Axiomatic stable homotopy 5 — John Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he…

Definitions and mathematical statement

For graded-commutative algebras of characteristic $p$, a conventional Quillen $F$-isomorphism (written "f-isomorphism" in the source) is a homomorphism with nilpotent kernel and with power-surjectivity: for each homogeneous element $b$ of the target, some $b^{p^r}$ lies in its image. Neither a uniform exponent nor an actual ring isomorphism is required. The source does not state its exact convention.

Axiomatic stable-homotopy problem 5. Can Palmieri's methods work in other specific non-Noetherian stable homotopy categories? In particular, following Haynes Miller's suggestion, use them to calculate group cohomology of $\operatorname{GL}(\infty)$ up to f-isomorphism. A precise version must specify the coefficient/base ring and topology of that group, the cohomology coefficients, and an explicit computable comparison ring and f-isomorphism or appropriate comparison diagram.

For example, only after a ring $R$ is selected can one define a standard stable linear group $\operatorname{GL}_\infty(R)=\varinjlim_m\operatorname{GL}_m(R)$, using block stabilization. Discrete group cohomology and a topological-group version are different targets. The excerpt gives none of these choices, and $\operatorname{GL}(\infty)$ is not silently identified with $\operatorname{GL}_\infty({\mathbb{F}}_p)$.

The source reports that John Palmieri found a computable ring f-isomorphic to the Adams spectral sequence's $E_2$-term, describing this as determination up to nilpotence; Hovey qualifies the prime by "I think this is only at $p=2$." In the usual mod-two sphere setup that term is $\operatorname{Ext}_{\mathcal A}^{s,t}({\mathbb{F}}_2,{\mathbb{F}}_2)$, but the exact spectral-sequence setting and comparison ring are absent. The report concerns $E_2$ up to f-isomorphism, not its full ring structure, differentials, abutment, or an explicit computation of sphere homotopy groups. The other categories and the sense in which they are non-Noetherian also require specification.

Short English statement

Extend Palmieri's nilpotence/f-isomorphism method to other non-Noetherian stable settings, especially the cohomology of a specified stable general linear group. The group, coefficients, and comparison ring are missing, and the source's prime-two scope is explicitly tentative.

Sources

• Local UnsolvedMath registry, permanent ID 11100037, source code catalogue entry; proposer Mark Hovey, "Axiomatic stable homotopy," item 5. John Palmieri is credited for the Adams $E_2$ comparison and Haynes Miller for the $\operatorname{GL}(\infty)$ suggestion.
• Hovey's Algebraic Topology Problem List, surviving mirror: https://www-users.cse.umn.edu/~tlawson/hovey/. Recorded access 2026-07-29; extraction html-ordered-list. The mirror has 65 top-level items across nine category pages.
• The supplied summary refers to Palmieri's "Quillen stratification for the Steenrod algebra" and a memoir on Adams $E_2$, and mentions Friedlander--Mislin and Duy Nguyen in related detection work. No complete bibliography or exact matching theorem is supplied, and its final discussion is truncated.
• Supplied status partially_solved and PARTIAL-PROGRESS. The summary describes the framework as established but the specific stable-GL target as not definitively resolved in its audit. Current primary-literature status and redistribution terms remain NEEDS_REVIEW.