Fixed-ambient extraction of roots of finite Frobenius modules

Frobenius roots: a terminating kernel certificate and an exact one-variable family

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

In characteristic p, the Frobenius map sends $r$ to $r^p$; $I^{[p]}$ is generated by pth powers of elements of I. A unit F-module carries an isomorphism to its Frobenius pullback. A root is finite data whose iterated Frobenius maps generate the stated unit module, including local-cohomology modules in the regular-ring setting of the source.

Statement recorded in the supplied source.

Local cohomology in characteristic p

If R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.

Given an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\mathrm{Ext}^i(R/I,R)\to \mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$? [S2]

Scope qualification.

The regular-ring assumption and the supplied Ext map are retained. The source does not prescribe a ring encoding or a polynomial complexity bound.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness.

Short English statement

Can finite Frobenius-module data be converted efficiently into a root description of a local-cohomology module?

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002190 (AIM-INFRASTRUCTURE-0090). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 AIM Workshop Problem Lists, original problem source linked by catalogue record AIM-INFRASTRUCTURE-0090 http://aimpl.org/macaulay2efie/2/.

Verbatim supplied catalogue question

Local cohomology in characteristic p

If R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.

Given an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\mathrm{Ext}^i(R/I,R)\to \mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$?

Attempt and results

We first state the hypotheses needed for a kernel algorithm, then test it on an explicit regular ring. These standard constructions overlap with the supplied research summary; complete proofs below replace reliance on that summary. They do not supply the requested efficient general implementation.

A fixed-module kernel iteration under exact Frobenius pullback.

Let $R$ be Noetherian, $F$ an exact additive functor on its module category, and $\beta:M\to F(M)$ a map with $M$ finitely generated. In the application, $F$ is Frobenius pullback. Write $\beta_e=F^{e-1}(\beta)\cdots F(\beta)\beta$ for $e\ge1$ and $\beta_0=1_M$. Put $K_e=\ker\beta_e$. Since $\beta_{e+1}=F(\beta_e)\beta$, exactness gives \[ K_0=0,\qquad K_{e+1}=\beta^{-1}(F(K_e)). \] Here $F(K_e)$ is identified with its injected image in $F(M)$. The kernels form an increasing chain because each iterate factors through the preceding one. Noetherianity gives $K_t=K_{t+1}$ for some finite $t$. The displayed recursion then implies by induction that all later kernels equal $K_t=K$. Further, $\beta(K)\subseteq F(K)$, and exactness identifies $F(M/K)$ with $F(M)/F(K)$. Thus the induced map $\bar\beta:M/K\to F(M/K)$ has kernel $K_{t+1}/K_t=0$.

The original and quotient directed systems have the same direct limit. Indeed their stage maps are the quotient maps $F^j(M)\to F^j(M/K)$, so they induce a surjection on limits. If a stage-$j$ element maps to zero in the quotient limit, at some later stage $l$ its image lies in $F^l(K)$. The map from stage $l$ to stage $l+t$ is $F^l(\beta_t)$ and annihilates $F^l(K)$ by exactness, so the element is already zero in the original limit. This proves injectivity. It also proves that an equality of successive computed kernels is a valid stopping certificate. ACC alone supplies no numerical bound on $t$ or on the sizes of the presentations.

Exactness in a stated regular example.

For $R=\mathbb F_p[x]$, every polynomial has a unique expression $\sum_{i=0}^{p-1}x^i f_i(x)^p$, by sorting exponents modulo $p$ and using that coefficients have unique $p$th roots in $\mathbb F_p$. Thus $R$, regarded as a right $R$-module by $a\cdot r=ar^p$, is free with basis $1,x,\ldots,x^{p-1}$. Tensoring with this module is exact: after choosing the basis it is a direct sum of $p$ copies of the underlying exact sequence. This establishes the needed hypothesis here without silently using flatness for arbitrary rings. It also identifies $F(R/(x^m))=R/(x^{pm})$.

A complete scalar test family.

Fix $m\ge1$ and define $\beta:R/(x^m)\to R/(x^{pm})$ by multiplication by $x^a$, where $a\ge(p-1)m$ is an integer. This inequality makes the map well defined. At stage $e$ the module is $R/(x^{mp^e})$, and the transition multiplies by $x^{ap^e}$. The $e$-fold map from stage zero multiplies by $x^{D_e}$, where $D_e=a(p^e-1)/(p-1)$. Consequently a monomial $x^j$, $0\le j<m$, belongs to $K_e$ exactly when $j+D_e\ge mp^e$. If $a=(p-1)m$, then $D_e=m(p^e-1)$ and no such $j$ is killed. The module at stage $e$ embeds in $R_x/R$ by \[ \bar v\longmapsto v/x^{mp^e}+R. \] The map is well defined and injective, since its kernel consists exactly of multiples of $x^{mp^e}$, and it respects the transition maps. Every class of a fraction with power-of-$x$ denominator occurs once $mp^e$ exceeds that power, so the direct limit is $R_x/R$. If $a>(p-1)m$, then $D_e-mp^e=(a/(p-1)-m)p^e-a/(p-1)$ eventually becomes nonnegative. All of $M$ is then in $K_e$, and the stable quotient and direct limit are zero. Thus a single extra exponent can change a nonzero injective root to an ultimately killed generating module. An implementation that simply calls all finite generating data a root would miss this distinction.

Context, adversarial audit, and gap

The source names a particular Ext map and a local-cohomology module. No proof identifying that map's direct limit with the stated local cohomology, or algorithm constructing its presentations, is supplied here; such identification must not be inferred from the abstract kernel lemma alone. The original AIMPL page failed retrieval, and the primary workshop report supplies only context. The audit checked functor exactness, the direction of the quotient map, stage indices, equality as a stopping condition, and the threshold exponent. No numerical experiments, package implementation, or complexity bound is claimed.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The kernel theorem is conditional on effective exact pullback; efficient general root extraction and the canonical Ext-to-local-cohomology identification are not established here.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• AIM, Macaulay2: expanded functionality and improved efficiency, official workshop summary; local-cohomology, DG, and direct-summand discussion read on 11 October 2026. The original AIMPL problem pages failed retrieval; their wording is available here only from the supplied snapshot. https://aimath.org/pastworkshops/macaulay2efierep.pdf.

Final status

Unresolved in this attempt. The kernel theorem is conditional on effective exact pullback; efficient general root extraction and the canonical Ext-to-local-cohomology identification are not established here.