An arXiv preprint dated 20 August 2026 gives a structural account of relative topological Hochschild homology (THH), a way of organizing algebraic information degree by degree for a map of rings. Under the paper’s stated assumptions, THH has homotopy groups only in even degrees and those groups are canonically identified with an associated graded Nygaard object.
One framework for three theories
The result extends to two related constructions, TC⁻ and TP. Their homotopy groups are also confined to even degrees. TC⁻ is identified with a direct sum of Nygaard-filtered twists, while TP is identified with a direct sum of Frobenius-twisted prismatic objects.
The paper also tracks the maps connecting these theories. The canonical map from TC⁻ to TP becomes the Nygaard-filtration inclusions in the graded description, while the cyclotomic Frobenius on homotopy groups is represented by the stated c∘ϕ/Bₙ^∧ map between the displayed direct sums.
The assumptions matter
The theorem is framed for a specific algebraic setting. It starts with a surjective map Bₙ→R where R is p-complete and has bounded p-power torsion; the kernel must contain a distinguished ideal, and the cotangent complex must have p-complete Tor-amplitude [1,1].
In the principal distinguished-ideal case, when that ideal is generated by one element, the preprint gives an explicit graded-ring presentation for relative THH with polynomial and divided-power pieces.
A worked algebraic example
The application focuses on R = Zₚ[x]/(px), viewed as an E∞-algebra over the p-completed sphere polynomial ring Sₚ[z,x], with z sent to p and x sent to x.
To study its p-completed THH, the paper recovers the theory as the limit of an augmented cosimplicial relative-THH construction. Filtering that construction by its coskeleton produces a multiplicative, second-quadrant spectral sequence—a staged calculation whose pages narrow the possible homotopy groups—that converges to the homotopy of p-complete THH.
The E² calculation is reduced first to an Ext computation over a graded Hopf algebroid and then to a chain complex with D′_z and D′_x maps. The spectral sequence collapses at E²: entries vanish unless the row is i = 0, −1 or −2 and the degree j is even.
Three rows carry the answer
The surviving rows are described explicitly. At i = 0, degree zero is R, while higher even degrees are the Fₚ[x]xεₗ modules specified in the paper.
At i = −1, the entries are zero in degree 0 and Fₚ[x]dx in degree 2. For l ≥ 1, the row is described by non-split exact sequences, with one form when p does not divide l + 1 and another when p divides l + 1; the latter includes an additional (R/(l + 1,x^{p+1}))uˡ dz term.
At i = −2, the entries in degrees 0, 2 and 4 vanish. From there, the paper gives an explicit direct-sum description involving Fₚ[x]/(xᵖ) and the indicated t_zx, u and dz∧dx factors.
A precise result, not a finished calculation
The collapse is not a complete ring calculation. The extension problem—how the surviving graded pieces fit together—and the multiplicative structure remain unresolved, so the preprint does not give a complete explicit presentation of π_*THH(Zₚ[x]/(px))^∧ₚ.
The structural theorem remains tied to its stated ring-theoretic assumptions, and the explicit descent application concerns the single ring R = Zₚ[x]/(px).
Paper data and sources
Original title: A Note on Topological Hochschild Homology Relative to $\Sphere_{W(k)}[x_0,x_1,\ldots,x_n]$
Authors: Jingbang Guo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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