# Deligne-Mumford Stacks and exactness of products

Let me start by saying that this is probably a very ingenuous question but I still need to digest most definitions before being able to formulate a more precise question.

In this paper: http://arxiv.org/abs/0902.4016 one can read the following result:

Theorem. Let $S$ be a separated Noetherian scheme and $X/S$ be a separated Deligne-Mumford stack of ﬁnite type. Assume

(1) The coarse moduli space of $X$ is a scheme;

(2) $X$ is covered by Zariski open substacks each of which admits a ﬁnite ´etale cover from a scheme.

Then $QC(X)$ satisﬁes $AB.4^*$-$n$ for some positive integer $n$.

Here $AB.4^*$-$n$ refers to a weak version of the correspondent axiom listed by Grothendieck for Abelian categories ($AB.4^*$ says that products are exact, $AB.4^*$-$n$ says that the the $m$-th derived functor of the product is trivial for $m>n$).

I do not have experience with stacks but I have a result that gives conditions on the localization of an $AB.4^*$-$k$ Grothendieck category for being $AB.4^*$-$n$ (where $n=k+d$, with $d$ a suitable notion of relative Gabriel dimension). Now, I would like to know if this applies here to give the above theorem as an easy application.

My question is the following: is it possible, under the hypotheses of the theorem, to view the category $QC(X)$ as a localization of an $AB.4^*$-$k$ Grothendieck category (possibly $QC(S)$?) in some natural way? If so, is there a natural way to describe the kernel of the localization functor?

As I said this may be trivial, trivially false, or not trivial at all. Depending on comments and answers I hope to edit the question in order to ask more. Thanks in advance!

EDIT: Of course any Grothendieck category is the localization of a category of modules. But above I'm asking for a localization of a better and somehow "closer" category. The ideal situation would be that $QC(X)$ is the localization (which for me, in this context, means quotient category over a hereditary torsion class) of a locally Noetherian, $AB.4^*$-$n$, Grothendieck category (with some more technical properties) and such that the hereditary torsion class which forms the kernel of the localization is itself a Grothendieck category with finite (or co-finite) Gabriel dimension (in the sense that its Gabriel filtration (or the Gabriel filtration of the quotient category) stops after finitely many steps).

EDIT2: Let me remark that one is interested in knowing that a Grothendieck category has the $AB.4^*$-$n$ property essentially because it allows for an inductive construction of injective resolutions of unbounded complexes (which has non-trivial consequences on the derived category). This procedure is based on the notion of homotopy colimit (see the paper quoted above or the classical paper by Bockstedt and Neeman [Homotopy limits in triangulated categories. Compositio Math. 86 (1993). 209-234])

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## migrated from math.stackexchange.comJul 8 '13 at 17:29

This question came from our site for people studying math at any level and professionals in related fields.

Perhaps your question is more suitable for MO? – BenjaLim Jul 8 '13 at 15:05
I do not know if this is really research in the sense that it may be common knowledge for people working in the area, I do not know. Anyway the question can be migrated, I have no problems with it – Simone Virili Jul 8 '13 at 15:11