Remark 4. When $n=1$ , Theorem 2 is part of the main result of [Reference Deligne and IllusieDI87]Footnote ⁴ : in this case, condition (b) in Theorem 2 is asking for a lifting to $\mathrm {BP}\langle {0}\rangle = \mathbf {Z}_p$ . As mentioned above, Sasha Petrov recently constructed in [Reference PetrovPet23] a $(p+1)$ -dimensional smooth and proper $\mathbf {Z}_p$ -scheme $\mathfrak {X}$ such that the Hodge-de Rham spectral sequence for its special fiber $\mathfrak {X}_{p=0}$ does not degenerate at the $\mathrm{E}_1$ -page. If the HKR spectral sequence degenerates at the $\mathrm{E}_2$ -page for Petrov’s $\mathfrak {X}_{p=0}$ , then $\operatorname {{\mathrm {QCoh}}}(\mathfrak {X})$ provides an example of a $\mathbf {Z}_p$ -linear $\infty $ -category which cannot lift to a $\mathrm {ku}$ -linear $\infty $ -category.

We view Theorem 2 as a step towards a positive answer of Deligne and Illusie’s question in some generality. Note that condition (b) in Theorem 2 is significantly weaker than asking that X itself admit some sort of lifting as a spectral scheme. Note, also, that we do not prove anything nearly as refined as [Reference Deligne and IllusieDI87]: namely, we do not provide any sort of correspondence between liftings and splittings of truncations of the de Rham complex. For instance, it would be very interesting if, for a $\mathbf {Z}_p$ -scheme $\mathfrak {X}$ , there were a relationship between splittings of the mod p reduction of the zeroth generalized eigenspace of the diffracted Hodge complex (see [Reference Bhatt and LurieBL22a, Remark 4.7.20] for this notion) and liftings of $\operatorname {{\mathrm {QCoh}}}(\mathfrak {X})$ to $\mathrm {BP}\langle {1}\rangle $ .

Remark 5. Let $\text{I} = (p^2, v_1^2, \cdots , v_{n-1}^2)$ . If $\mathrm {BP}\langle {n-1}\rangle /\text{I}$ were to admit the structure of an ${\mathbf {E}_{{2}}}$ -ring, Theorem 2 (and Proposition 3) would continue to hold with $\mathrm {BP}\langle {n-1}\rangle $ replaced by $\mathrm {BP}\langle {n-1}\rangle /\text{I}$ . This is because one can prove that Lemma 10 continues to hold for $\mathrm {BP}\langle {n-1}\rangle /\text{I}$ .

Some preliminary calculations seem to suggest that Petrov’s first Sen class (see [Reference PetrovPet23, Reference IllusieIll22]) is related to the obstruction in Hochschild cohomology to lifting a $\mathbf {Z}_p$ -scheme $\mathfrak {X}$ along the map $\mathrm {BP}\langle {1}\rangle /v_1^2 \to \mathbf {Z}_p$ (and even along the map $\tau _{\leq 2p-3} j \to \mathbf {Z}_p$ , where j is the connective complex image-of-J spectrum). For instance, the first k-invariant of $\mathrm {BP}\langle {1}\rangle /v_1^2$ is given by the map $\mathbf {Z}_p \to \mathbf {Z}_p[2p-1]$ defined via the composite

$$ \begin{align*}\mathbf{Z}_p \to \mathbf{F}_p \xrightarrow{{\text{P}^1}} \mathbf{F}_p[2p-2] \xrightarrow{{\beta}} \mathbf{Z}_p[2p-1],\end{align*} $$

where $\text{P}^1$ is a Steenrod operation and $\beta $ is the Bockstein. In other words, $\mathrm {BP}\langle {1}\rangle /v_1^2$ is equivalent to the fiber of the above composite. However, the extension class for is computed in [Reference PetrovPet23, Lemma 6.5] to be the composite

$$ \begin{align*}\text{L}\Omega^p_{\mathfrak{X}}[-p] \to \text{L}\Omega^p_{\mathfrak{X}_{p=0}/\mathbf{F}_p} [-p] \xrightarrow{{c_{\text{X},p}}} \mathcal{O}_{\mathfrak{X}_{p=0}} \xrightarrow{{\beta}} \mathcal{O}_{\mathfrak{X}}[1],\end{align*} $$

where the ‘first Sen class’ $c_{\text{X},p}$ can be defined using Steenrod operations on cosimplicial algebras via [Reference PetrovPet23, Theorem 7.1]. We hope to explore this further to obtain a tighter connection between the results in this article and those of Petrov’s.

Remark 6. Theorem 2 has the following counter-intuitive consequence: if the HKR spectral sequence for X degenerates at the $\mathrm{E}_2$ -page, then the differentials in the Hodge-de Rham spectral sequence obstruct the lifting of $\operatorname {{\mathrm {QCoh}}}(\text{X})$ to a smooth and proper left $\mathrm {BP}\langle {n-1}\rangle $ -linear $\infty $ -category. In particular, taking $n = \infty $ , the condition in Proposition 3 that $\pi _j \mathrm {HH}(\mathcal {C}/\mathbf {F}_p) = 0$ for $j\not \in [-p^n,p^n]$ is vacuous; so we find that if $\mathcal {C}$ is a smooth and proper $\mathbf {F}_p$ -linear $\infty $ -category which admits a smooth and proper lift to $\mathrm {BP}$ , then its Tate spectral sequence collapses at the $\mathrm{E}_2$ -page.

This was already known if $\mathcal {C}$ lifts all the way to $\text{S}^0$ ; see [Reference MathewMat20, Example 3.5]. In particular, therefore, one class of X for which $\operatorname {{\mathrm {QCoh}}}(\text{X})$ does satisfy the hypotheses of Proposition 3 and Theorem 2 are toric varieties; but in those cases, degeneration was already known for X of arbitrary dimension (since they are F-liftable). Interesting examples of Theorem 2 and Proposition 3 are currently lacking, but one would be most welcome.

Remark 7. One could also ask the following question: if $n\geq 0$ , is there an example of a smooth and proper $\mathrm {BP}\langle {n-1}\rangle $ -linear $\infty $ -category $\mathcal {C}$ which does not lift to a smooth and proper left $\mathrm {BP}\langle {n}\rangle $ -linear $\infty $ -category?

Remark 13. It follows from Proposition 12 that lifting an arbitrary-dimensional X to a $\mathrm {ku}^\wedge _p$ -scheme suffices to conclude Hodge-de Rham degeneration; in particular, this assumption is significantly stronger than those of Theorem 2. One intermediate between the assumptions of Proposition 12 and Theorem 2 is the following: one could assume that $\mathcal {O}_{\text{X}}$ only admit a lift to a sheaf of ${\mathbf {E}_{{m}}}$ - $\mathrm {BP}\langle {n-1}\rangle $ -algebras (whenever this makes sense). Proposition 12 corresponds to the case $n=2$ and $m=\infty $ , while Theorem 2 roughly corresponds to the case $m=1$ (and n arbitrary). What constraints does such a lifting impose on the Hodge-de Rham spectral sequence for X? For instance, if p is an odd prime, and $\mathcal {O}_{\text{X}}$ admits a flat lift to a sheaf of ${\mathbf {E}_{{2n+1}}}$ - $\mathrm {ku}^\wedge _p$ -algebras, then the general construction of power operations (following [Reference HopkinsHop14]) along with the equivalence $\text{L}_{\text{K}(1)} \mathrm {Conf}^{\mathrm {un}}_p(\mathbf {R}^{2n+1}) \simeq \text{L}_{\text{K}(1)} \text{S}^{-1}/p^n$ of [Reference DavisDav86] shows that $\mathfrak {X}_0$ has a lift of Frobenius modulo $p^{n+1}$ . In particular, if $\mathcal {O}_{\text{X}}$ admits a flat lift to a sheaf of ${\mathbf {E}_{{3}}}$ - $\mathrm {ku}^\wedge _p$ -algebras, and $\dim (\text{X})<p$ , then [Reference Deligne and IllusieDI87] implies that the Hodge-de Rham spectral sequence degenerates for X.

Remark 14. Finally, one might wonder whether a lifting of X to $\mathrm {BP}\langle {n-1}\rangle $ , or $\mathrm {ku}^\wedge _p$ , or even the sphere spectrum can be used to prove that the HKR spectral sequence degenerates. Unfortunately, it seems that there is no clear relationship between HKR degeneration and liftings to the sphere. For instance, the stack $\text{B}\mu _p$ over $\mathbf {Z}_p$ lifts to the p-complete sphere spectrum (by writing $\mu _p = \operatorname {\mathrm {Spec}} \text{S}[\mathbf {Z}/p]$ ), but the HKR spectral sequence for $\text{B}\mu _p$ does not degenerate by [Reference Antieau, Bhatt and MathewABM21, Theorem 4.6]. Nevertheless, there are some liftability and torsion-freeness criteria, such as those described by Antieau-Vezzosi in [Reference Antieau and VezzosiAV20, Remark 1.6 and Example 1.7], which do guarantee HKR degeneration.