About Loewy length of syzygy












1














Suppose $Lambda$ is an Artin algebra with $DeclareMathOperator{rad}{rad}rad^3Lambda=0 $, and $ M $ any finite $Lambda$-module with projective dimension finite. Proof that $Omega M $ has Loewy length at most 2.



This question is from a paper: On the finitistic global dimension conjecture for Artin algebras - Igusa and Todorov. It is a corollary.



My idea is: we can guarantee that $rad^3M=0$ because $rad M=MradLambda$, and
take the projective cover of $M$:
$$Omega M rightarrow P rightarrow M rightarrow 0$$



$ΩM=ker(f)$ is a submodule of $rad(P)$ (where $P$ is the first term of projective cover). And thinking use exactness of this for make something.










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  • 1




    Any own ideas? Where did you get this question from?
    – A. Pongrácz
    Nov 27 '18 at 18:20












  • This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
    – Júlio César M. Marques
    Nov 27 '18 at 18:29


















1














Suppose $Lambda$ is an Artin algebra with $DeclareMathOperator{rad}{rad}rad^3Lambda=0 $, and $ M $ any finite $Lambda$-module with projective dimension finite. Proof that $Omega M $ has Loewy length at most 2.



This question is from a paper: On the finitistic global dimension conjecture for Artin algebras - Igusa and Todorov. It is a corollary.



My idea is: we can guarantee that $rad^3M=0$ because $rad M=MradLambda$, and
take the projective cover of $M$:
$$Omega M rightarrow P rightarrow M rightarrow 0$$



$ΩM=ker(f)$ is a submodule of $rad(P)$ (where $P$ is the first term of projective cover). And thinking use exactness of this for make something.










share|cite|improve this question




















  • 1




    Any own ideas? Where did you get this question from?
    – A. Pongrácz
    Nov 27 '18 at 18:20












  • This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
    – Júlio César M. Marques
    Nov 27 '18 at 18:29
















1












1








1







Suppose $Lambda$ is an Artin algebra with $DeclareMathOperator{rad}{rad}rad^3Lambda=0 $, and $ M $ any finite $Lambda$-module with projective dimension finite. Proof that $Omega M $ has Loewy length at most 2.



This question is from a paper: On the finitistic global dimension conjecture for Artin algebras - Igusa and Todorov. It is a corollary.



My idea is: we can guarantee that $rad^3M=0$ because $rad M=MradLambda$, and
take the projective cover of $M$:
$$Omega M rightarrow P rightarrow M rightarrow 0$$



$ΩM=ker(f)$ is a submodule of $rad(P)$ (where $P$ is the first term of projective cover). And thinking use exactness of this for make something.










share|cite|improve this question















Suppose $Lambda$ is an Artin algebra with $DeclareMathOperator{rad}{rad}rad^3Lambda=0 $, and $ M $ any finite $Lambda$-module with projective dimension finite. Proof that $Omega M $ has Loewy length at most 2.



This question is from a paper: On the finitistic global dimension conjecture for Artin algebras - Igusa and Todorov. It is a corollary.



My idea is: we can guarantee that $rad^3M=0$ because $rad M=MradLambda$, and
take the projective cover of $M$:
$$Omega M rightarrow P rightarrow M rightarrow 0$$



$ΩM=ker(f)$ is a submodule of $rad(P)$ (where $P$ is the first term of projective cover). And thinking use exactness of this for make something.







abstract-algebra modules representation-theory






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edited Nov 28 '18 at 18:52









Xander Henderson

14.1k103554




14.1k103554










asked Nov 27 '18 at 18:15









Júlio César M. Marques

565




565








  • 1




    Any own ideas? Where did you get this question from?
    – A. Pongrácz
    Nov 27 '18 at 18:20












  • This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
    – Júlio César M. Marques
    Nov 27 '18 at 18:29
















  • 1




    Any own ideas? Where did you get this question from?
    – A. Pongrácz
    Nov 27 '18 at 18:20












  • This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
    – Júlio César M. Marques
    Nov 27 '18 at 18:29










1




1




Any own ideas? Where did you get this question from?
– A. Pongrácz
Nov 27 '18 at 18:20






Any own ideas? Where did you get this question from?
– A. Pongrácz
Nov 27 '18 at 18:20














This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
– Júlio César M. Marques
Nov 27 '18 at 18:29






This is from a paper: On the finitistic global dimension conjecture for Artin algebras. My ideia is: we can guarantee that $ rad^3 M=0 $, and $Omega M $ is a submodule of $ rad(P)$ (where P is the first term of projective cover). Use exactness of this for make something.
– Júlio César M. Marques
Nov 27 '18 at 18:29












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I discuss this question to my professor today, the answer is simple.



Take $M$ a f.g. $Lambda$-module with $pdim M < infty$.



Since $MradLambda = radM$, and we have $Omega M subseteq rad P$ (and is a submodule), then $$rad^2Omega M = rad(radOmega M)=radOmega M radLambda=Omega M rad^2Lambda leq rad P rad^2Lambda=Prad^3Lambda=0$$
Then, the Loewy length of $Omega M$ is less or equal 2.






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    I discuss this question to my professor today, the answer is simple.



    Take $M$ a f.g. $Lambda$-module with $pdim M < infty$.



    Since $MradLambda = radM$, and we have $Omega M subseteq rad P$ (and is a submodule), then $$rad^2Omega M = rad(radOmega M)=radOmega M radLambda=Omega M rad^2Lambda leq rad P rad^2Lambda=Prad^3Lambda=0$$
    Then, the Loewy length of $Omega M$ is less or equal 2.






    share|cite|improve this answer


























      2














      I discuss this question to my professor today, the answer is simple.



      Take $M$ a f.g. $Lambda$-module with $pdim M < infty$.



      Since $MradLambda = radM$, and we have $Omega M subseteq rad P$ (and is a submodule), then $$rad^2Omega M = rad(radOmega M)=radOmega M radLambda=Omega M rad^2Lambda leq rad P rad^2Lambda=Prad^3Lambda=0$$
      Then, the Loewy length of $Omega M$ is less or equal 2.






      share|cite|improve this answer
























        2












        2








        2






        I discuss this question to my professor today, the answer is simple.



        Take $M$ a f.g. $Lambda$-module with $pdim M < infty$.



        Since $MradLambda = radM$, and we have $Omega M subseteq rad P$ (and is a submodule), then $$rad^2Omega M = rad(radOmega M)=radOmega M radLambda=Omega M rad^2Lambda leq rad P rad^2Lambda=Prad^3Lambda=0$$
        Then, the Loewy length of $Omega M$ is less or equal 2.






        share|cite|improve this answer












        I discuss this question to my professor today, the answer is simple.



        Take $M$ a f.g. $Lambda$-module with $pdim M < infty$.



        Since $MradLambda = radM$, and we have $Omega M subseteq rad P$ (and is a submodule), then $$rad^2Omega M = rad(radOmega M)=radOmega M radLambda=Omega M rad^2Lambda leq rad P rad^2Lambda=Prad^3Lambda=0$$
        Then, the Loewy length of $Omega M$ is less or equal 2.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 28 '18 at 23:46









        Júlio César M. Marques

        565




        565






























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