Coherent obstruction theory of spectral algebras

Square-Zero Extensions of Discrete Rings Linearize

A central phenomenon uncovered in the deformation/obstruction theory of discrete rings (in the tradition of Artin, Illusie, Hartshorne, …) is that the a priori geometric / non-linear theory of square-zero extensions linearizes:

$$\{\text{Square zero extensions of }R\text{ by }I\}\simeq {\mathrm{Map}}_{D(R)}({\mathbb{L}}_R,\Sigma I)$$

To drive home the point about linearity, observe that the LHS is a priori a groupoid. The RHS, being a mapping space in a stable $\infty$-category, has the structure of an $E_{\infty }$-group.
This equivalence tells us that it makes sense to “add” square-zero extensions and “scale” square-zero extensions by elements of $R$—there’s an operation building new square-zero extensions from old ones by taking $R$-linear superpositions / weighted sums.

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“Bounded” Extensions of Ring Spectra Also Linearize

The same phenomenon occurs when the square-zero ideal $I$ has bounded Postnikov amplitude, by work of Basterra–Mandell and Lurie.

Theorem ([Lurie, HA 7.4.1.23, Def. 7.4.1.18]):

$$\{\text{Square zero extensions }A\to R\text{ by }I\}\simeq {\mathrm{Map}}_{D(R)}({\mathbb{L}}_R,\Sigma I)$$

granted that $A$ is connective and $I$ has homotopy groups concentrated in the degree range $[n,2n]$.

This note explores what happens when the strong connectivity and truncation constraints here do not hold.
Examples: $MU$, $THH({\mathbb{F}}_p)$, …

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Coherent Square-Zero Extensions

Definition (Square Zero Extensions)

Let:

• $\mathcal{O}$ be a coherent stable operad (in the sense of HA, Chapter 5),
• $V^{\otimes }\in \mathrm{CAlg}({\mathrm{Mod}}_{Sp}({\mathrm{Pr}}^L))$ be a presentably symmetric monoidal stable $\infty$-category,
• $A\in {\mathrm{Alg}}^{\mathcal{O}}(V)$ be an $\mathcal{O}$-algebra in $V$.

A (coherent) square-zero extension of $A$ is the data of a pair $(\mathcal{R},\theta )$, where:

• $\mathcal{R}\in {\mathrm{Alg}}^{\mathcal{O}}({\widehat{\mathrm{Fil}}}_V)$
  — i.e. an $\mathcal{O}$-algebra in complete filtered objects of $V$ (a filtered $\mathcal{O}$-algebra in $V$),
• $\theta :{\mathrm{gr}}^0(\mathcal{R})\stackrel{\sim }{\to }A$ is an equivalence in ${\mathrm{Alg}}^{\mathcal{O}}(V)$,
• and $\mathcal{R}$ satisfies: $${\mathrm{gr}}^q(\mathcal{R})=0\forall q\notin \{0,1\}.$$

In this situation, we say that $(\mathcal{R},\theta )$ is a square-zero extension of $A$ by ${\mathrm{gr}}^1\mathcal{R}$.

Footnotes

• Presentability. I’m not sure if the presentability assumption on $V$ is strictly necessary. It’s possible that I’m asking for it because

  1. the simple expression ${\mathrm{Fil}}_V\simeq \mathrm{Fil}{\otimes }_{Sp}V$ simplifies our arguments,
  2. all our examples will be presentable.

• Attribution. I learned this from Tyler Lawson, but it might have a longer history, e.g. in model-structure approaches to Smith ideals.

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What Do Square-Zero Extensions Look Like?

Remark (Underlying filtrations of square-zero extensions)

The combination of the conditions

• that the associated graded of $\mathcal{R}$ is concentrated in degrees 0 and 1, and
• that the filtration on $\mathcal{R}$ is complete,

forces the underlying filtered object of $\mathcal{R}$ to be of the form

$$\cdots \stackrel{\sim }{\to }0\to 0\to I\stackrel{f}{\to }R\stackrel{\sim }{\to }R\stackrel{\sim }{\to }\cdots$$

where $I$ sits in degree 1, mapping to an $R$ in degree 0 via $f$, and we have an equivalence in $V$:

$$\mathrm{coFib}(f)\simeq A.$$

The objects $I$ and $R$ here have additional structures:

• ${\mathrm{gr}}^1(\mathcal{R})\simeq I$ has the structure of a module over the algebra ${\mathrm{gr}}^0(\mathcal{R})\simeq A$, hence $I\in V$ sitting in degree 1 has the structure of an $A$-module.

  • Here and already before this point, we’ve used extensively that ${\mathrm{gr}}^0$ and ${\mathrm{gr}}^{\bullet }$ are symmetric monoidal functors, among other salient properties about filtrations and gradings.
    These salient properties can be summarized by saying that
    $\mathrm{gr}:\mathrm{Fil}\to \mathrm{Gr}$ is the universal perverse schober — a theorem of Gammage–Hilburn–Mazel-Gee.

• The $\mathcal{O}$-algebra structure on $\mathcal{R}$ induces a $\mathcal{O}$-algebra structure on the 0-th degree term $\mathcal{R}(0)\simeq R$, as the functor $(-)(0)$ is lax symmetric monoidal.

  • We can also see this from the fact that the “converges-to” functor $(-)(-\infty )$ is lax symmetric monoidal.

In short, a square-zero extension of an $\mathcal{O}$-algebra $A$ by an ( $\mathcal{O}$-operadic) $A$-module $I$ is given by

• an $\mathcal{O}$-algebra $R$,
• a map of algebras $R\to A$ whose kernel is the $A$-module $I$,
• some additional coherence data (we’ll see later that this data precisely encodes what it means for $I$ to be “square-zero” in higher algebra).

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Remark (Square-zero extensions as 1-term coherent cochains)

By the correspondence between complete filtered spectra and homotopy coherent cochains (whose strongest modern version was proved by Ariotta), the above description of the filtration tells us that a coherent extension of $A$ is an algebra in coherent cochains whose underlying cochain object looks like

$$\cdots \stackrel{\sim }{\to }0\stackrel{\sim }{\to }0\stackrel{\sim }{\to }A\stackrel{d}{\to }\Sigma I\to 0\stackrel{\sim }{\to }0\stackrel{\sim }{\to }\cdots$$

This encodes the idea of a derivation valued in $I$.

TODO: Explain the degree shift in terms of the universal property of $\mathbb{S}(1)[1]\in \mathrm{Fil}$.

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Linearization of Coherent Square-Zero Extensions

For discrete rings, we saw that

• the non-linear / geometric / “commutative” data of a square-zero thickening ring map
• is converted to
• the linear / representation-theoretic / “non-commutative” data.

By our discussion above, we see that

• the non-linear data is the filtered algebra description (encodes coherences on a ring map),
• the linear data is the coherent cochains description (encodes structure on a derivation),
• and the conversion is performed by the Ariotta correspondence.

We’ll use this philosophy as a hint to prove a precise theorem about a linear classification of coherent square-zero extensions of ring spectra.

Lesson: By passing to a more coherent version of the notion of square-zero extension, we get to classify square-zero extensions of arbitrary ring spectra, using an argument where neither connectivity nor truncatedness assumptions are needed.
Contrast this to the situation in Higher Algebra, where square-zeroness is defined for maps with connective fibers, and there it’s defined as a condition on ${\pi }_0(I)$.