# How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation? Is there any reference for such staff?

My example of functors underlying this question is functors of the form $$\mathcal{E}xt^1_S(A,\mathbb{G}_m)$$ for $A$ some group scheme or even some fppf-sheaf of geometric meaning over some base scheme $S$, and $\mathcal{E}xt^1_S(,)$ denotes the 1st fppf extension sheaf in the category of abelian fppf-sheaves over $S$.

• What precisely do you mean by that Ext functor? Are you considering functors in the category of sheaves for some Grothendieck topology (fppf, etale, etc.), and if so, which one? Do you mean "extensions of group schemes"? Do you really want to look at some relative Picard functor of $A$ over $S$? At any rate, many such results follow directly from Sections 8 and 11 of EGA IV. – Jason Starr Jan 2 '15 at 14:38
• @JasonStarr: I mean the first fppf-sheaf extension functor in the category of abelian fppf-sheaves. In my case, $A$ is a log abelian variety over $S$ in Kato's sense, usually not a group scheme (although closely related to some abelian scheme). Thanks for the reference, I will try to find something useful. – Heer Jan 2 '15 at 14:52
• @JasonStarr: thanks for pointing out the ambiguity, I have reedited the question. – Heer Jan 2 '15 at 14:56
• @JasonStarr: In SGA 7-1 Exp. VII, 1.3.4, under the condition of 1.3.5, we could realize the category $EXT(A,G)$ as a full subcategory of $BITORSRIG(A,\mathrm{G}_m)$. So if the second category is locally of finite presentation, would it follows that the second is so too? – Heer Jan 2 '15 at 15:03
• The above question is stupid. What I meant should be: if some functor (relative Picard over $A$?) associated to $BITORSRIG(A,\mathrm{G}_m)$ is locally of finite presentation, would it follows that the functor $\mathcal{E}xt^1_S(A,\mathbb{G}_m)$ is so too? – Heer Jan 2 '15 at 15:22