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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 ring (non-commutative) and we let $Mod(R)$ be the category of right $R$-modules.

Recall the following theorem due to H. Krause (for its proof Brown's representability is used).

Theorem 1. Let $\mathcal T$ be a triangulated category with small coproducts which is well generated. Let $H : \mathcal T \to \mathcal A$ be a cohomological functor into an abelian category $\mathcal A$ which has small coproducts and exact $\alpha$-filtered colimits for some regular cardinal $\alpha$. Suppose also that $H$ preserves small coproducts. Then there exists an exact localization functor $L:\mathcal T \to\mathcal T$ such that for each object $X$ we have $LX=0$ if and only if $H(X[n])=0$ for all $n \in \mathbb Z$.

Fix now a hereditary torsion theory $\tau$ on $Mod(R)$. We go on defining in three steps an exact localization functor of the derived category $L_\tau:{ D}(R)\to { D}(R)$.

(1) Denote by $$H^n:{\bf D}(R)\to Mod (R)$$ the usual $n$-th cohomology, for every $n\in\mathbb Z$. It is clear that each $H^n(-)$ is cohomological and preserves coproducts.

(2) Fix a hereditary torsion theory $\tau$ on $Mod(R)$. The $\tau$-localization functor $$Q_\tau:Mod(R)\to \mathcal A_\tau=Mod(R)/\mathcal T_{\tau} ,$$ where $\mathcal T_{\tau}$ is the hereditary torsion class associated to $\tau$, is exact and preserves coproducts. Furthermore, $\mathcal A_\tau$ is a Grothendieck category and so it has small coproducts and exact colimits.

(3) For every $n\in\mathbb Z$ denote by $$H_\tau^n:{\bf D}(R)\to \mathcal A_\tau$$ the composition of the above two functors, that is $H_\tau^n(-)=Q_\tau H^n(-)$. By (1) and (2) one can easily derive that $H_\tau^n(-)$ is cohomological and preserves coproducts.

Now we have all the instruments to construct the localization functor $L_\tau(-)$:

Corollary. Let $\tau$ be a hereditary torsion theory on $Mod(R)$. Then there exists an exact localization functor $L_\tau:{\bf D}(R)\to {\bf D}(R)$ such that $L_\tau(X)=0$ if and only if the $n$-th cohomology of $X$ is $\tau$-torsion for every $n\in\mathbb Z$.

Proof. Consider the functor $\prod_{n\in\mathbb Z}H^n_\tau:{\bf D}(R)\to \prod_{n\in\mathbb Z}\mathcal A_\tau$. By the above discussion, this functor is a cohomological functor preserving coproducts from ${\bf D}(R)$ to a bicomplete abelian category with exact colimits. Let $X\in {\bf D}(R)$, then $\prod_{n\in\mathbb Z}H^n_\tau(X)=0$ if and only if $H_\tau^n(X)=0$ for every $n\in\mathbb Z$, if and only if $Q_\tau H^n(X)=0$ for every $n\in\mathbb Z$. This is equivalent to say that all the cohomologies of $X$ are in the kernel of $Q_\tau(-)$ that is, they are $\tau$-torsion. Now, to prove the existence of $L_\tau(-)$ it is enough to apply Theorem 1.


Question. In the above notation, is it possible to prove that for a given object $X\in{\bf D}(R)$, $L_\tau(X)$ belongs to the smallest localizing subcategory of ${\bf D}(R)$ containing $X$?


More specifically, consider an indecomposable injective module $E$ and let $\tau$ be the hereditary torsion theory cogenerated by $E$. In the commutative case, there is a unique prime ideal associated to $E$ and localizing with respect to $\tau$ is the same as localizing at that prime ideal. In particular, localized modules can be constructed as a direct limit of $R$-modules and this construction can be "lifted" to the derived category to answer positively to the above question.

In the non-commutative case, there is no prime ideal associate in general but we can suppose it if we suppose our ring to be right FBN (in the general case $E=E(C)$ with $C$ a cocritical module which can be chosen of the form $R/I$ with $I$ an irreducible ideal). But even in the case when the localization at $\tau$ is Ore, the localized modules are constructed as direct limits in the category of Abelian groups and after that they are induced with the structure of modules. This makes no sense (to me) in ${\bf D}(R)$.

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I think the answer is no in the following very simple example.

Let $R$ be the path algebra of the quiver $\bullet\rightarrow\bullet$ over a field $k$. Then since $R$ is hereditary and of finite representation type, the objects of ${\bf D}(R)$ are just coproducts of shifts of copies of the three indecomposable representations $[k\rightarrow0]$, $[k\rightarrow k]$ and $[0\rightarrow k]$.

Let $E$ be the injective representation $[k\rightarrow 0]$, so the $\tau$-torsion objects for the hereditary torsion theory $\tau$ cogenerated by $E$ are the representations of the form $[0\rightarrow V]$, and the localization functor $L_{\tau}$ just takes $[U\rightarrow V]$ to $[U\rightarrow 0]$.

In particular, if $X=[k\rightarrow k]$. then $L_{\tau}(X)=[k\rightarrow0]$, but this is not in the localizing subcategory generated by $X$, which contains only coproducts of shifts of copies of $X$.

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yes I agree, thanks for the nice "minimal" example. This happens essentially because there is a link between the indecomposable injective object which cogenerates $\tau$ and the other indecomposable injective. Now, suppose that you are in a semi-artinian Grothendieck category (just to make it simpler) and that, instead of being cogenerated by a single indecomposable injective, $\tau$ is cogenerated by an entire clique of indecomposable injectives (all the injectives that have non-trivial morphisms to or from a given indecomposable injective). Any thought in this more restricted setting? –  Simone Virili Oct 27 '13 at 11:53
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I suspect you may have mis-written the definition of "clique" (it's not terminology I'm familiar with in this context, so I'm not sure). But with the definition you've given, my example can be adjusted to the quiver $1\leftarrow 2\rightarrow 3$, the clique determined by the injective associated with vertex 1, and $X$ the representation $[0\leftarrow k\rightarrow k]$. –  Jeremy Rickard Oct 27 '13 at 13:31
    
yes you are right, it's not just what I have written. Again in a semi-artinian Grothendieck category, you have a link between two indecomposable injectives $E$ and $E′$ if $Hom(E,E′)\neq 0$ or $Hom(E′,E)\neq 0$. The Gabriel spectrum becomes a (undirected) graph where the edges are the links I have just defined. The clique of E is the connected component of E in this graph. Sorry for the mistake before... (this is not exactly the usual terminology of, say, Jategaonkar but I tried to give some natural notion of links and cliques in more "categorical" terms) –  Simone Virili Oct 27 '13 at 14:09
    
let me also say that it would be safe to assume that the category is also locally Noetherian (otherwise this notion of link is not sufficient...) –  Simone Virili Oct 27 '13 at 14:10
    
In all the examples I've thought of that satisfy your conditions, every object of the Grothendieck category has a unique decomposition as the direct sum of a torsion and a torsion-free object (in which case this property is also true of the derived category, and for any $X$, $L_{\tau}X$ is a direct summand of $X$, and therefore is in the localizing subcategory generated by $X$). Do you know examples where this is not true? –  Jeremy Rickard Oct 27 '13 at 16:40

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