Deformation of (locally) ringed spaces and of their abelian categories of modules

Question

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.

Answer 1

As Jon Pridham notes in the comments, the quote should be understood noncommutatively. In fact, in the introduction Lowen and Van den Bergh write

Deformation theory of abelian categories is important for non-commutative algebraic geometry. One of the possible goals of non-commutative algebraic geometry is to understand the abelian (or triangulated) categories which have properties close to those of the (derived) category of (quasi-)coherent sheaves on a scheme.

As you observe, this is a natural point of view when identifying a ringed space $(X, \mathcal O_X)$ with its category of quasi-coherent sheaves by considering spectra of Abelian categories as in the Gabriel–Rosenberg reconstruction theorem.

To answer your second question, the deformations of the Abelian category $\mathrm{Qcoh} (X)$ of quasi-coherent sheaves contain classical deformations of $X$ as a special case, but in general admit more deformations than $X$.

The difference becomes quite clear when looking at the space of first-order deformations up to equivalence (i.e. the tangent space of the deformation functor).

For example, for smooth $X$ one gets

• $\mathrm H^1 (\mathcal T_X)$ for deformations of $X$
• $\mathrm H_{\mathrm{Ab}}^2 (\mathrm{Qcoh} (X)) \simeq \mathrm{HH}^2 (X) \simeq \mathrm H^0 (\Lambda^2 \mathcal T_X) \oplus \mathrm H^1 (\mathcal T_X) \oplus \mathrm H^2 (\mathcal O_X)$ for deformations of $\mathrm{Qcoh} (X)$, i.e. you get two new types of deformations: the direct summand $\mathrm H^0 (\Lambda^2 \mathcal T_X)$ corresponds to quantizations of algebraic Poisson structures and $\mathrm H^2 (\mathcal O_X)$ to twists of the structure sheaf.

For affine $X = \mathrm{Spec} (A)$ with $A$ commutative one gets

• $\mathrm{Har}^2 (A, A)$ for deformations of $X$ (Harrison / André–Quillen cohomology which controls commutative deformations of $A$ which are trivial if $A$ is smooth)
• $\mathrm{HH}^2 (A, A)$ for deformations of $\mathrm{Qcoh} (X)$ (Hochschild cohomology which controls associative deformations of $A$).

If you want to compare deformations of $\mathrm{Qcoh} (X)$ and $X$ on equal footing, you can look at deformations of $\mathcal O_X \vert_{\mathfrak U}$ for some affine open cover $\mathfrak U$ closed under $\cap$ either

• as a twisted presheaf of associative algebras — this is equivalent to deformations of $\mathrm{Qcoh} (X)$ as Abelian category by the results of Lowen and Van den Bergh
• as a presheaf of commutative algebras — this is equivalent to classical deformations of $X$ (see for example [1]).

The "insight" by Gerstenhaber and Schack was that the former type of deformations of the diagram $\mathcal O_X \vert_{\mathfrak U}$ give a useful generalization of "deformation" of a ringed space, and Lowen and Van den Bergh show that this corresponds to the deformation theory of $\mathrm{Qcoh} (X)$ as Abelian category.

[1] Lepri, Emma; Manetti, Marco, On deformations of diagrams of commutative algebras, Colombo, Elisabetta (ed.) et al., Birational geometry and moduli spaces. Collected papers presented at the INdAM workshop, Rome, Italy, June 11–15, 2018. Cham: Springer. Springer INdAM Ser. 39, 77-107 (2020). ZBL1440.13066.

Answer 2

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 $U\in\mathcal B$, $H^1(U,\mathcal O_U)=H^2(U,\mathcal O_U)=0$).