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2022 Volume 47 Issue 2
Article Contents

WANG Xiaomei, WANG Zhanping. Gorenstein Injective Phantom Morphism[J]. Journal of Southwest China Normal University(Natural Science Edition), 2022, 47(2): 11-15. doi: 10.13718/j.cnki.xsxb.2022.02.003
Citation: WANG Xiaomei, WANG Zhanping. Gorenstein Injective Phantom Morphism[J]. Journal of Southwest China Normal University(Natural Science Edition), 2022, 47(2): 11-15. doi: 10.13718/j.cnki.xsxb.2022.02.003

Gorenstein Injective Phantom Morphism

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  • Received Date: 23/03/2021
    Available Online: 20/02/2022
  • MSC: O153.3

  • In this thesis, the concepts of Gorenstein injective Phantom morphism and higher Gorenstein injective Phantom morphism have been introduced, some properties of Gorenstein injective Phantom morphism been studied, and the class of Gorenstein injective Phantom morphisms been closed under direct product and ME-extensions in H(R). It is proved that a morphism φ: XX′ of R-modules is higher Gorenstein injective Phantom morphism if and only if there is a Gorenstein injective resolution of φ in H(R) such that its n-th cosyzygy is a Gorenstein injective Phantom morphism.
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Gorenstein Injective Phantom Morphism

Abstract: In this thesis, the concepts of Gorenstein injective Phantom morphism and higher Gorenstein injective Phantom morphism have been introduced, some properties of Gorenstein injective Phantom morphism been studied, and the class of Gorenstein injective Phantom morphisms been closed under direct product and ME-extensions in H(R). It is proved that a morphism φ: XX′ of R-modules is higher Gorenstein injective Phantom morphism if and only if there is a Gorenstein injective resolution of φ in H(R) such that its n-th cosyzygy is a Gorenstein injective Phantom morphism.

  • Phantom态射起源于代数拓扑CW-复形之间的映射[1]. 文献[2]首次定义了三角范畴上的Phantom态射. 在文献[3-6]中,Phantom态射的理论在有限群环的稳定模范畴中得到了发展. 文献[7]利用有限表示模和Tor1R(-,-)将Phantom态射的概念推广到了任意环的模范畴上. 文献[8]引入了n-Phantom态射和n-Ext-Phantom态射的概念,证明了:在R-Mod中,fMNn-Phantom态射(n>1)当且仅当存在态射范畴中的正合序列0→kn-1pn-2pn-1→…→p0f→0,其中pi是投射的,kn-1是Phantom态射. 文献[9]定义了Gorenstein平坦Phantom态射和高维Gorenstein平坦Phantom态射,并研究了其同调性质,证明了:R-模态射φXX′是高维Gorenstein平坦Phantom态射当且仅当在态射范畴中存在φ的Gorenstein平坦分解,使得它的n次合冲是Gorenstein平坦Phantom态射. 与其相关的研究课题还得到了许多其他有意义的结论(参见文献[10-11]).

    受以上结论的启发,本文主要研究Gorenstein内射Phantom态射.

1.   预备知识
  • 本文中所提到的环均指有单位元的结合环,模均指左R-模. R-Mod(Mod-R)表示左(右)R-模范畴.

    定义1  用H(R)表示R-Mod的态射范畴,其中:

    (a) H(R)中的对象是左R-模同态;

    (b) H(R)中从$ (A\xrightarrow{{{f}_{1}}}B)$$ (A^{\prime} \text{ }\xrightarrow{{{f}_{2}}}B^{\prime} )$的态射为R-Mod中态射的对子(g1g2)

    使得图

    交换.

    由文献[12]可得,态射范畴H(R)是局部有限表示的Grothendieck范畴.

2.   Gorenstein内射Phantom态射
  • 文献[13]在一般环上引入了Gorenstein内射模的概念.

    定义2[13]  如果存在内射模的正合列…→I1I0I0I1→…,使得M$ \cong $Ker(I0I0),且对任意的内射模I,该序列在函子HomR(I,-)作用下仍是正合的,则称R-模M是Gorenstein内射模.

    Gorenstein内射模的类记为ΓI.

    由此我们引入Gorenstein内射Phantom态射的概念.

    定义3  如果对任意的内射左R-模E,ExtR1(Eφ)=0,其中ExtR1(Eφ):ExtR1(EA)→ExtR1(EB),则称R-模态射φAB是Gorenstein内射Phantom,简称为ΓI-Phantom态射.

    ΓI-Phantom态射的类记为ΦΓI. 由定义可知:ΦΓIR-Mod的理想,并包含所有R-模同态MG,其中G为Gorenstein内射模.

    命题1  在H(R)中,ΓI-Phantom态射的类关于直积封闭.

      设(fiMiNi)iI是一族ΓI-Phantom态射. 下证ΠiI:ΠiIMi→ΠiINiΓI-Phantom态射. 对任意的内射模E,考虑交换图

    因为Ext1(Efi)=0,所以ΠiIExtn(Efi)=0,即Ext1(E,ΠiIfi)=0. 故ΠiI:ΠiIMi→ΠiINiΓI-Phantom态射.

    命题2  设R是环,fMNR-Mod中的ΓΦ-Phantom态射当且仅当f+N+M+是Mod-R中的ΓI-Phantom态射.

      对任意的内射左R-模E,考虑交换图

    由文献[14]知,αβ是同构的,所以Tor1(Nf)+=0当且仅当Ext1(Ef+)=0. 因此fMNR-Mod中的ΓΦ-Phantom态射当且仅当f+N+M+是Mod-R中的ΓI-Phantom态射.

    Λ表示R-模的所有短正合序列0→XYZ→0的类,使得对任意的内射左R-模E,当函子HomR(E,-)作用时仍保持正合. 由文献[15]的引理1.1知,Λ是Ext的加法子函子.

    命题3  对R-模态射φAB,以下条件等价:

    (ⅰ) φΓI-Phantom态射;

    (ⅱ) 如果AIA的态射包,那么对任意的内射左R-模E,沿着φ的推出

    是HomR(E,-)-正合的.

      (ⅰ)$ \Rightarrow $(ⅱ)  若AIA的态射包,则存在短正合序列ξ:0→AIC→0,其中I是内射. 又由态射φAB,则有推出图

    因为φABΓI-Phantom态射,所以对任意的内射左R-模E,Ext1(Eφ)=0. 因此有交换图

    这就表明0→BGC→0是HomR(E,-)-正合的.

    (ⅱ)$ \Rightarrow $(ⅰ)  假设(ⅱ)成立,则对任意的内射左R-模E,考虑交换图

    δExt1(Eφ)=0. 因为δ是单的,所以Ext1(Eφ)=0,即φΓI-Phantom态射.

3.   高维Gorenstein内射Phantom态射
  • 下面引入高维Gorenstein内射Phantom态射的概念,即n-Gorenstein内射Phantom态射(n$ {{\mathbb{N}}_{\text{+}}}$).

    定义4  如果对任意的内射左R-模E,ExtRn(Eφ)=0,其中ExtRn(Eφ):ExtRn(EA)→ExtRn(EB),则称R-模态射φABn-Gorenstein内射Phantom,简称为n-ΓI-Phantom态射.

    n-ΓI-Phantom态射的类记为Φn-ΓI.

    注1  当n=1时,1-ΓI-Phantom态射就叫作Gorenstein内射Phantom态射,即ΓI-Phantom态射.

    众所周知,态射范畴H(R)等价于三角矩阵环$ \left( \begin{matrix} R & 0 \\ R & R \\ \end{matrix} \right)$上的模范畴. 由文献[16]的定理2.6知,H(R)中的对象$ \mathit{\Gamma }={{G}_{1}}\xrightarrow{f}{{G}_{2}}$是Gorenstein内射当且仅当f是满的,且G1G2和Ker f是Gorenstein内射左R-模.

    定理1  设R是一个环且n>1,对H(R)中的对象Ξ$ X\xrightarrow{\varphi }X^{\prime} $,以下条件等价:

    (ⅰ) φn-ΓI-Phantom态射;

    (ⅱ) 若有H(R)中的任意正合序列0→ΞΓ0Γ1→…→Γn-2Kn-1→0,其中Γi是Gorenstein内射,则Kn-1ΓI-Phantom态射;

    (ⅲ) 若有H(R)中的任意正合序列0→ΞI0I1→…→In-2Kn-1→0,其中Ii是内射,则Kn-1ΓI-Phantom态射;

    (ⅳ) 存在H(R)中的正合序列0→ΞI0I1→…→In-2Kn-1→0使得Ii是内射,则Kn-1ΓI-Phantom态射;

    (ⅴ) 存在H(R)中的正合序列0→ΞΓ0Γ1→…→Γn-2Kn-1→0使得Γi是Gorenstein内射,Kn-1ΓI-Phantom态射.

      (ⅰ)$ \Rightarrow $(ⅱ)  设任意i∈{0,1,…,n-1},Γi$ {{G}_{i}}\xrightarrow{{{\varphi }_{i}}}~G ^{\prime}{_{i}}$Kn-1$ {{K}_{n-1}}~\xrightarrow{{{k}_{n-1}}}~K ^{\prime}{_{n-1}}$,考虑在H(R)中的正合序列

    其中态射φiGiGi是Gorenstein内射. 令k1K1K1ΞΓ0的余核. 对任意的内射左R-模E,我们有行正合的交换图

    因为φi是Gorenstein内射态射,所以G0G0是Gorenstein内射模,即Extn-1(EG0)=0,Extn-1(EG0)=0. 又因为φn-ΓI-Phantom态射,所以对任意的内射左R-模E,Extn(Eφ)=0. 则αExtn-1(Ek1) =0. 由于α是单射,故Extn-1(Ek1)=0. 重复上述过程,可得Kn-1ΓI-Phantom态射.

    (ⅱ)$ \Rightarrow $(ⅲ)$ \Rightarrow $(ⅳ)$ \Rightarrow $(ⅴ)  显然.

    (ⅴ)$ \Rightarrow $(ⅰ)  考虑行正合的交换图

    使得对任意i∈{0,1,…,n-1},态射φiGiGi是Gorenstein内射,并且kn-1Kn-1Kn-1ΓI-Phantom态射. 因此对任意的内射左R-模E,Ext1(Ekn-1)=0. 令k1K1K1ΞΓ0的余核. 考虑行正合的交换图

    因为φn-2是Gorenstein内射态射,所以Gn-2Gn-2是Gorenstein内射模,即Ext2(EGn-2)=0,Ext2(EGn-2)=0. 则β是满射,所以Ext2(Ekn-2)β=0. Ext2(Ekn-2)=0. 重复上述过程,有Extn(Eφ) =0. 所以φn-ΓI-Phantom态射.

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