The classical Schlessinger conditions for the existence of a formally versal object are formulated for functors $$ F : \mathrm{Art}_k \to \mathrm{Set}, $$ with $F(k) = \{ * \}$. One requires that for every pair of morphisms of local Artin $k$-algebras $$ A' \to A, \quad A'' \to A, $$ and every object $X \in F(A)$, the natural map $$ F(A' \times_A A'') \;\longrightarrow\; F(A') \times F(A'') \tag{$\ast$} $$ satisfies: - (S1a) Surjective when $A'' \to A$ is a small extension. - (S1b) Bijective when $A = k$, $A'' = k[\epsilon] := k[\epsilon]/(\epsilon^2)$. In this case $F(k[\epsilon])$ acquires a $k$-vector space structure. The last condition is - (S2) $\dim_k F(k[\epsilon]) < \infty.$ In practice, these conditions are often checked by identifying $F$ with a cohomology group (e.g. for curves, $H^1(C, T_{C/k})$), and obstructions with $H^2=0$. This works because every surjection of local, Artin rings can be factored into a finite composite of square-zero extension, and square-zero extensions are encoded by these cohomology groups.
In Versal deformations and algebraic stacks, Artin generalizes these conditions: he requires that $(\ast)$ be surjective for every pair $(A'' \to A, A' \to A)$ where $A' \to A$ is a square-zero extension of Noetherian $k$-algebras with kernel $I$ satisfying $I^2 = 0$ and $A'' \to A_{\mathrm{red}}$ is a surjection of Noetherians factoring as $A'' \to A \to A_{\mathrm{red}}$.
Here is my problem: the condition on $A'' \to A$ is so weak that there is no obvious connection to square-zero extensions and so the link with cohomology seems lost. For instance, for the moduli functor of smooth projective curves, I do not see how to prove surjectivity of $(\ast)$ without the usual identification with $H^1(C, T_{C/k})$.
To add context, Artin ends his paper by stating that these conditions hold for the moduli stack of surfaces of general type, calling this “standard,” but without a references or proof, so maybe I am missing something obvious or did not understand his definition.
Question. What is the general strategy for verifying Artin’s version of Schlessinger’s conditions? Can these checks somehow be reduced to the nilpotent case, or is there a different conceptual approach?
