and plant landscape data management. Int J Geo-Inf 3: 853-867.
    [18] Olson DM, Dinerstein E (2002) The Global 200: Priority Ecoregions for Global Conservation. Annals of the Missouri Botanical Garden 89: 199-224.
    [19] Jongman R HG, Külvik M, Kristiansen I (2004) European ecological networks and greenways. Landscape Urban Plan 68: 305-319.
    [20] Scolozzi R, Morri E, Santolini R (2012) Delphi-based change assessment in ecosystem service values to support strategic spatial planning in Italian landscapes. Ecol Indic 21: 134-144.
    [21] Sargolini M (2013) Urban Landscapes, Environmental Networks and Quality of Life. Springer, Milano, 177.
    [22] Larson LR, Keith SJ, Fernandez M, et al. (2016) Ecosystem services and urban greenways: What's the public's perspective? Ecosys Ser 22: 111-116.
    [23] Leone F, Zoppi C (2016) Conservation Measures and Loss of Ecosystem Services: A Study Concerning the Sardinian Natura 2000 Network. Sustainability 8.
    [24] Pungetti G, Romano B (2004) Planning the future landscapes between nature and culture, Ecological Networks and Greenways, Cambridge University Press, UK, 107-127.
    [25] Hennig EI. Schwick C, Soukup T, et al. (2015) Multi-scale analysis of urban sprawl in Europe: Towards a European de-sprawling strategy. Land Use Policy 49: 483-498.
    [26] Epifani R, Amato F, Murgante B, et al. (2017) Quantitative Measure of Habitat Quality to Support the Implementation of Sustainable Urban Planning Measures. Computational Science and Its Applications - ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science, Springer, Cham, 10409: 585-600.
    [27] Zanfi F (2013) The città abusiva in contemporary southern Italy: illegal building and prospects for change. Urban Stud 50: 3428-3445.
    [28] Romano B, Zullo F, Marucci A, et al. (2018) Vintage Urban Planning in Italy: Land Management with the Tools of the Mid-Twentieth Century. Sustainability 10: 4125.
    [29] Bossard, M, Feranec J, Otahel J (2000) Corine Land Cover Technical Guide, EEA Technical report 40: 105.
    [30] Büttner G (2014) CORINE Land Cover and Land Cover Change Products., Land Use and Land Cover Mapping in Europe, Springer, Dordrecht, 55-74.
    [31] Battisti C (2003) Habitat fragmentation, fauna and ecological network planning: Toward a theoretical conceptual framework. Ital J Zool 70: 241-247.
    [32] Boitani L, Falcucci A, Maiorano L, et al. (2007) Ecological Networks as Conceptual Frameworks or Operational Tools in Conservation. Conserv Biol 21: 1414-1422.
    [33] Romano B, Zullo F (2012) Landscape fragmentation in Italy. Indices implementation to support territorial policies. Planning Support Tools: Policy analysis, Implementation and Evaluation, Franco Angeli, 399-414.
    [34] Malcevschi S (2011) Reti ecologiche polivalenti ed alcune considerazioni sui sistemi eco-territoriali. Territorio 58: 54-60.
    [35] European Commission (2006) Urban Sprawl in Europe: the Ignored Challenge, EEA Report 10, 2006, 60.
    [36] The Worldwatch Institute (2007) State of the World, Our Urban Future. Norton, NY, 250.
    [37] Ewing RH (2008) Characteristics, Causes, and Effects of Sprawl: A Literature Review. Urban Ecology Springer, Boston, MA. 519-535.
    [38] Pileri P, Maggi M (2010) Sustainable planning? First results in land uptakes in rural, natural and protected areas: the Lombardia case study (Italy). J Land Use Sci 5: 105-122.
    [39] Munafò M, Salvati L, Zitti M (2013) Estimating soil sealing rate at national level—Italy as a case study. Ecol Indic 26: 137-140.
    [40] Romano B, Zullo F (2013) Models of urban land use in Europe: assessment tools and criticalities. Int J Agri Environ Infor S 4: 80-97.
    [41] ISPRA (2017) Consumo di suolo, dinamiche territoriali e servizi ecosistemici. Rapporto 2017. ISPRA, Roma, 186.
    [42] ISTAT (2017) Forme, livelli e dinamiche dell'urbanizzazione in Italia, ISTAT, 350.
    [43] Romano B, Zullo F, Fiorini L, et al. (2017) Land transformation of Italy due to half a century of urbanisation. Land Use Policy 67: 387-400.
    [44] Agostini A (1930) Il problema dei rimboschimenti in Italia, Libreria del Littorio. Roma.
    [45] Romano D (1986) I rimboschimenti nella politica forestale italiana. Monti e Boschi 37: 7-12.
    [46] Lazzarini A (2002) Disboscamento montano e politiche territoriali. Alpi e Appennini dal Settecento al Duemila. Franco Angeli, Milano. 608.
    [47] Tasser E, Walde J, Tappeiner U, et al. (2007) Land-use changes and natural reforestation in the Eastern Central Alps. Agriculture. Ecosyst Environ 118: 115-129.
    [48] D'Ippolito A, Ferrari E, Iovino F, et al. (2013) Reforestation and land use change in a drainage basin of southern Italy. iForest - Biogeo For 6: 175-182.
    [49] Paolinelli G (2015) Crosscutting Issues in Treating the Fragmentation of Ecosystems and Landscapes. Nature Policies and Landscape Policies. Urban and Landscape Perspectives, Springer, Cham. 18: 283-290.
    [50] Marucci A, Zullo F, Morri E, et al. (2017) Spatial Methods to Measure Natura 2000 Sites Insularization in Italy. ICCSA 2017, Part Ⅳ, LNCS 10407: 437-450.
    [51] Salvati L, Biasi R, Carlucci M, et al. (2015) Forest transition and urban growth: exploring latent dynamics (1936-2006) in Rome, Italy, using a geographically weighted regression and implications for coastal forest conservation. Rendiconti Lincei 26: 577-585
    [52] Vizzarri M, Tognetti R, Marchetti M (2015) Forest Ecosystem Services: Issues and Challenges for Biodiversity, Conservation, and Management in Italy. Forests 6: 1810-1838.
    [53] Corona P, Chirici G, Travaglini D (2004) Forest ecotone survey by line intersect sampling. Can J Forest Res 34: 1776-1783.
    [54] Falcucci A, Maiorano L, Boitani L (2007) Changes in land-use/land-cover patterns in Italy and their implications for biodiversity conservation. Landscape Ecol 22: 617-631.
    [55] Orsi F, Geneletti D, Borsdorf A (2013) Mapping wildness for protected area management: A methodological approach and application to the Dolomites UNESCO World Heritage Site (Italy). Landscape Urban Plan 120: 1-15.
    [56] Petrillo PL, Di Bella O, Di Palo N (2015) The UNESCO World Heritage Convention and the Enhancement of Rural Vine-Growing Landscapes. Cultural Heritage and Value Creation. Springer, Cham. 127-169.
    [57] Meier K, Kuusemets V, Luig J, et al. (2005) Riparian buffer zones as elements of ecological networks: Case study on Parnassius mnemosyne distribution in Estonia. Ecol Eng 24: 531-537.
    [58] Theobald DM, Reed SE, Fields K, et al. (2012) Connecting natural landscapes using a landscape permeability model to prioritize conservation activities in the United States. Conservation Letters 5: 123-133.
    [59] Hepcan S, Hepcan CC, Bouwma IM, et al. (2009) Ecological networks as a new approach for nature conservation in Turkey: A case study of İzmir Province. Landscape Urban Plan 90: 143-154.
    [60] Deodatus F, Kruhlov I, Protsenko L, et al. (2013) Creation of ecological corridors in the Ukrainian Carpathians. The Carpathians: Integrating Nature and Society Towards Sustainability, Springer, Berlin, Heidelberg, 701-717.
    [61] Izakovičová Z, Świąder M (2017) Building Ecological Networks in Slovakia And Poland. Ekológia (Bratislava) 36: 303-322.
    [62] Kozieł M, Michalczuk W, Jędrzejewski W, et al. (2010) Protection of ecological corridors in spatial planning documents in Poland implementation problems. Europa, XXI: 77-89.
    [63] Gambino R (2003) APE, Appennino Parco d'Europa. Ministero dell'Ambiente e della Tutela del Territorio, Alinea Firenze. 240.
    [64] Gambino R, Romano B (2004) Territorial strategies and environmental continuity in mountain systems, the case of the Apennines (Italy), Managing Mountain Protected Areas: challenges and responses for the 21st Century, IUCN, Andromeda, 66-77.
    [65] Saura S, Santini L, Rondinini C (2015) Connectivity of the global network of protected areas. Biod Res 22: 199-211.
    [66] Saura S, Bertzky B, Bastin L, et al. (2018) Protected area connectivity: Shortfalls in global targets and country-level priorities. Biol Conserv 219: 53-67.
    [67] Saura S, Bertzky B, Bastin L, et al. (2019) Global trends in protected area connectivity from 2010 to 2018. Biol Conserv 238: 108183.
    [68] Thrush, SF, Hewitt, JE, Lohrer, et al. (2013) When small changes matter: the role of cross‐scale interactions between habitat and ecological connectivity in recovery. Ecol Appl 23: 226-238.
    [69] La Point S, Balkenhol N, Hale J, et al. (2015) Ecological connectivity research in urban areas. Funct Ecol 29: 868-878.
    [70] Rosengarten F (2009) The contemporary relevance of Gramsci's views on Italy's "Southern question". Perspectives on Gramsci, Politics, culture and social theory, Routledge, UK, 134-144.
    [71] Romano B, Fiorini L, Di Dato C, et al. (2020) Latitudinal Gradient in Urban Pressure and Socio Environmental Quality: The "Peninsula Effect" in Italy. Land 9: 126.
    [72] Forman RTT, Sperling D, Bissonette JA, et al. (2002) Road Ecology: Science and Solutions. Island Press, 479.
    [73] Jaeger AG, Schwarz-von Raumer HG, Esswein H, et al. (2007) Time series of landscape fragmentation caused by transportation infrastructure and urban development: a case study from Baden-Württemberg, Germany. Ecol Soc 12: 22.
    [74] Karlson M, Mörtberg U, Balfors B (2014) Road ecology in environmental impact assessment. Environ Impact Assess 48: 10-19.
    [75] Zurlini G, Amadio V, Rossi O (1999) A Landscape Approach to Biodiversity and Biological Health Planning: The Map of Italian. Ecosyst Health 5: 294-311
    [76] Maiorano L, Falcucci A, Boitani L (2006) Gap analysis of terrestrial vertebrates in Italy: Priorities for conservation planning in a human dominated landscape. Biol Conserv 133: 455-473.
    [77] Rondinini C, Boitani L (2007) Systematic Conservation Planning and the Cost of Tackling Conservation Conflicts with Large Carnivores in Italy. Conserv Biol 21: 1455-1462.
    [78] Peruzzi L, Conti F, Bartolucci F (2014) An inventory of vascular plants endemic to Italy. Phytotaxa 168: 1-75.
    [79] Baldwin RF, Reed SE, McRae BH, et al. (2012) Connectivity Restoration in Large Landscapes: Modeling Landscape Condition and Ecological Flows. Ecol Restor 30: 274-279.
    [80] Foster ML, Humphrey SR (1995) Use of Highway Underpasses by Florida Panthers and Other Wildlife. Wildlife Soc B 23: 95-100.
    [81] Rosillon F (2004) Valley landscape management: the context of the "river contract" in the Semois Valley (Belgium). Landscape Res 29: 413-422.
    [82] Brun A (2010) Les contrats de rivière en France: un outil de gestion concertée de la ressource en local. La Découverte 498.
    [83] Peng J, Zhao H, Liu Y (2017) Urban ecological corridors construction: A review. Acta Ecologica Sinica 37: 23-30.
    [84] Marucci A, Zullo F, Fiorini L, et al. (2019) The role of infrastructural barriers and gaps on Natura 2000 functionality in Italy A case study on Umbria region. Rendiconti Lincei 30: 223-235.
  • This article has been cited by:

    1. Wenjun Guo, Honghui Jia, Renyu Zhao, Strongly VW-Gorenstein N-complexes, 2024, 1306-6048, 10.24330/ieja.1559027
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4217) PDF downloads(6) Cited by(1)

Figures and Tables

Figures(8)  /  Tables(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog