Questions tagged [torsion-theory]

For questions about torsion theories in abelian categories and related concepts.

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Subclasses of abelian categories, that are closed under extensions and their complement as well and a construction of torsion pairs using them

Let $\mathcal{A}$ be a length abelian category. A subclass $\mathcal{S}$ of $\mathcal{A}$ is called super-closed if $0\in \mathcal{S}$, $\mathcal{S}$ is closed under extensions and $\mathcal{A}\...
4
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1answer
136 views

On the relation between two definitions of torsion functors

Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module we consider the sub-$R$-modules $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{...
6
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1answer
466 views

On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
2
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1answer
395 views

Torsion theory for quasi-coherent sheaves?

In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies: (1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$. (2) If $Hom(T,F)=0$ for ...
2
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0answers
219 views

Intuition for hereditary torsion theories

I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts: Definition. A torsion theory $(\mathcal T,\...
4
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2answers
341 views

Torsion pairs and projective dimension

Let $A$ denote an algebra finite dimensional, basic, and connected algebra over a algebraically closed field $K$. We denote by $mod A$ the abelian category whose objects are finitely generated right ...
2
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2answers
293 views

torsion theories localizing the base ring to the same ring

If two torsion theories on a ring localize the ring to the same extension ring, I can find no reason that their "meet" in the lattice of torsion theories must also localize to the same ring. I cannot ...
9
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1answer
1k views

Definition of torsion sheaf on reducible spaces

I need to discuss torsion-free sheaves on reduced, but possibly reducible spaces. Here "torsion" means "element is annihilated by a non-zero-divisor". The standard references (EGA, Hartshorne, ...) ...
6
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1answer
527 views

(Co)localization of the derived category

Let me start saying that a similar question can be stated for general locally Noetherian Grothendieck categories but I state it for categories of modules as it is simpler. So we fix a right Noetherian ...
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1answer
597 views

Looking for reference talking about torsion theory on coherent sheaves on projective space

I am looking for reference talking about how torsion theory play roles in algebraic geometry. I will be really happy to see some concrete examples. Say, talking about torsion theory in $Coh(P^{1})$. ...
12
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2answers
3k views

What is the relationship between t-structure and Torsion pair?

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 $(\...