Return to Question

I am away from Torsion theory in abelian category for some while. So It might be a stupid question.

The definition of torsion pair in category of modules is as follows:

Definition: A pair $\[(T,F)\]$ of full subcategories of $\[A-mod\]$ is called a torsion pair if following conditions hold:

1. $\[Hom_{A}(M,N)=0\]$ for all $\[M\epsilon T,N\epsilon F\]$

2. $\[Hom_{A}(M,-)|_{F}=0\Rightarrow M\epsilon T\]$

3. $\[Hom_{A}(-,N)|_{T}=0\Rightarrow N\epsilon F\]$

condition 2) and 3) means that the pair $\[(T,F)\]$ is maximal for $\[Hom_{A}(M,N)=0\]$

This definition is from the book elements of representation theory of associative algebras

I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from dimca sheaves in topology as follows:

A t-structure on a triangulated category $\[D\]$ consists in two strictly full subcategories: $\[D^{\leq 0},D^{\geq 0}\]$ such that the following condition holds:

1)$\[Hom(X,Y)=0\]$ if $\[X\epsilon D^{\leq 0},Y\epsilon D^{\geq 1}\]$

2. $\[D^{\leq 0}\subseteq D^{\leq 1},D^{\geq 1}\subseteq D^{\geq 0}\]$

3. For any $\[X\epsilon D\]$,there is a distinguished triangle:

$\[A\rightarrow X\rightarrow B\rightarrow A[+1],A\epsilon D^{\leq 0},B\epsilon D^{\geq 1}\]$

Although the axioms 3) for t-structures looks different to the axioms of torsion pairs. However, there is a propositions of torsion pairs establishing the similar formula:

Let a pair $\[(T,F)\]$ be a torsion pair in $\[A-mod\]$ , M is an A-module. Then there exists a short exact sequence: $\[0\rightarrow tM\rightarrow M\rightarrow M/tM\rightarrow 0,tM\epsilon T,M/tM\epsilon F\]$

where t is the idempotent radical(it behaves like radical of module)

#My question#

1. Is there any relationship between these two constructions?
2. Is there a torsion theories defined in triangulated category? If there exists, is it coincide with t-structures in triangulated category?
3. t-structures played important roles in reconstruction schemes(or go back to abelian category) from derived category. So, is torsion theory in abelian category playing the similar roles?(I suspected very much, so it might be stupid)

Thank you in advance!

I am away from Torsion theory in abelian category for some while. So it might be a stupid question.

The definition of a torsion pair in the category of modules is as follows:

Definition: A pair $(\mathcal T,\mathcal F)$ of full subcategories of $A-\mathrm{mod}$ is called a torsion pair if following conditions hold:

1. $\mathrm{Hom}_{A}(M,N)=0$ for all $M \in \mathcal T, N \in \mathcal F$.
2. $\mathrm{Hom}_{A}(M,-)|_{\mathcal F}=0 \Rightarrow M \in \mathcal T$.
3. $\mathrm{Hom}_{A}(-,N)|_{\mathcal T}=0 \Rightarrow N \in \mathcal F$.

Condition 2) and 3) means that the pair $(\mathcal T,\mathcal F)$ is maximal for $\mathrm{Hom}_{A}(M,N)=0$.

This definition is from the book elements of representation theory of associative algebras

I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from Dimca, Sheaves in topology as follows:

A t-structure on a triangulated category $\mathcal D$ consists in two strictly full subcategories: $\mathcal D^{\leq 0}, \mathcal D^{\geq 0}$ such that the following conditions hold:

1. $\mathrm{Hom}(X,Y)=0$ if $X \in \mathcal D^{\leq 0}, Y \in \mathcal D^{\geq 1}$.
2. $\mathcal D^{\leq 0} \subseteq \mathcal D^{\leq 1}, \mathcal D^{\geq 1} \subseteq \mathcal D^{\geq 0}$.
3. For any $X \in \mathcal D$, there is a distinguished triangle $$A\rightarrow X\rightarrow B\rightarrow A[+1], \qquad A \in \mathcal D^{\leq 0}, B \in \mathcal D^{\geq 1}.$$

Although the axiom 3) for t-structures looks different to the axioms of torsion pairs. However, there is a proposition of torsion pairs establishing the similar formula:

Let a pair $(\mathcal T,\mathcal F)$ be a torsion pair in $A-\mathrm{mod}$, and let $M$ be an $A$-module. Then there exists a short exact sequence $$0 \rightarrow tM \rightarrow M \rightarrow M/tM \rightarrow 0, \qquad tM \in \mathcal T, M/tM \in \mathcal F,$$ where $t$ is the idempotent radical  (it behaves like radical of module).

#My questions#

1. Is there any relationship between these two constructions?
2. Is there a definition of torsion theory in triangulated categories? If there exists, does it coincide with t-structures in triangulated categories?
3. t-structures played important roles in reconstruction schemes  (or go back to abelian category) from derived category. So, is torsion theory in abelian category playing similar roles?  (I suspected very much, so it might be stupid.)

Thank you in advance!

I am away from Torsion theory in abelian category for some while. So It might be a stupid question.

The definition of torsion pair in category of modules is as follows:

Definition: A pair $\[(T,F)\]$ of full subcategories of $\[A-mod\]$ is called a torsion pair if following conditions hold:

1. $\[Hom_{A}(M,N)=0\]$ for all $\[M\epsilon T,N\epsilon F\]$

2. $\[Hom_{A}(M,-)|_{F}=0\Rightarrow M\epsilon T\]$

3. $\[Hom_{A}(-,N)|_{T}=0\Rightarrow N\epsilon F\]$

condition 2) and 3) means that the pair $\[(T,F)\]$ is maximal for $\[Hom_{A}(M,N)=0\]$

This definition is from the book elements of representation theory of associative algebras

I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from dimca sheaves in topology as follows:

A t-structure on a triangulated category $\[D\]$ consists in two strictly full subcategories: $\[D^{\leq 0},D^{\geq 0}\]$ such that the following condition holds:

1)$\[Hom(X,Y)=0\]$ if $\[X\epsilon D^{\leq 0},Y\epsilon D^{\geq 1}\]$

2. $\[D^{\leq 0}\subseteq D^{\leq 1},D^{\geq 1}\subseteq D^{\geq 0}\]$

3. For any $\[X\epsilon D\]$,there is a distinguished triangle:

$\[A\rightarrow X\rightarrow B\rightarrow A[+1],A\epsilon D^{\leq 0},B\epsilon D^{\geq 1}\]$

Although the axioms 3) for t-structures looks different to the axioms of torsion pairs. However, there is a propositions of torsion pairs establishing the similar formula:

Let a pair $\[(T,F)\]$ be a torsion pair in $\[A-mod\]$ , M is an A-module. Then there exists a short exact sequence: $\[0\rightarrow tM\rightarrow M\rightarrow M/tM\rightarrow 0,tM\epsilon T,M/tM\epsilon F\]$

where t is the idempotent radical(it behaves like radical of module)

##My question##

1. Is there any relationship between these two constructions?
2. Is there a torsion theories defined in triangulated category? If there exists, is it coincide with t-structures in triangulated category?
3. t-structures played important roles in reconstruction schemes(or go back to abelian category) from derived category. So, is torsion theory in abelian category playing the similar roles?(I suspected very much, so it might be stupid)

Thank you in advance!

I am away from Torsion theory in abelian category for some while. So It might be a stupid question.

The definition of torsion pair in category of modules is as follows:

Definition: A pair $\[(T,F)\]$ of full subcategories of $\[A-mod\]$ is called a torsion pair if following conditions hold:

1. $\[Hom_{A}(M,N)=0\]$ for all $\[M\epsilon T,N\epsilon F\]$

2. $\[Hom_{A}(M,-)|_{F}=0\Rightarrow M\epsilon T\]$

3. $\[Hom_{A}(-,N)|_{T}=0\Rightarrow N\epsilon F\]$

condition 2) and 3) means that the pair $\[(T,F)\]$ is maximal for $\[Hom_{A}(M,N)=0\]$

This definition is from the book elements of representation theory of associative algebras

I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from dimca sheaves in topology as follows:

A t-structure on a triangulated category $\[D\]$ consists in two strictly full subcategories: $\[D^{\leq 0},D^{\geq 0}\]$ such that the following condition holds:

1)$\[Hom(X,Y)=0\]$ if $\[X\epsilon D^{\leq 0},Y\epsilon D^{\geq 1}\]$

2. $\[D^{\leq 0}\subseteq D^{\leq 1},D^{\geq 1}\subseteq D^{\geq 0}\]$

3. For any $\[X\epsilon D\]$,there is a distinguished triangle:

$\[A\rightarrow X\rightarrow B\rightarrow A[+1],A\epsilon D^{\leq 0},B\epsilon D^{\geq 1}\]$

Although the axioms 3) for t-structures looks different to the axioms of torsion pairs. However, there is a propositions of torsion pairs establishing the similar formula:

Let a pair $\[(T,F)\]$ be a torsion pair in $\[A-mod\]$ , M is an A-module. Then there exists a short exact sequence: $\[0\rightarrow tM\rightarrow M\rightarrow M/tM\rightarrow 0,tM\epsilon T,M/tM\epsilon F\]$

where t is the idempotent radical(it behaves like radical of module)

#My question#

1. Is there any relationship between these two constructions?
2. Is there a torsion theories defined in triangulated category? If there exists, is it coincide with t-structures in triangulated category?
3. t-structures played important roles in reconstruction schemes(or go back to abelian category) from derived category. So, is torsion theory in abelian category playing the similar roles?(I suspected very much, so it might be stupid)

Thank you in advance!