I've recently been learning about the quotient of an abelian category by a Serre (thick) subcategory, specifically in the context of module categories.

A thought occurred to me today about classifying Noetherian objects. I've taken a modules course at the honours level so I'm relatively familiar with Noetherian modules, and I know there is a generalisation to arbitrary categories of Noetherian objects, so I was wondering if there is any way of describing the Noetherian objects of a quotient category in terms of the Noetherian objects of the original category.

For example, if $R$ is some integral domain and we take the quotient $\mathcal Q:=R\text{-Mod}/R\text{-Tor}$, where $R\text{-Tor}$ is the Serre subcategory of torsion $R$-modules (a torsion module being one in which every element has non-zero annihilator).

Is there a nice way of describing the Noetherian objects of $\mathcal Q$ in terms of the Noetherian $R$-modules in $R\text{-Mod}$? I'm prepared to accept that there is no nice way of describing them, given the nature of the definition of the quotient, but I'm curious nonetheless.

  • $\begingroup$ Hint: in this case, $\mathcal{Q}$ can equivalently be realised as a sub-category of $R\mathrm{-mod.}$, namely, the full sub-category consisting of torsion-free $R$-modules. In more categorical terms, for the example at hand, there is a torsion theory https://ncatlab.org/nlab/show/torsion+theory lurking in the background; for such, it should be easy to describe the noetherian objects. $\endgroup$ – Ben Jul 21 '17 at 15:23

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