About Loewy length of syzygy
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
add a comment |
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
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
add a comment |
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
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
abstract-algebra modules representation-theory
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
add a comment |
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
add a comment |
1 Answer
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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.
add a comment |
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1 Answer
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1 Answer
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active
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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.
add a comment |
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.
add a comment |
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.
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.
answered Nov 28 '18 at 23:46
Júlio César M. Marques
565
565
add a comment |
add a comment |
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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