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][1]) 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 

<cite authors="Lowen, Wendy; van den Bergh, Michel">_Lowen, Wendy; van den Bergh, Michel_, [**Deformation theory of abelian categories**](http://dx.doi.org/10.1090/S0002-9947-06-03871-2), Trans. Am. Math. Soc. 358, No. 12, 5441-5483 (2006). [ZBL1113.13009](https://zbmath.org/?q=an:1113.13009).</cite>

>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 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 <cite authors="Lowen, Wendy; van den Bergh, Michel">_Lowen, Wendy; van den Bergh, Michel_, [**Hochschild cohomology of Abelian categories and ringed spaces**](http://dx.doi.org/10.1016/j.aim.2004.11.010), Adv. Math. 198, No. 1, 172-221 (2005). [ZBL1095.13013](https://zbmath.org/?q=an:1095.13013).</cite>)

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: 

<cite authors="Gerstenhaber, M.; Schack, S. D.">_Gerstenhaber, M.; Schack, S. D._, [**On the deformation of algebra morphisms and diagrams**](http://dx.doi.org/10.2307/1999369), Trans. Am. Math. Soc. 279, 1-50 (1983). [ZBL0544.18005](https://zbmath.org/?q=an:0544.18005).</cite>
and, 

<cite authors="Gerstenhaber, Murray; Schack, Samuel D.">_Gerstenhaber, Murray; Schack, Samuel D._, [**The cohomology of presheaves of algebras. I: Presheaves over a partially ordered set**](http://dx.doi.org/10.2307/2001114), Trans. Am. Math. Soc. 310, No. 1, 135-165 (1988). [ZBL0706.16021](https://zbmath.org/?q=an:0706.16021).</cite>  



  [1]: https://stacks.math.columbia.edu/tag/08UX
  [2]: https://www.ams.org/journals/tran/2006-358-12/S0002-9947-06-03871-2/home.html