I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the picture clearer before complicating too much the setting and hope things will not be that different then.

Let me consider a smooth projective variety $X$ defined over $\mathbb{C}$ (these hypotheses can obviously be relaxed), and a surjection $A \to A_0$ of commutative algebras over $\mathbb{C}$ defined by a nilpotent ideal. Taking an $E\in \mathrm{Coh}(X\times \mathrm{Spec} A_0)$, I guess a deformation of $E$ should be some $\mathcal{E}\in \mathrm{Coh}(X\times \mathrm{Spec} A)$ such that $i^\ast \mathcal{E}\simeq E$ where $i\colon X\times \mathrm{Spec} A_0 \hookrightarrow X\times \mathrm{Spec} A$ is the inclusion.

My question is: one can associate to $E$ various gadgets like $\Gamma(X,E)$, $H^\bullet (X, E)$, $Ext^\bullet(E,E)$: which is the relation between these objects calculated for $E$ and for $\mathcal{E}$? this would be equivalent to asking whether the functions $dim\Gamma(X,-)$, etc.. are locally constant on a moduli space of such sheaves.

Looking forward to thank you for giving me some examples and references as well