A Note on Topological Hochschild Homology Relative to $\Sphere_{W(k)}[x_0,x_1,\ldots,x_n]$

Abstract: We explain the relation between the relative topological Hochschild homology $\THH(R/\Sphere_{W(k)}[x_0,\ldots,x_n])$ and the Nygaard completed Frobenius twisted relative prismatic cohomology $\widehat{\Prism}<sup>{(1)}_{R/W(k)[x_0,\ldots,x_n]<sup>\wedge}$, where $W(k)[x_0,x_1,\ldots,x_n]\rightarrow R$ is relatively quasiregular semiperfectoid. As an application, for $R=\Z_p[x]/(px)$, we compute $π_*\THH(R)<sup>\wedge_p$ by descent along $\THH(R)^{\wedge_p\rightarrow} \THH(R/\Sphere_p[z,x])$, where $R=\Z_p[x]/(px)$ is regarded as an $\Einfty$-$\Sphere_p[z,x]$-algebra through $\Sphere_p[z,x]\xrightarrow{z\mapsto p,x\mapsto x}\Z_p[x]/(px)$.

• The paper identifies relative THH, TC−, and TP over spherical polynomial bases with graded pieces of Nygaard-completed, Frobenius-twisted relative prismatic cohomology.
• The paper develops explicit prismatic-envelope generators and uses them to describe relative THH homotopy groups through divided-power algebras under quasiregular semiperfectoid hypotheses.
• The paper computes a multiplicative E²-page for the descent spectral sequence of THH(ℤp[x]/(px))p̂, which collapses by degree reasons but retains unresolved extension and multiplicative-structure problems.

This paper by Jingbang Guo establishes an identification between relative topological Hochschild homology over the spherical polynomial base $B_n = S_{W(k)}[x_0,\ldots,x_n]$ and Nygaard-completed Frobenius-twisted relative prismatic cohomology, and applies this machinery to compute a descent spectral sequence for $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$. The work synthesizes techniques from Bhatt–Morrow–Scholze, Bhatt–Scholze, Antieau–Krause–Nikolaus, and Liu–Wang into a uniform statement for relatively quasiregular semiperfectoid maps out of $B_n$.

Relative THH and cyclotomic bases

The foundational setup concerns $THH(-/E)$ for $E$-rings. The author recalls that while $THH(A) = THH(A/\mathbb{S})$ carries a canonical cyclotomic structure, relative $THH$ generally does not; the obstruction is the absence of a Tate diagonal for general bases (e.g., $\mathbb{D}(\mathbb{Z})$, by Nikolaus–Scholze). The relevant class of bases admitting a cyclotomic lift of $THH(-/E)$ is that of cyclotomic bases in the sense of Liu–Wang. The key observation is that if $E$ is a $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$0-complete connective $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$1-ring with $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$2 a perfect ring concentrated in degree zero — as holds for spherical Witt vectors $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$3 and for their polynomial extensions $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$4, whose cyclotomic Frobenius sends $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$5 — then $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$6 is an equivalence, so $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$7 is an equivalence for any $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$8-algebra $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$9. This justifies treating $B_n$0 as a stand-in for the sphere spectrum in prismatic computations, with the $B_n$1-ring $B_n$2 carrying $B_n$3.

The main identification

The central result identifies homotopy groups of relative $B_n$4, $B_n$5, and $B_n$6 with graded pieces of relative prismatic cohomology data. For a map $B_n$7 that is relatively quasiregular semiperfectoid — meaning $B_n$8 has bounded $B_n$9-power torsion and is $THH(-/E)$0-complete, the map is surjective, $THH(-/E)$1 contains a distinguished ideal $THH(-/E)$2, and $THH(-/E)$3 has $THH(-/E)$4-complete Tor-amplitude concentrated in degree $THH(-/E)$5 — the paper proves:

$THH(-/E)$6

with analogous statements for $THH(-/E)$7, and with the canonical map and cyclotomic Frobenius corresponding respectively to Nygaard filtration inclusions and the composition $THH(-/E)$8.

The proof strategy is descent along Frobenius perfection. Base-changing along $THH(-/E)$9 reduces to the case of ordinary quasiregular semiperfectoids, where the BMS/BS identification of $E$0 with $E$1 applies. Two base-change compatibilities drive the argument: prismatic cohomology satisfies base change in the prism (by AKN), and $E$2 satisfies flat base change since $E$3 is $E$4-completely faithfully flat. Flat descent then recovers the absolute statement. Notably, the author attributes the key idea to BMS Proposition 11.10 and notes that Krause has announced a more general version allowing arbitrary bases and full generality of prismatic cohomology relative to $E$5-rings; the present note is thus positioned as an expository consolidation with concrete applications rather than maximal generality.

Prismatic envelopes and explicit generators

To make the abstract identification computable, the paper analyzes prismatic envelopes $E$6 for quotients $E$7 with $E$8 distinguished principal and the $E$9 Koszul regular mod $THH(A) = THH(A/\mathbb{S})$0. Two technical lemmas are established: first, the elements $THH(A) = THH(A/\mathbb{S})$1 defined recursively via the $THH(A) = THH(A/\mathbb{S})$2-structure satisfy $THH(A) = THH(A/\mathbb{S})$3 inside the envelope, hence lie in $THH(A) = THH(A/\mathbb{S})$4; second, they obey the divided-power-type relation $THH(A) = THH(A/\mathbb{S})$5. Under either of two hypotheses — $THH(A) = THH(A/\mathbb{S})$6 $THH(A) = THH(A/\mathbb{S})$7-torsion-free with $THH(A) = THH(A/\mathbb{S})$8 for all $THH(A) = THH(A/\mathbb{S})$9, or $THH$0 with $THH$1 — the author concludes:

$THH$2

where $THH$3 corresponds to the class of $THH$4 and $THH$5 to $THH$6 up to units. The second hypothesis is handled by a reduction to the first via adjoining variables $THH$7 with $THH$8 and mapping $THH$9 to $\mathbb{D}(\mathbb{Z})$0. When $\mathbb{D}(\mathbb{Z})$1 has $\mathbb{D}(\mathbb{Z})$2-torsion, the divided power structures require the specification given by these generators, which the author notes explicitly.

Application: the descent spectral sequence for $\mathbb{D}(\mathbb{Z})$3

Following the Liu–Wang strategy for $\mathbb{D}(\mathbb{Z})$4, the paper computes $\mathbb{D}(\mathbb{Z})$5 for $\mathbb{D}(\mathbb{Z})$6 via descent along $\mathbb{D}(\mathbb{Z})$7, where $\mathbb{D}(\mathbb{Z})$8, $\mathbb{D}(\mathbb{Z})$9. The coskeleton filtration yields a multiplicative second-quadrant spectral sequence whose $THH(-/E)$0-page is computed as $THH(-/E)$1 over the Hopf algebroid

$THH(-/E)$2

The right unit acts by $THH(-/E)$3 and a nontrivial formula on $THH(-/E)$4 involving both difference classes. Since $THH(-/E)$5 is concentrated in even degrees, the spectral sequence collapses at $THH(-/E)$6 for degree reasons. The author constructs a short resolution by relative injectives (in Ravenel's sense), reducing the cobar complex to a three-term complex governed by explicit differential operators $THH(-/E)$7 (an $THH(-/E)$8-linear derivation-like operator mixing $THH(-/E)$9-degree and $E$0-degree through multiplication by $E$1) and $E$2 ($E$3-linear, sending $E$4).

The resulting $E$5-page exhibits genuinely intricate structure. In summary form:

Position	Value
$E$6	$E$7
$E$8, $E$9	$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$00
$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$01	$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$02
$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$03	$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$04
$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$05, $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$06	non-split extension of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$07 by torsion terms
$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$08	$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$09
$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$10, $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$11	$THH(\mathbb{Z}_p[x]/(px))_p^\wedge$12

Here $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$13 is an explicit alternating sum involving denominators $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$14, reflecting the interplay between the $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$15-adic valuation of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$16 and division by $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$17 in $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$18. The $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$19 column fits non-split exact sequences; when $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$20, an additional summand $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$21 appears, recording the failure of invertibility of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$22 in $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$23. The author also observes that from the $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$24-page onward the spectral sequence depends only on $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$25 itself, being identifiable with the motivic-filtration spectral sequence à la BMS Proposition 7.13, i.e., computing $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$26.

Limitations and open questions

Two significant gaps are conceded directly. First, the extension problem for the spectral sequence remains unsolved, and the multiplicative structure on the $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$27-page is not specified; consequently the computation determines $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$28 only up to hidden extensions. The author states that a forthcoming joint work with Wei Yang will resolve both by constructing a suitable DGA model. Second, the main theorem is stated under a deliberately restricted definition of relative quasiregular semiperfectoid requiring surjectivity of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$29; the more general framework of AKN would remove this hypothesis, and the fully general statement (arbitrary bases, motivic filtrations on all of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$30, $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$31, $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$32) is deferred to Krause's announced work. Additionally, the identification of the spectral sequence with the motivic filtration relies on results cited from ABBK and AKN rather than being proved self-containedly here.

Conclusion

The paper provides a clean, usable bridge between relative $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$33 over spherical polynomial bases and relative prismatic cohomology, together with explicit generator-level control of the relevant prismatic envelopes. Its concrete payoff is a fully collapsed, explicitly computed $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$34-page for the descent spectral sequence converging to $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$35, exhibiting nontrivial torsion phenomena tied to the arithmetic of $THH(\mathbb{Z}_p[x]/(px))_p^\wedge$36. The remaining extension problem and multiplicative structure constitute the natural next target, which the author addresses in subsequent work.