Prismatic Cohomology of Local Complete Intersections

Definitions and mathematical statement

Let $p$ be a prime, let $n\geq 0$ be an integer, and let $Y\hookrightarrow{\mathbb{P}}^n_{{\mathbb{Z}}_p}$ be a closed subscheme, where ${\mathbb{Z}}_p$ is the ring of $p$-adic integers. Write \[ Y_{{\mathbb{F}}_p}:=Y\times_{\operatorname{Spec}{\mathbb{Z}}_p}\operatorname{Spec}{\mathbb{F}}_p. \] Assume that $Y_{{\mathbb{F}}_p}$ is a local complete intersection over ${\mathbb{F}}_p$: locally it admits a closed immersion into a smooth ${\mathbb{F}}_p$-scheme with ideal generated by a regular sequence. The supplied hypothesis concerns the special fiber; no additional flatness or local complete-intersection hypothesis over ${\mathbb{Z}}_p$ is imposed here.

The source introduces an ambient infinitesimal object $T_n$ in a relative periodic-cyclic-homology setup and uses the notation \[ A:=TP_0(\mathcal O_Y/\mathcal O_{T_n}),\qquad \operatorname{Ext}^{j}_{\mathrm{P\text{-}Mod}}\left(A,A\{i\}\right). \] Here $\mathrm{P\text{-}Mod}$, the relative $TP$ construction, and the twist $\{i\}$ are retained as source notation; their definitions are not supplied in the record.

Question. Under the stated special-fiber hypothesis, do these Ext groups identify with the corresponding prismatic cohomology of $Y$, and does the resulting spectral sequence with abutment $TP_{2i-j}(Y)$ identify with the Bhatt--Morrow--Scholze spectral sequence?

Unresolved qualifications. The record does not specify the construction of $T_n$, the category $\mathrm{P\text{-}Mod}$, the relative $TP$ theory or its completion conventions, or the meaning of $\{i\}$. It also omits the ranges of $i,j$, the page and indexing of the spectral sequence, and its convergence conditions. The base prism, completion, cohomological degree, and twist of the proposed prismatic cohomology are unspecified. Consequently, the question above preserves the extracted target but does not determine a precise isomorphism or a fully specified comparison of spectral sequences.

Short English statement

For a closed subscheme of ${\mathbb{P}}^n_{{\mathbb{Z}}_p}$ whose special fiber is a local complete intersection, the question asks whether certain relative $TP$ Ext groups compute prismatic cohomology and whether their spectral sequence is the Bhatt--Morrow--Scholze spectral sequence. Essential constructions and comparison conventions are missing from the supplied record, so a precise formulation requires mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004168$, source problem code catalogue entry, title Prismatic Cohomology of Local Complete Intersections; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Algebraic K-theory (2020), Oberwolfach Reports, pp. 1737--1790, DOI 10.4171/owr/2019/29. Supplied source URL and literature-source link labeled "OWR primary report": https://doi.org/10.4171/owr/2019/29.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the conjectured identification with prismatic cohomology and the Bhatt--Morrow--Scholze spectral sequence remains unresolved in the exact extracted form. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a supplied verification note claiming comparison against the official Oberwolfach report and preservation of its hypotheses, notation, and target. That claim is recorded as provenance and has not been independently verified here.