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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 all $F\in \mathcal F$, then $T\in \mathcal T$.

(3) If $Hom(T,F)=0$ for all $T\in \mathcal T$, then $F\in \mathcal F$.

For modules over a commutative integral domain $A$, torsion modules and torsion free modules form such a pair.

For sheaves of $\mathcal O_X$-modules over an integral scheme, we can take locally torsion and locally torsion free $\mathcal O_X$-modules, i.e., torsion and torsion free respectively on each affine.

My question is : does this still hold for quasi-coherent sheaves on an integral scheme? In other words, do locally torsion and locally torsion free quasi-coherent sheaves still give a torsion theory on $QCoh(X)$?

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  • $\begingroup$ I guess that $\mathcal{T}$ and $\mathcal{F}$ are meant to be sub-collections of objects. $\endgroup$
    – YCor
    Commented May 18, 2015 at 14:55
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    $\begingroup$ for coherent sheaves on a noetherian scheme this is pretty easy to see: take any coherent module M and consider all its torsion subsheaves. By noetherianness there will be a maximal one Mt and the quotient M/Mt will be torsion free. (this is a general fact: for a noetherian abelian category, whenever you have a subcategory closed under quotients and extensions it gives rise to a torsion pair) $\endgroup$
    – bananastack
    Commented May 18, 2015 at 14:55
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    $\begingroup$ You need at least that $C$ has zero morphisms or else I don't know what $\text{Hom}(T, F) = 0$ means. Really you should restrict to something like abelian categories or else I doubt the notion is useful for much. $\endgroup$
    – Qiaochu Yuan
    Commented May 18, 2015 at 19:32

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Yes, it is still a torsion theory (by the way, you forget to mention the condition that each object $A$ can be represented as an extension $0 \to T(A) \to A \to F(A) \to 0$). To show you just take $T(A)$ to be the union of all torsion subsheaves of $A$. A similar technique is used in the paper Tarrío, Leovigildo Alonso, Ana Jeremías López, and María José Salorio. "Construction of 𝑡-structures and equivalences of derived categories." Transactions of the American Mathematical Society 355, no. 6 (2003): 2523-2543.

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