Kodaira-Spencer spectra

Construction

Let $A\to R$ be a map of $E_{\infty }$-rings, $M\in Mo{d}_A$, the $\infty$-category of deformations of the $A$-algebra $R$, along the trivial extension $A\oplus M\to A$, is the pullback of the diagram

$\ast \stackrel{R}{\to }CAl{g}_A\stackrel{-{\otimes }_{A\oplus M}A}{\leftarrow }CAl{g}_{A\oplus M}$ We denote the result by

$De{f}_A(R,M):=\{R\}{\times }_{CAl{g}_A}CAl{g}_{A\oplus M}$ Furthermore, there’s a natural choice of a basepoint: namely the trivial/constant deformation $R{\otimes }_A(A\oplus M),R{\otimes }_A(A\oplus M){\otimes }_{A\oplus M}A\stackrel{\sim }{\to }R$ We’ll henceforce denote by $De{f}_A(R,M)$ the above space as a pointed anima ($\in {\mathcal{S}}_{\ast }$). Our first observation is that this pointed space is an infinite loop space.

Lemma There’s an equivalence $De{f}_A(R,M)\simeq \Omega De{f}_A(R,\Sigma M)$ Proof: This is a slight generalization of a theorem in DAG-XIV. Using “clutching” for $CAl{g}_{(-)}$.

Construction/Notation

Let $K{S}_A(R,M)\in Sp$ denote the spectrum with ${\Omega }^{\infty -n}K{S}_A(R,M)\simeq De{f}_A(R,{\Sigma }^{n}M)$ Remark $K{S}_A(R,M)$ is a stable homotopy type classifying deformation theory of rings.

• It is non-connective in general.
• A priori, even if $R,A$ are classical rings and $M$ is a discrete module, $K{S}_A(R,M)$ is still non-discrete: the Postnikov amplitude (range of non-vanishing homotopy groups) of $K{S}_A(R,M)$ measures non-smoothness of the map $A\to R$

Remark The above works verbatim for any suitably nice $\infty$- operad, we didn’t use anything special about the $E_{\infty }$-operad. In particular, we have $E_n$-versions.

Caution: Given the $E_{\infty }$ input as before, we have an $E_n$-Kodaira-Spencer spectrum $K{S}_A^{(n)}(R,M)$ for each $n$. These are all distinct in general since deformations of an $E_{\infty }$-algebra is different from those of its underlying $E_n$ algebra.

Relationship to $E_{\infty }$ cotangent complex

The underlying linearity of the situation has a even more direct manifestation, we have

Lemma ${\Omega }^{\infty }K{S}_A(R,M)\simeq Ma{p}_{Mo{d}_R(Sp)}({\mathbb{L}}_{R/A},\Sigma M{\otimes }_{A}R)$ Proof: Using the equivalence$De{f}_A(R,M)\simeq \Omega De{f}_A(R,\Sigma M)$ The RHS can be unwound in terms of automorphisms of objects in categories of algebras, unpacking this description leads to the result, as in DAG-XIV.

Remark: We can see that the RHS is pointed by the 0 map and lifts to a infinite loop space, as it’s the mapping space in a stable $\infty$-category. We also see immediately by substituting $M\mapsto {\Sigma }^{n}M$ that it agrees with the spectrum constructed in the last section, as we have $De{f}_A(R,{\Sigma }^{n}M)\simeq Ma{p}_{Mo{d}_R(Sp)}({\mathbb{L}}_{R/A},{\Sigma }^{n+1}M{\otimes }_{A}R)$

Examples

Tangent complexes as Kodaira-Spencer spectra Taking $M=\Omega A\in Mo{d}_A$, we recover the relative tangent complex: $K{S}_A(R,\Omega A)\simeq \text{map}({\mathbb{L}}_{R/A},R)\simeq {\mathbb{T}}_{R/A}$ Remark: Another useful/suggestive form of this example is that the Kodaira-Spencer spectrum for the unit $A$-module $A$ is the shifted cotangent complex $\Sigma {\mathbb{T}}_{R/A}$, which suggests the whole picture could be interpreted via Koszul duality, as in the picture in Paper - formal moduli via fractal infinity categories.