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> Let $\mathcal{X} \to B$ be a flat family with some fibre $X\_b \to b$. Suppose I have a coherent sheaf $F\_b$ on $X\_b$. When does it spread out to a sheaf $\mathcal{F}$ on $\mathcal{X}$ flat over $B$? 
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What about a subscheme $z \subset X\_b$? Arbitrary diagrams of sheaves? 


(I am only concerned with the case where everything is defined over $\mathbf{C}$, and moreover in the local case where $B$ is a disc and I am perfectly happy to take $\mathcal{X}$ to be affine. However $\mathcal{X}$ should not be assumed smooth, nor $X\_b$ to even be reduced.) 



When do sheaves deform over a family?
Suppose that $B\_n$ it the $n^{\rm th}$ infinitesimal neighborhood of $b$ in $B$; that is, if $\frak m$ is the maximal ideal of $b$ in $B$, we set $B\_n := \mathop{\rm Spec} \mathcal O\_B/{\frak m}^{n+1}$. If $\mathcal F\_n$ is as extension of $\mathcal F\_b$ to $B\_n$, there is a canonically defined element of $({\frak m}^{n+2}/{\frak m}^{n+1})\otimes\_{\mathbb C}\mathop{\rm Ext}\_{\mathcal O\_{X\_b}}(\mathcal F, \mathcal F)$, called the *obstruction*; if this is zero, then the sheaf $\mathcal F\_n$ extends to $B\_{n+1}$. This depends on $\mathcal F\_n$, not only on $\mathcal F\_b$.


If this obstructions are always 0 (for example, if $\mathop{\rm Ext}\_{\mathcal O\_{X\_b}}(\mathcal F, \mathcal F) = 0$), then $\mathcal F\_b$ will extend to some étale neighborhood of $b$ in $B$. I don't think you can say much more in this generality.


There is a whole subject devoted to the study of this kind of problems (not only for sheaves, but for much more general objects), called *deformation theory*.


