Questions tagged [torsion-theory]
For questions about torsion theories in abelian categories and related concepts.
14 questions
15
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2
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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 $(\...
- 5,515
9
votes
1
answer
1k
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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
votes
1
answer
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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 $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
- 16.9k
6
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1
answer
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(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 ...
- 2,452
5
votes
2
answers
325
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Model of an elliptic curve with p-torsion
Suppose I have an elliptic curve $E$ defined over a number field $K$.
I know that if it has
a $2$ $K$-torsion, it has a model of the form:
$E: Y^2=X^3+aX^2+bX$
a $3$ $K$-torsion, it has a model of ...
- 637
5
votes
1
answer
172
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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{...
- 6,700
4
votes
2
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373
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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 ...
- 237
3
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1
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527
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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 ...
- 109
2
votes
2
answers
301
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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 ...
- 403
2
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0
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100
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Pairs of ideals in an abelian category similar to torsion pairs
Let $\mathcal{A}$ be an abelian category. In the context of my work I am considering pairs of ideals $(\mathcal{I}, \mathcal{J})$ in $\mathcal{A}$ with the following properties:
$\quad \mathcal{I} \...
- 696
2
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0
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141
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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}\...
- 696
2
votes
0
answers
252
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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,\...
- 935
1
vote
1
answer
646
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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 $Coh(P^{1})$.
...
- 5,515
1
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0
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When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory
Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under ...
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