Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Question

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-Quillen) cohomology $$ T^i_R(M)=Ext^i(\mathbb{L}_R, M), T^i_R=T^i_R(R). $$ I want to use tangent cohomology to study infinitesimal deformations of $R$.

If $A$ is an artinian ring over $\mathbb{C}$ then there are obstruction for lifting of deformations over $A$ to deformations over $\tilde{A}$, where $\tilde{A} \in Ex_{\mathbb{C}}(A, \mathbb{C})\cong T^1_A(\mathbb{C})$. In such settings, how obstruction map $$ o : Ex_{\mathbb{C}}(A, \mathbb{C}) \to T^2_R, $$ could be defined? I know construction for such map in case of explicit description of $T^2_R$, but it should be possible to define it just using properties of tangent cohomology.

Answer 1

Sorry, I don't know quite how to define this just in terms of the cotangent complex alone (one needs to give it, or its ``dual'', an additional structure). In characteristic zero, the idea is that there exists a (somewhat canonical) differential graded Lie algebra D whose cohomology gives T^i = Ext^i(L_R, R). For example, see for example Remark 6.4 of this article I just found. The Lie bracket of D then gives the pairing T^1 x T^1 ---> T^2 (as well as other pairings). As you may well imagine, there is a huge literature about this kind of thing, but often to study a particular singularity one simply uses the direct descriptions of the obstruction pairing that you know about.