TITLE: Families of ideal sheaves: What's the correct definition?
QUESTION [10 upvotes]: I'm looking at Bridgeland's paper "Flops and Derived categories" and I got confused on what he meant by a family of ideal sheaves.
Let $Y$ be a scheme, and let $S$ be another scheme. A family of sheaves $\mathcal E_Y$ on $Y$ over $S$ is a sheaf on $S\times Y$, flat over $S$. Two such families are equivalent if they differ by tensoring by pullback of a line bundle on $S$.
When we talk about ideal sheaves $\mathcal{I}$ of $Y$, usually an inclusion to the structure sheaf $\mathcal{O}_Y$ is taken as part of the data. Now when we talk about a family of ideal sheaves, what do we really mean by that hidden part of the data? At least a family of sheaves which are flat over $S$, but there should be more. More specifically, when we say "let $\mathcal E_Y$ be a family of ideal sheaves on $Y$ over $S$", is there an inclusion of $\mathcal E_Y$ into $\mathcal O_{S\times Y}$ given part of the data?
(EDIT: according to MartinG's answer, this attempt of definition doesn't seem to be right, some functoriality is missing. Now let me replace my question by the following:


What is the functor $M_I(X)$? What's the definition and why is it functorial? (I'm guessing MartinG's suggestion is right, namely rank 1 sheaves with trivial determinant line bundle, but I'm not 100% sure.)

Why does $M_I(X)$ exist as a scheme under that definition? (In the paper he seems need this fact in a crucial way.)

What's the relation between $M_I(X)$ and $\text{Hilb}(X)$?



End of EDIT)

REPLY [2 votes]: Bridgeland uses another point of view on moduli spaces. His definition of a moduli functor (Def. 3.7) does not uses any assumption of flatness. Instead, he considers DERIVED restriction of the family object to fibers, and the condition which replaces flatness is that these derived restrictions are PURE in the t-structure he considers. If the t-structure is the usual one then this definition is equivalent to the standard definition (flatness = no higher derived pullbacks). 
The moduli space $M_I(X)$ of ideal sheaves is defined by the functor which associates to any scheme $S$ the set of (equivalence classes of) all sheaves on $S\times X$ which after (derived) restriction to fibers over $S$ are ABSTRACTLY isomorphic to sheaves of ideals. On a contrary, the Hilbert scheme functor associates to a scheme $S$ the set of sheaves on $S\times X$ with a FIXED EMBEDDING into $O_{S\times X}$ (again up to an appropriate embedding). Forgetting the embedding we obtain a morphism $Hilb(X) \to M_I(X)$. 
Of course, if $S$ is a closed point, and $I$ is isomorphic to an ideal sheaf, then there is a unique (up to a constant) embedding of $I$ into $O_X$. This shows that the map above is
a bijection on closed points. On the other hand, globally it is not clear why a family of sheaves isomorphic to ideal sheaves has an embedding into $O_{S\times X}$. I think that Bridgeland just does not want to deal with this subtle question, which has nothing to do with his goal, so he just ignores it.