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# Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration
in characteristic $p>0$

S. K. Devalapurkar  1 Oxford St, Cambridge, MA 02139
<EMAIL_ADDRESS>
###### Abstract.

The goal of this note is to prove that Hodge-de Rham degeneration holds for
smooth and proper $\mathbf{F}_{p}$-schemes $X$ with $\dim(X)<p^{n}$ as soon as
its category of quasicoherent sheaves admits a lift to the truncated Brown-
Peterson spectrum $\mathrm{BP}\langle{n-1}\rangle$, and the Hochschild-
Kostant-Rosenberg spectral sequence for $X$ degenerates at the $E_{2}$-page.
This is obtained from a noncommutative version, whose proof is essentially the
same as Mathew’s argument in [Mat20].

Part of this work was done when the author was supported by the PD Soros
Fellowship and NSF DGE-2140743. I’m grateful to Ben Antieau, Bhargav Bhatt,
and Jeremy Hahn for suggestions which improved this note.

Let $X$ be a smooth and proper scheme over a perfect field $k$ of
characteristic $p>0$. In [DI87], Deligne and Illusie proved that the Hodge
decomposition holds for the de Rham cohomology of $X$ under certain
hypotheses: namely, if $\dim(X)<p$ and $X$ admits a smooth and proper lift to
the truncated Witt vectors $W_{2}(k)=W(k)/p^{2}$, they showed that the Hodge-
de Rham spectral sequence

$E_{1}^{\ast,\ast}=\mathrm{H}^{\ast}(X;\Omega^{\ast}_{X/k})\Rightarrow\mathrm{H}^{\ast}_{\mathrm{dR}}(X/k)$

collapses at the $E_{1}$-page.

In [DI87, Remarque 2.6(iii)] (see also [Ill96, Problem 7.10]), Deligne and
Illusie asked if the Hodge-de Rham spectral sequence could degenerate for a
smooth proper $k$-scheme $X$ with a lift to $W(k)/p^{2}$ (or even to $W(k)$),
without any dimension assumptions. This remarkable question has recently been
answered (in the negative) by Sasha Petrov in [Pet23]. Our goal in this note
is to study conditions on $X$ arising from chromatic homotopy theory which do
guarantee Hodge-de Rham degeneration if $\dim(X)>p$.

######  Recollection 1.

Let $X$ be a smooth scheme over a commutative ring $k$. One then has the HKR
and de-Rham-to-$\mathrm{HP}$ spectral sequences (see [ABM21, Definition 3.1]):

$\displaystyle E_{2}^{s,t}=\mathrm{H}^{s}(X;\wedge^{-t}L_{X/k})$
$\displaystyle\Rightarrow\pi_{-(s+t)}\mathrm{HH}(X/k),$ $\displaystyle
E_{2}^{s,t}=\mathrm{H}^{s-t}_{\mathrm{dR}}(X/k)$
$\displaystyle\Rightarrow\pi_{-(s+t)}\mathrm{HP}(X/k).$

Fix an ${\mathbf{E}_{{3}}}$-form of the ($p$-completed) truncated Brown-
Peterson spectrum $\mathrm{BP}\langle{n-1}\rangle$ of height $n-1$, which
exists thanks to [HW20, Theorem A]. By construction,
$\pi_{\ast}\mathrm{BP}\langle{n-1}\rangle\cong\mathbf{Z}_{p}[v_{1},\cdots,v_{n-1}]$
for classes $v_{i}$ in degree $2p^{i}-2$. By convention,
$\mathrm{BP}\langle{-1}\rangle=\mathbf{F}_{p}$. We also have
$\mathrm{BP}\langle{0}\rangle=\mathbf{Z}_{p}$, and
$\mathrm{BP}\langle{1}\rangle$ can be identified with the connective cover of
the Adams summand of $p$-completed complex K-theory. There is also a tight
relationship between $\mathrm{BP}\langle{2}\rangle$ and elliptic cohomology.

Our goal in this note is to prove:

######  Theorem 2.

Let $X$ be a smooth and proper scheme over111Here, $\mathbf{F}_{p}$ could be
replaced by any perfect field of characteristic $p>0$; we only use
$\mathbf{F}_{p}$ to avoid introducing conceptually unnecessary notation.
$\mathbf{F}_{p}$ of dimension $<p^{n}$. Suppose that:

  1. (a)

The HKR spectral sequence degenerates at the $E_{2}$-page; and

  2. (b)

$\operatorname{\mathrm{QCoh}}(X)$ lifts to a smooth and proper left
$\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category222Recall that at the
beginning of this article, we picked an ${\mathbf{E}_{{3}}}$-form of
$\mathrm{BP}\langle{n-1}\rangle$, which exists by [HW20, Theorem A]. Then, a
“left $\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category” is simply a
left $\mathrm{LMod}_{\mathrm{BP}\langle{n-1}\rangle}$-module in
$\mathrm{Pr^{L}}$, where $\mathrm{LMod}_{\mathrm{BP}\langle{n-1}\rangle}$ is
equipped with the ${\mathbf{E}_{{2}}}$-monoidal structure arising from the
${\mathbf{E}_{{3}}}$-structure on $\mathrm{BP}\langle{n-1}\rangle$. See
[Lur17, Variant D.1.5.1]..

Then the Hodge-de Rham spectral sequence

$E_{1}^{\ast,\ast}=\mathrm{H}^{\ast}(X;\Omega^{\ast}_{X/\mathbf{F}_{p}})\Rightarrow\mathrm{H}^{\ast}_{\mathrm{dR}}(X/\mathbf{F}_{p})$

collapses at the $E_{1}$-page, and the de-Rham-to-$\mathrm{HP}$ spectral
sequence collapses at the $E_{2}$-page.

The discussion in [ABM21, Remark 3.6] implies that if the HKR and Tate
spectral sequences both degenerate, then both the Hodge-de Rham and de Rham-
to-$\mathrm{HP}$ spectral sequences must also degenerate. It therefore
suffices to prove the following noncommutative statement333Our original proof
used the higher chromatic topological Sen operators from our forthcoming
article [Dev23] to argue in a manner similar to [BL22a, Example 4.7.17], but
we soon realized that the argument could be simplified much further. In
[Dev23, Remark C.14], we also phrase an analogue of Proposition 3 in stacky
language via the Sen operator of [BL22a] and the stack $BW^{\times}[F^{n}]$.
The expected isomorphism, which we hope to study in joint work with Jeremy
Hahn and Arpon Raksit, between $BW^{\times}[F^{n}]$ and the stack associated
to the motivic filtration of
$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)^{t{\mathbf{Z}/p}}/(p,\cdots,v_{n-1})$
was the original motivation for our result.:

######  Proposition 3.

Let $\mathscr{C}$ be a smooth and proper $\mathbf{F}_{p}$-linear
$\infty$-category such that $\pi_{j}\mathrm{HH}(\mathscr{C}/\mathbf{F}_{p})=0$
for $j\not\in[-p^{n},p^{n}]$. If $\mathscr{C}$ lifts to a smooth and proper
left $\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category, then the Tate
spectral sequence

$E_{2}^{\ast,\ast}=\mathrm{HH}(\mathscr{C}/\mathbf{F}_{p})[\hbar^{\pm
1}]\Rightarrow\mathrm{HP}(\mathscr{C}/\mathbf{F}_{p})$

collapses at the $E_{2}$-page.

######  Remark 4.

When $n=1$, Theorem 2 is part of the main result of [DI87]444As the reader may
have noticed, the title of this work is a tribute to the inspirational paper
[DI87].: 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 [Pet23] 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
$E_{1}$-page. If the HKR spectral sequence degenerates at the $E_{2}$-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 [DI87]: 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
$\widehat{\Omega}^{\not{D}}_{\mathfrak{X},0}\otimes_{\mathbf{Z}_{p}}\mathbf{F}_{p}$
of the zeroth generalized eigenspace of the diffracted Hodge complex (see
[BL22a, Remark 4.7.20] for this notion) and liftings of
$\operatorname{\mathrm{QCoh}}(\mathfrak{X})$ to
$\mathrm{BP}\langle{1}\rangle$.

######  Remark 5.

Were $\mathrm{BP}\langle{n-1}\rangle/(p^{2},v_{1}^{2},\cdots,v_{n-1}^{2})$ 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
$\mathrm{BP}\langle{n-1}\rangle/(p^{2},v_{1}^{2},\cdots,v_{n-1}^{2})$. This is
because one can prove that Lemma 9 continues to hold for
$\mathrm{BP}\langle{n-1}\rangle/(p^{2},v_{1}^{2},\cdots,v_{n-1}^{2})$.

Some preliminary calculations seem to suggest that Petrov’s first Sen class
(see [Pet23, Ill22]) is related to the obstruction in Hochschild cohomology to
lifting a $\mathbf{Z}_{p}$-scheme $\mathfrak{X}$ along the map
$\mathrm{BP}\langle{1}\rangle/v_{1}^{2}\to\mathbf{Z}_{p}$. 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

$\mathbf{Z}_{p}\to\mathbf{F}_{p}\xrightarrow{{P^{1}}}\mathbf{F}_{p}[2p-2]\xrightarrow{{\beta}}\mathbf{Z}_{p}[2p-1],$

where $P^{1}$ 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. On the other hand, the extension class for
$\mathscr{O}_{\mathfrak{X}}\to\mathrm{F}^{p}\widehat{\Omega}^{\not{D}}_{\mathfrak{X},0}\to
L\Omega^{p}_{\mathfrak{X}}[-p]$ is computed in [Pet23, Lemma 6.5] to be the
composite

$L\Omega^{p}_{\mathfrak{X}}[-p]\to
L\Omega^{p}_{\mathfrak{X}_{p=0}/\mathbf{F}_{p}}[-p]\xrightarrow{{c_{X,p}}}\mathscr{O}_{\mathfrak{X}_{p=0}}\xrightarrow{{\beta}}\mathscr{O}_{\mathfrak{X}}[1],$

where the “first Sen class” $c_{X,p}$ can be defined using Steenrod operations
on cosimplicial algebras via [Pet23, 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 $E_{2}$-page, then the differentials in
the Hodge-de Rham spectral sequence obstruct the lifting of
$\operatorname{\mathrm{QCoh}}(X)$ to a smooth and proper left
$\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category. In particular, one
consequence of Proposition 3 is the fact that if $\mathscr{C}$ is a smooth and
proper $\mathbf{F}_{p}$-linear $\infty$-category which admits a smooth and
proper lift to $\mathrm{BP}$, then the Tate spectral sequence collapses at the
$E_{2}$-page (with no further assumption on
$\mathrm{HH}(\mathscr{C}/\mathbf{F}_{p})$ vanishing outside a certain range!).

This was already known if $\mathscr{C}$ lifts all the way to $S^{0}$; see
[Mat20, Example 3.5]. In particular, therefore, one class of $X$ for which
$\operatorname{\mathrm{QCoh}}(X)$ 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 $\mathbf{F}_{p}$-linear $\infty$-category $\mathscr{C}$
which lifts to a smooth and proper left
$\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category, but not to a smooth
and proper left $\mathrm{BP}\langle{n}\rangle$-linear $\infty$-category?

The idea to prove Proposition 3 is essentially the argument of [Mat20], so we
recommend reading that paper first. Recall Bökstedt’s calculation that
$\pi_{\ast}\mathrm{THH}(\mathbf{F}_{p})\cong\mathbf{F}_{p}[\sigma]$, where
$\sigma$ lives in degree $2$. By [Mat20, Proposition 3.4], Proposition 3 is a
consequence of:

######  Proposition 8.

Let $\mathscr{C}$ be a smooth and proper $\mathbf{F}_{p}$-linear
$\infty$-category such that $\pi_{j}\mathrm{HH}(\mathscr{C}/\mathbf{F}_{p})=0$
for $j\not\in[-p^{n},p^{n}]$. If $\mathscr{C}$ lifts to a smooth and proper
left $\mathrm{BP}\langle{n-1}\rangle$-linear $\infty$-category, then
$\mathrm{THH}(\mathscr{C})$ is $\sigma$-torsionfree.

To prove Proposition 8, we need a lemma. The following result is essentially
[Mat20, Proposition 3.7]; it could also be proved using the methods of
[Dev23].

######  Lemma 9.

Let $M$ be a perfect $\mathrm{THH}(\mathbf{F}_{p})$-module such that
$\pi_{i}(M)=0$ for $i<a$. If $M$ lifts to a perfect
$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)$-module $\widetilde{M}$, then
$\sigma$-multiplication $\sigma:\pi_{i-2}M\to\pi_{i}M$ is injective for $i\leq
a+2p^{n}-1$.

###### Proof.

Let $I_{n}=(p,\cdots,v_{n-1})$. First, observe that the map
$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)\to\mathrm{THH}(\mathbf{F}_{p})$
factors through a map

$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)\otimes_{\mathrm{BP}\langle{n-1}\rangle}\mathbf{F}_{p}\simeq\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)/I_{n}\to\mathrm{THH}(\mathbf{F}_{p}).$

By [ACH21, Proposition 2.9] (see also [Dev23, Remark 2.2.5]), there is an
isomorphism

$\pi_{\ast}\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)/I_{n}\cong\mathbf{F}_{p}[\sigma^{2}(v_{n})]\otimes\Lambda(\sigma(t_{1}),\cdots,\sigma(t_{n})).$

Here, $|\sigma^{2}(v_{n})|=2p^{n}$ and $|\sigma(t_{i})|=2p^{i}-1$. Moreover,
the map
$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)/I_{n}\to\mathrm{THH}(\mathbf{F}_{p})$
induces a map of motivically filtered ring spectra (see [HRW22, Example
4.2.4]), and is given on motivic associated graded by a graded map of rings:

$\mathbf{F}_{p}[\sigma^{2}(v_{n})]\otimes\Lambda(\sigma(t_{1}),\cdots,\sigma(t_{n}))\to\mathbf{F}_{p}[\sigma].$

Since $\sigma^{2}(v_{n})$ lives in weight $p^{n}$, while $\sigma(t_{i})$ lives
in weight $p^{i}$ and degree $1$, we see that $\sigma(t_{i})\mapsto 0$ and
$\sigma^{2}(v_{n})\mapsto\sigma^{p^{n}}$. Therefore, $\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)\to\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p})$ factors through the map $\tau_{\leq
0}\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)/I_{n}\simeq\mathbf{F}_{p}\to\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p})$ of ring spectra.

To prove the result of the lemma, we can assume without loss of generality
that $a=0$. Then, there is a map

$M\to\tau_{\leq 2p^{n}-1}\widetilde{M}\otimes_{\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)}\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p}),$

which is an equivalence on $\tau_{\leq 2p^{n}-1}$. But the map $\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)\to\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p})$ factors through
$\mathbf{F}_{p}\to\tau_{\leq 2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p})$, so we see
that $\tau_{\leq 2p^{n}-1}M$ is a free $\tau_{\leq
2p^{n}-1}\mathrm{THH}(\mathbf{F}_{p})$-module on classes in nonnegative
degrees. Therefore, $\sigma$-multiplication is injective through the stated
range. ∎

Proposition 8 is now a consequence of the following, which is essentially
[Mat20, Proposition 3.8].

######  Proposition 10.

Let $M$ be a perfect $\mathrm{THH}(\mathbf{F}_{p})$-module with Tor-amplitude
in $[-p^{n},p^{n}]$. If $M$ lifts to a perfect
$\mathrm{THH}(\mathrm{BP}\langle{n-1}\rangle)$-module $\widetilde{M}$, then
$M$ is free.

###### Proof.

The argument is the same as in [Mat20, Proposition 3.8]. Indeed, $M$ is a
direct sum of $\mathrm{THH}(\mathbf{F}_{p})$-modules which are free or of the
form $M_{i,j}=\Sigma^{i}\mathrm{THH}(\mathbf{F}_{p})/\sigma^{j}$ (see [Mat20,
Proposition 3.3]). Since $M_{i,j}$ has Tor-amplitude in $[i,i+2j+1]$, the
condition on $M$ implies that $M_{i,j}$ could appear as a summand of $M$ if
and only if $-p^{n}\leq i\leq i+2j+1\leq p^{n}$.

The class $\sigma^{j-1}[i]\in\pi_{i+2j-2}M_{i,j}$ is killed by $\sigma$, so
taking $a=-p^{n}$ in Lemma 9, we see that

$i+2j>-p^{n}+2p^{n}-1=p^{n}-1.$

In particular, $i+2j+1>p^{n}$, which contradicts $i+2j+1\leq p^{n}$.
Therefore, no $M_{i,j}$ can be a summand of $M$, so that $M$ is free. ∎

In the remainder of this note, we will clarify the relationship between
liftings of $X$ itself and Hodge-de Rham degeneration. First, observe that
assumption (b) in Theorem 2 is only a condition on
$\operatorname{\mathrm{QCoh}}(X)$, which is essentially why Proposition 3 is
the more natural noncommutative statement. One might hope that assumption (a)
in Theorem 2 could be removed if we strengthened (b) to assume that $X$ itself
lifted as a spectral scheme to $\mathrm{BP}\langle{n-1}\rangle$. (The
appropriate assumption on $X$ is probably a statement at the level of stacks
over $\mathscr{M}_{\mathrm{FG}}$ defined via the even filtration of [HRW22].)

Unfortunately, the question of lifting $X$ often does not make sense in the
current setup [Lur17] of spectral algebraic geometry, since
$\mathrm{BP}\langle{n-1}\rangle$ is generally not an
${\mathbf{E}_{\infty}}$-ring [Law18, Sen17]. Nevertheless, the question does
make sense if, for instance, $n=2$ (since $\mathrm{BP}\langle{1}\rangle$ is an
${\mathbf{E}_{\infty}}$-ring). In this case, requiring that $X$ lift is
significantly stronger than the assumptions of Theorem 2, as shown by the
following.

######  Proposition 11.

Let $X$ be a smooth and proper $\mathbf{F}_{p}$-scheme. If $X$ lifts to a
$p$-adic flat $\mathrm{ku}^{\wedge}_{p}$-scheme $\mathfrak{X}$, then the
Hodge-de Rham spectral sequence for $X$ degenerates at the $E_{1}$-page.

###### Proof.

The lift $\mathfrak{X}$ defines a lift of $X$ to $\mathbf{Z}_{p}$ via
$\mathfrak{X}_{0}:=\mathfrak{X}\otimes_{\mathrm{ku}^{\wedge}_{p}}\mathbf{Z}_{p}$.
It suffices to show that $\mathfrak{X}_{0}$ admits a $\delta$-ring structure;
then, the Hodge-Tate gerbe over $\mathfrak{X}_{0}$ (from [BL22b, Proposition
5.12]) splits, so that the conjugate (and hence Hodge-de Rham) spectral
sequence for $X$ degenerates. The fact that $\mathfrak{X}$ is assumed to be
flat implies that
$\pi_{0}L_{K(1)}\mathscr{O}_{\mathfrak{X}}\cong\pi_{0}\mathscr{O}_{\mathfrak{X}}=\mathscr{O}_{\mathfrak{X}_{0}}$.
By [Hop14], if $R$ is any $K(1)$-local ${\mathbf{E}_{\infty}}$-ring, then
$\pi_{0}(R)$ admits a $\delta$-ring structure (functorially in $R$).
Globalizing, we see that
$\pi_{0}L_{K(1)}\mathscr{O}_{\mathfrak{X}}=\mathscr{O}_{\mathfrak{X}_{0}}$ has
a $\delta$-ring structure, which implies the desired claim. ∎

######  Remark 12.

It follows from Proposition 11 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 11
and Theorem 2 is the following: one could assume that $\mathscr{O}_{X}$ only
admit a lift to a sheaf of
${\mathbf{E}_{{m}}}$-$\mathrm{BP}\langle{n-1}\rangle$-algebras (whenever this
makes sense). Proposition 11 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 $\mathscr{O}_{X}$
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 [Hop14]) shows that
$\mathfrak{X}_{0}$ has a lift of Frobenius modulo $p^{n+1}$. In particular, if
$\mathscr{O}_{X}$ admits a flat lift to a sheaf of
${\mathbf{E}_{{3}}}$-$\mathrm{ku}^{\wedge}_{p}$-algebras, and $\dim(X)<p$,
then [DI87] implies that the Hodge-de Rham spectral sequence degenerates for
$X$.

######  Remark 13.

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
$B\mu_{p}$ over $\mathbf{Z}_{p}$ lifts to the $p$-complete sphere spectrum (by
writing $\mu_{p}=\operatorname{Spec}S[\mathbf{Z}/p]$), but the HKR spectral
sequence for $B\mu_{p}$ does not degenerate by [ABM21, Theorem 4.6].

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