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If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-sheaf axioms" and then use

Theorem 1: the B-sheaf extends uninquely to the whole of X.

I was wondering if there's a similar thing for more general sites, and actually not just for sheaves on a given site but for stacks.

The question I'm really interested in is the following:

If one has a fibred category over Schemes (say Schemes over some fixed field with the fppf topology) and one wants to check that descent is effective, would it be sufficient to check it on some subcategory of schemes (using perhaps some vague analogue of Theorem 1)?

Thanks.

EDIT

For example one might want to construct the stack M of coherent sheaves on some scheme X. One way to do it is to define the functor which associates with each scheme S the groupoid of coherent sheaves on $S\times X$ flat over S $$ M(S) = \{ E \in Coh\ S\times X,\ E \text{ flat over S} \}.$$

Let's say I want to use a different characterization of $M(S)$, perhaps using Lemma 3.31 on page 82 of Huybrechts' Fourier-Mukai book.

My ignorance prevents me from knowing if the that lemma is valid for a general scheme S (no matter how nice my X might be). This is why I'd like to work over some nice subcategory of schemes (where the lemma is valid) and then extend.

The stack I'd be interested in defining would be a stack of perverse sheaves on X, where the matters would be a bit worse.

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Let's address the question you say is of real interest. If your fibered category satisfies the limit criterion for locally finite presentation then for effective descent it suffices to check over base schemes of finite presentation over (or ground field, or whatever). I am vague since you were vague about finiteness hypotheses on your fibered category, so it isn't a precise answer. This principle is implemented in a precise way and is used all the time by those who look under the hood. It is an important application of the massive theory of limits of schemes in EGA IV3, sections 8--12, 17. –  Boyarsky Jun 23 '10 at 0:23
    
Would it be too much to ask for an example (an easy one?) where the technique you describe is applied? –  babubba Jun 23 '10 at 8:35
    
@angoleirovero: please name a fibered category for which you want to know effective descent for the fppf topology, such as a stack of interest to you which is not a scheme. –  Boyarsky Jun 23 '10 at 8:57
    
@Boyarksy: I'll edit my question. –  babubba Jun 29 '10 at 18:44
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@angoleirovero: you mean "quasi-coherent and finitely presented", not "coherent". The general limit formalism implies that if $\{S_i\}$ is an inverse system of affine schemes with limit $S$ then $\varinjlim M(S_i) \rightarrow M(S)$ is an equivalence in a sense I hope is evident. If $S' \rightarrow S$ is an fppf cover then it arises from an fppf cover of some $S_i$ via base change. So if effective descent holds with $M$ for $S$ of finite type over $\mathbb{Z}$, it holds in general. (This example is crazy, since fppf descent for all quasi-coherent sheaves works directly on arbitrary schemes.) –  Boyarsky Jun 30 '10 at 5:03

1 Answer 1

up vote 4 down vote accepted

Let $S$ be your Grothendieck site. What you want is a subcategory $j:B \hookrightarrow S$ such that the Grothendieck topology of $S$ restricts to $B$ in the sense that every covering sieve of $b \in B$ can be refined by one coming from a family $(U_i \to b)_i$ with each $U_i \in B$, AND such that $j^*:Sh(B) \to Sh(S)$ is an equivalence of topoi. This is exactly what makes Theorem 1 work.

Concretely, you want the topology to restrict and every $s$ in $S$ to have a covering family $(b_i \to s)$ with $b_i \in B$, so that you can then say:

$$F(s):=\varprojlim \left(\prod_{i}F(b_i)\rightrightarrows \prod_{i,j}F(b_i \times_{s} b_j)\right).$$

However, you need to make sure this doesn't depend on the covering family you chose.

Suppose only that every $s$ in $S$ to have a covering family $(b_i \to s)$ with $b \in B$ and that the Grothendieck topology on $S$ restricts to $B$ in the sense described above. Then, since the Grothendieck topology is subcanonical, $s$ is a colimit of $b_i$s, hence $s \mapsto Hom(blank,s)$ embeds $S$ into $Sh(B)$ (note that the left-Kan extension of this embedding is precisely $j^*$, which is literally restriction).

I claim $j^*$ is fully-faithful. This is essentially because $Hom(j^*F,j^*G)$ for two sheaves on $S$ determines $Hom(F,G)$ since the value of $F(s)$ is determined by the value of $F$ on $b_i$s by the cover $(b_i \to s)$, by descent.

Now, if $F$ is a $B$-sheaf (i.e. an element of $Sh(B)$), then $j_*F(s)=Hom(j^*s,F).$ Hence, $$j_*j^*(F)(s)\cong Hom(s,j^*j_*F)\cong Hom(j^*s,j^*F)\cong Hom(s,F)\cong F(s),$$

by Yoneda, adjointness, and full an faithfulness.

Note also that $j^*j_*(G) \cong G$ for all $G \in Sh(B)$ pretty much by definition. Hence the adjoint pair $j_*,j^*$ is an equivalence.

So, what does this mean concretely? You need to find a subcategory $B$ of schemes such that

1.)every cover in the fppf topology of an element of $b \in B$ can be refined by one with domains in $B$ (at least you need to be able to find a family of morphisms whose SIEVE is in the topology GENERATED by the fppf pretopology)

2.) Every scheme can be covered by elements of $B$.

I'll leave it to you to find such a subcategory, as I don't know much AG.

P.S., everything I said will hold for stacks as well.

EDIT: Condition 2.) implies condition 1.), so this becomes simpler:

You just need a category subcategory $B$ of schemes such that every scheme can be covered by elements of $B$.

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I can't, for the life of me, figure out why this bit of Latex won't compile. I DIDN'T use "underleftarrow" anywhere, and I used backticks.... anyway, the thing that's not displaying is essentially THIS upload.wikimedia.org/math/c/d/6/… (but with the correct variables and intersection replaced with fibred-product). –  David Carchedi Jun 23 '10 at 1:01
    
I assume you meant "...can be refined by one with \emph{domains} in $B$..." in your condition 1 above, anyway it seems to me that condition 2 implies 1: if you have a covering of $b$ by arbitrary objects of $S$, then by 2 you can find a covering of each one of those by elements of $B$, and putting these together will give you a refinement with domains in $B$... right? –  Mattia Talpo Jun 23 '10 at 22:39
    
@Mattia: The domain/codomain thing was of course a typo due to the fact that I answered this at 3 in the morning. I'll fix it. As to your other comment, I totally agree. I should've made this simplification. Please let me also blame this on it having been 3 in the morning. –  David Carchedi Jun 23 '10 at 23:00
    
Ok thanks, I only asked because I thought I was missing something.. Oh, and I totally understand about late hours :) –  Mattia Talpo Jun 23 '10 at 23:19

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