I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example of what is written down in the stacks project (for example Tag 08UX) but I'm failing to find a more through treatment specially one that would put in contrast the differences with the special case of schemes/varieties. In my literature review, however, I stumbled into the following quote in

*Lowen, Wendy; van den Bergh, Michel*, **Deformation theory of abelian categories**, Trans. Am. Math. Soc. 358, No. 12, 5441-5483 (2006). ZBL1113.13009.

These results confirm the fundamental insight of Gerstenhaber and Schack [6, 8] that one should define the deformations of a ringed space $(X, \mathcal{O}_X)$ not as the deformations of $\mathcal{O}_X$ as a sheaf of k-algebras, but rather as the deformations of the k-linear category $\mathfrak{u}$ (or of the “diagram” $(\mathcal{B}, \mathcal{O}_{\mathcal{B}})$ in case $X\in \mathcal{B}$). These “virtual” deformations are nothing but the deformations of the abelian category $Mod(\mathcal{O}_{X})$.

The cited category $\mathfrak{u}$ is an appropriately defined category which is used to show that the deformations of the category of presheaves of modules over an appropriate basis $\mathcal{B}$ are equivalent to deformations of sheaves of modules over $\mathcal{O}_{X}$

I have skimmed through the literature including the cited papers and while I have found some indication of the exact meaning of this claim I am still confused. My interpretation is that the claim is either strictly noncommutative in nature ( so passing through a reconstruction theorem ), or deformations of this abelian category directly give the 'correct' deformation theory of the space in some sense ( for example the relationship with Hochschild cohomology in *Lowen, Wendy; van den Bergh, Michel*, **Hochschild cohomology of Abelian categories and ringed spaces**, Adv. Math. 198, No. 1, 172-221 (2005). ZBL1095.13013.)

My first question would then be, could somebody clarify what exactly is the quote saying?

My second question is then, if the space is a (sufficiently nice) scheme then am I to understand that the deformation theory of the category of quasi-coherent sheaves controls the deformations of the scheme as I would find it written in classical texts?

Thanks in advance

The cited papers on the quote are, respectively:

*Gerstenhaber, M.; Schack, S. D.*, **On the deformation of algebra morphisms and diagrams**, Trans. Am. Math. Soc. 279, 1-50 (1983). ZBL0544.18005.
and,

*Gerstenhaber, Murray; Schack, Samuel D.*, **The cohomology of presheaves of algebras. I: Presheaves over a partially ordered set**, Trans. Am. Math. Soc. 310, No. 1, 135-165 (1988). ZBL0706.16021.