TITLE: Deformation of (locally) ringed spaces and of their abelian categories of modules
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: The answer to your second question is "no", I think.
Let's assume that sufficiently nice means that it is a smooth algebraic variety over a field of characteristic $0$. Then, as written in Severin Barmeier's answer, first order deformations of the abelian category of modules are given by $HH^2(X)\simeq H^0(X,\wedge^2 T_X)\oplus H^1(X,T_X)\oplus H^2(X,\mathcal O_X)$ [note that this isomorphism is a consequence of the Hochschild-Kostant-Rosenberg iso]. Now, only $H^1(X,T_X)$ amounts for classical first order deformations of $X$ as a scheme. $H^0(X,\wedge^2 T_X)$ provides first order (possibly noncommutative) deformations of the product of $\mathcal O_X$, while $H^2(X,\mathcal O_X)$ provides first order deformations of the glueing data of $\mathcal O_X$ (these are gerby/stacky deformations of $\mathcal O_X$).
All together, these three parts provide first order deformations of $\mathcal O_X$ as an algebroid stack, in the sense of Kontsevich (see Appendix A of this paper, as well as papers by Yekutieli or Kashiwara-Schapira).
As for your first question, the claims says that deforming the structure sheaf $\mathcal O_X$ as a sheaf of (possibly noncommutative) algebras is not enough to get all deformations of $Mod(\mathcal O_X)$. Indeed, for first order deformation this would only give $H^0(X,\wedge^2 T_X)\oplus H^1(X,T_X)$, and one would miss the part $H^2(X,\mathcal O_X)$. To get all deformations of $Mod(\mathcal O_X)$, the claim says that one shall deform the $k$-linear category $\mathfrak{u}$ (which is essentially the same as deformating $\mathcal O_X$ as an algebroid stack in the sense of Kontsevich- under the assumption that $X$ has a basis $\mathcal B$ so that for every $U\in\mathcal B$, $H^1(U,\mathcal O_U)=H^2(U,\mathcal O_U)=0$).