What is a skyscraper sheaf? Even for just sheaves on topological spaces there seem to be various definitions that one can use and that are used in the literature. Here are a few:
- It is an abelian sheaf F whose stalks F_x are nonzero for finitely many points x if X.
- It is an abelian sheaf F whose stalks F_x are nonzero for finitely many closed points x of X.
- It is an abelian sheaf of the form i_{x,*}A where i_x : {x} —> X is the inclusion of a point into X and A is an abelian group.
- It is an abelian sheaf of the form i_{x,*}A where i_x : {x} —> X is the inclusion of a closed point into X and A is an abelian group.
Of course the exact definition of any mathematical object is subject to minor variations. In each paper you have to look carefully what the definition really is. Moreover, sometimes authors have “hidden” assumptions (for example when an algebraic geometry paper says “everything is over C” they probably mean everything is at least locally of finite type over C and maybe even that everything is a variety over C).
Coming back to skyscraper sheaves, I think that for the stacks project the most natural choice is the one where a skyscraper sheaf is a sheaf of the form i_{x, *}A for any point x of X. An advantage of this choice is that we can also define skyscraper sheaves in the category of sheaves of sets.
As an aside we note that a sheaf satisfying 1 is not necessarily a finite direct sum of abelian skyscraper sheaves (think of the combinatorial circle and a nontrivial local system).
For a topos Sh(C) we define a skyscraper sheaf to be any sheaf of the form p_*A where p is a point of the site C.
This has the perhaps unfortunate consequence that if X is a topological space then the topos Sh(X) may have skyscraper sheaves which are not skyscraper sheaves on X. Namely, the points of the topos Sh(X) correspond 1-1 with irreducible closed subsets of X, and hence there may be more points of the topos than points of the space. This does not happen for schemes, which have underlying sober topological spaces.
It also means that the logical choice, in the stacks project, for a skyscraper sheaf on the small etale site S_{etale} of a scheme S is to require it to be a sheaf of the form s_*A where s : Spec(k) —> S is a geometric point of S. The reason is that points of the small etale site of S do indeed correspond to geometric points of S.