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4
votes
1answer
273 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
votes
1answer
166 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 ...
1
vote
0answers
145 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 ...
2
votes
2answers
281 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 ...
4
votes
1answer
435 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, ...) ...
4
votes
1answer
430 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 ...
1
vote
1answer
566 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 $Coh(P^{1})$. ...
9
votes
2answers
2k 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 ...