Formal moduli problem on small derived bases

Fix a field $k$ of characteristic zero. Write ${\mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}}_k^{\mathrm{s}\mathrm{m}}$ for the infinity-category of small connective augmented $E_{\mathrm{\infty }}$ $k$-algebras over $k$, with its mapping spaces computed in the derived localization of simplicial commutative $k$-algebras. Here small means: ${\pi }_i(R)=0$ for $i<0$ and for all sufficiently large $i$; every ${\pi }_i(R)$ is finite-dimensional over $k$; and ${\pi }_0(R)$ is a local Artinian ring equipped with its specified residue identification ${\pi }_0(R)\mathrm{/}{\mathfrak{m}}_{{\pi }_0(R)}\cong k$. The augmentation is a map $R\to k$. The notation ${\pi }_i$ uses homotopy grading; when represented by a nonpositive cochain dg algebra, ${\pi }_i(R)=H^{-i}(R)$.

A formal moduli problem over $k$ is a functor $F:{\mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}}_k^{\mathrm{s}\mathrm{m}}\to \mathrm{S}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}$ such that $F(k)$ is contractible and such that, for every pullback square

in which both maps ${\pi }_0(R_0)\to {\pi }_0(R_{01})\leftarrow {\pi }_0(R_1)$ are surjective, the induced square

is a homotopy pullback of spaces. A space means a Kan complex up to weak equivalence, and all pullbacks of spaces in this definition are homotopy pullbacks. This is a derived, space-valued functor: it retains automorphisms and higher homotopies and is not just the set of isomorphism classes over ordinary Artinian rings.

Let ${\mathbb{E}}_k$ be the spectrum object in the stabilization of the category of augmented connective $E_{\mathrm{\infty }}$ $k$-algebras at $k$, corresponding under the square-zero stabilization to the $k$-module spectrum $k$. Thus its value on the pointed sphere $S^i$ is the square-zero algebra $k\oplus k[i]$ for $i\ge 0$. The functor $F\circ {\mathbb{E}}_k$ on finite pointed spaces is strongly excisive: it sends the base point to the contractible space $F(k)$, and its pullback condition follows from the formal-moduli condition above, as in Lurie, Proposition 1.2.4. Define the tangent spectrum of $F$ to be the object of spectra represented by this strongly excisive functor. Its value on $S^i$ is the pointed space $F(k\oplus k[i])$, based at the zero section. The scalar endomorphisms of the square-zero summand give this spectrum its canonical $k$-module structure. Let $T_F$ be the corresponding cochain complex of $k$-vector spaces, with ${\pi }_i(F\circ {\mathbb{E}}_k)\cong H^{-i}(T_F)$. With the cohomological shift convention $K[r{]}^j=K^{j+r}$, the characteristic-zero dg Lie correspondence uses ${\mathfrak{g}}_F\simeq T_F[-1]$. No nilcompleteness condition is included for an enlarged pro-Artinian domain; that domain requires a separately specified completion condition.

Simplicial objects, simplicial commutative rings and homotopy groups

Model categories and Quillen adjunctions

Simplicial sets, homotopies and trivial Kan fibrations

Model structures for variable simplicial modules and algebras