Lie algebroids describe infinitesimal geometric structures, and this article studies the relation between Lie algebroids and deformation theory. When \(k\) is a field with \(\operatorname{char}k=0\) and \(x:\operatorname{Spec}(A)\rightarrow\mathcal M\) is a map from an affine, possibly derived, scheme to a moduli space over \(k,\) a formal neighbourhood of \(\mathcal M\) around \(x\) is controlled by a Lie algebroid over \(\operatorname{Spec}(A).\) It is defined on a basis given by the vector fields on \(\operatorname{Spec}(A)\) tangent to the fibres of \(x\).
When \(x\) is a point, this is the usual correspondence between deformations and dg-Lie algebras from the work of Kodaira-Spencer in the differential geometric setting, formalized by Deligne and Drinfeld. The main point of those is to describe the formal neighbourhood of a moduli space in terms of derived geometry, using the functor of points and the ideas of Grothendieck. In the classical language of M. Schlessinger, a formal neighbourhood of \(x\) is described by the deformation functor \(\mathcal M^\wedge:\text{CAlg}^\text{sm}_k/A\rightarrow\mathcal S;\;B\mapsto\mathcal M(B)\times_{\mathcal M(A)}\{x\}.\) Here a derived infinitesimal thickening \(\tilde x:\operatorname{Spec}(A)\rightarrow\operatorname{Spec}(B)\) is sent to the space of extensions of \(x\) to a map \(\tilde x:\operatorname{Spec}(B)\rightarrow\mathcal M.\) These are dual to maps \(B\rightarrow A\) in the \(\infty\)-category of connective commutative \(k\)-algebras: The \(\infty\)-categorical localization of the category of homologically nonnegatively graded commutative dg-\(k\)-algebras at quasi-isomorphisms.
For reasonable moduli spaces, \(\mathcal M^\wedge\) satisfies a derived version of the Schlessinger conditions, encoding the usual obstruction theory for existence of deformations. V. Hinich [J. Pure Appl. Algebra 162, No. 2–3, 209–250 (2001; Zbl 1020.18007)], J. P. Pridham [Adv. Math. 224, No. 3, 772–826 (2010; Zbl 1195.14012)] and J. Lurie [“Derived algebraic geometry X: formal moduli problems”, (2011)] provided an equivalence between the homotopy theory of deformation functors and the homotopy theory of dg-Lie algebras. The main goal of the present article is to give an identification of the homotopy theory of dg-Lie algebroids with the homotopy theory of formal moduli problems.
For a connective (a homotopical invariant) commutative \(k\)-algebra \(A\), \(\operatorname{char}k=0,\) the \(\infty\)-category \(\mathbf{CAlg}^\text{sm}_k/A\) is defined as the smallest subcategory of the \(\infty\)-category of connective commutative \(k\)-algebras over \(A\) that contains \(A\) and that is closed under square zero extensions by \(A[n]\) with \(n\geq 0.\) A formal moduli problem on \(A\) is a functor \(X:\mathbf{CAlg}^\text{sm}_k/A\rightarrow\mathcal S\) to the category of stacks/\(k\) such that (a) \(X(A)\simeq\ast\) is contractible, and (b) \(X\) preserves pullback diagrams of small extensions of \(A\) such that a pullback \(B_\eta\) over \(A\) and \(B\) realizes \(B_\eta\) as a square zero extension of \(B\) by \(A[n].\)
The author views a formal moduli problem as a map of stacks \(x:\operatorname{Spec}(A)\rightarrow X\) realizing \(X\) as an infinitesimal thickening of \(\operatorname{Spec}(A).\) Such a stack is proved to determine a Lie algebroid \(T_{A/X},\) the fiberwise vector fields on \(\operatorname{Spec}(A)\) over \(X.\) This is one side of an equivalence with inverse sending a Lie algebroid \(\mathfrak g\) to the quotient of \(\operatorname{Spec}(A)\) by an infinitesimal \(\mathfrak g\)-action, verbatim:
Theorem 1.3. Suppose that \(A\) is eventually coconnective. Then there is an equivalence of \(\infty\)-categories \(\text{MC}:\mathbf{LieAlgd}_A\leftrightarrows\mathbf{FMP}_A:T_A/\) between the \(\infty\)-category of Lie algebroids over \(A\) and the \(\infty\)-category of formal moduli problems under \(A\).
The \(\infty\)-category of Lie algebroids can be explicitly described by homological algebra. It is the localization of the category of dg-Lie algebroids over \(A\) at the quasi-isomorphisms. The Theorem above is seen as a rectification result; any formal moduli problem admits a rigid description in terms of chain complexes endowed with an algebraic structure. This really says that formal moduli problems can be explicitly computed (by homology), rather than be studied by purely categorical theory as in the work (books) of D. Gaitsgory and N. Rozenblyum [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001)].
The proof of the theorem relies on a version of Koszul duality. The small extensions of \(A\) are Koszul dual to certain free Lie algebroids over \(A,\) sending a dg-Lie algebroid to its cohomology. That is, to a given coherent sheaf \(\mathcal F\) on (the functor) \(X,\) it is expected that the restriction \(x^\ast\mathcal F\) along \(x:\operatorname{Spec}(A)\rightarrow X\) carries a natural representation of the Lie algebroid \(T_{A/X}.\) In fact, the author proves that for \(A\) eventually coconnective and \(X\) a formal moduli problem on \(A\) with associated Lie algebroid \(T_{A/X},\) there is a fully faithful, symmetric monoidal left adjoint functor \(\Psi_X:\operatorname{QC}(X)\rightarrow\operatorname{Rep}_{T_{A/X}}\) inducing an equivalence on connective objects. This is then used to give a chain-level description of the formal moduli problem describing the deformations of a connective commutative \(A\)-algebra.
The functor \(\Psi_X\) is not an equivalence in general. E.g., for any map \(f:X\rightarrow Y\) of formal moduli problems, the restriction functor \(f^!:\operatorname{Rep}_{T_{A/Y}}\rightarrow\operatorname{Rep}_{T_{A/X}}\) preserves all limits, while the restriction functor \(f^\ast:\operatorname{QC}(Y)\rightarrow\operatorname{QC}(X)\) does not. This is the reason for D. Gaitsgory’s introduction of ind-coherent sheaves replacing quasi-coherent sheaves [Mosc. Math. J. 13, No. 3, 399–528 (2013; Zbl 1376.14023)]. However, in the present work, the dual notion of pro-coherent sheaves are considered. For derived schemes which are locally almost of finite type over a field, the pro-coherent sheaves are equivalent to the ind-coherent sheaves by (derived) Serre-duality. The \(\infty\)-category of pro-coherent sheaves on an eventually coconnective algebra \(A\) is an extension \(\mathbf{Mod}_A=\operatorname{QC}(A)\hookrightarrow\operatorname{QC}^!(A)=\operatorname{Ind}(\mathbf{Coh}^\text{op}_A)\) of the usual \(\infty\)-category of quasi-coherent sheaves, i.e. A-modules. Its behaviour with respect to deformation theory is convenient for relating pro-coherent sheaves to Lie algebroid representations. The author proves that if \(A\) is eventually coconnective and coherent, and if \(X\) is a formal moduli problem on \(A\) with associated Lie algebroid \(T_{A/X},\) then there is a fully faithful embedding \(\operatorname{Rep}_{T_{A/X}}\hookrightarrow\operatorname{QX}^!(X)\) whose essential image is the pro-coherent sheaves on \(X\) whose restriction to \(A\) is quasi-coherent.
Gaitsgory and Rozenblyum [Zbl 1408.14001] considered a larger \(\infty\)-category of small extensions of \(A\), the pro-coherent Lie-algebroids, which also contains square zero extensions of \(A\) by connective coherent \(A\)-modules. Then a Lie-algebroid is a functor \(X:\mathbf{CAlg}^\text{sm, coh}_k/A\rightarrow\mathcal S\) satisfying the Schlessinger conditions. The author calls this a pro-coherent formal moduli problem as the tangent space functor defines a pro-coherent sheaf on \(A\). Also, a representation of a Lie algebroid is defined to be a pro-coherent sheaf on a pro-coherent formal moduli problem. The main property this is that when \(A\) is eventually coconnective and almost of finte type, the formal completion of \(\mathcal M\) at \(\operatorname{Spec}(A)\) is completely determined.
Finally, the author proves that in some restricted situations, the rectification results applies to pro-coherent formal moduli problems, allowing to study formal completions of derived stacks algebraically in terms of dg Lie algebroids.
This is a highly specialized article, assuming full control on the theory of derived geometry. This given, it illustrates deep and new ideas bringing the geometry further, including important equivalences between useful \(\infty\)-categories. The article should be considered by anyone working in the field.