Finiteness of Nonzero Bockstein Maps in Local Cohomology

Definitions and mathematical statement

Let $n\geq 0$ be an integer, let $R={\mathbb{Z}}[x_1,\ldots,x_n]$, and fix an ideal $\mathfrak a\subseteq R$. For an $R$-module $M$, define \[ \Gamma_{\mathfrak a}(M)=\{m\in M:\mathfrak a^t m=0\text{ for some integer }t\geq 1\}, \qquad H^k_{\mathfrak a}(M)=R^k\Gamma_{\mathfrak a}(M)\quad(k\geq 0). \] For each positive prime integer $p$, the short exact sequence of $R$-modules \[ 0\longrightarrow R/pR\xrightarrow{\iota_p}R/p^2R\xrightarrow{q_p}R/pR\longrightarrow 0, \qquad \iota_p(r+pR)=pr+p^2R, \] where $q_p$ is reduction modulo $p$, defines connecting homomorphisms \[ \beta_k^p:H^k_{\mathfrak a}(R/pR)\longrightarrow H^{k+1}_{\mathfrak a}(R/pR). \] These are the Bockstein homomorphisms in the question; local cohomology on the quotient modules uses the image of $\mathfrak a$.

Question. For every such $R$ and $\mathfrak a$, is there a finite set $S=S(R,\mathfrak a)$ of positive prime integers such that \[ \beta_k^p=0\qquad\text{for every prime }p\notin S\text{ and every integer }k\geq 0? \] The supplied statement leaves the quantification of $k$ implicit. The displayed formulation uses all nonnegative degrees. It is equivalent here to asking for finitely many exceptional primes separately for each fixed degree: if $\mathfrak a$ has $r$ generators, then $H^j_{\mathfrak a}(M)=0$ for $j>r$, so only finitely many degrees can contribute.

The supplied curation labels this question open and describes it as an unresolved-context fragment concerning closure and local-cohomology statements over polynomial rings over the integers. It characterizes the report as supplying supporting evidence rather than a proof of a general resolution. These are supplied qualifications, not an independently verified status assessment.

Short English statement

For a fixed ideal in a polynomial ring in finitely many variables over ${\mathbb{Z}}$, must the Bockstein maps on local cohomology modulo $p$ vanish in every degree for all but finitely many primes $p$? The supplied open label and source-verification claims require independent review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001240$, source problem code catalogue entry, title "Finiteness of Nonzero Bockstein Maps in Local Cohomology." Collection: Oberwolfach Reports Open Problems; proposed year: 2009. No individual proposer is supplied.
• Supplied source citation: Kommutative Algebra (2009), Oberwolfach Reports, report 22, pp. 1157--1224, DOI 10.4171/owr/2009/22. https://doi.org/10.4171/owr/2009/22. The supplied background attributes the record to workshop reports and their open-problem discussions.
• Supplied literature source, "Oberwolfach Report 22, Kommutative Algebra": https://doi.org/10.4171/owr/2009/22.
• Supplied literature source, "original report and surrounding discussion": https://doi.org/10.4171/owr/2009/22.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. It labels the question open, calls the material an unresolved-context fragment concerning closure and local cohomology, and says the report supplies supporting evidence rather than a proof of a general resolution. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement status corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied metadata; no independent source consultation is asserted here.