Nonexistence of nonzero projectives in graded comodules over the dual Steenrod algebra
For every prime , let denote the mod- dual Steenrod algebra and let denote the full category of graded -comodules. The conjecture asserts that every projective object of is zero: .
References
Primary source
Additional references
- Ext-groups of graded comodules — arXiv — Andrew Salch
Progress summary
A September 2026 preprint claims to settle the question by proving that the full graded-comodule category has no nonzero projective objects, but the result has not been independently checked.
The problem asks whether every projective object in the full category of graded comodules over the dual Steenrod algebra is zero. Andrew Salch’s new preprint claims an affirmative answer using results about preservation of Ext-groups, with further mapping formulas as consequences.
Known results
- A 2016 result gives the connective graded-comodule category a projective generator for connective finite-type flat graded Hopf algebroids, including the mod- dual Steenrod algebra; this does not address the full graded category.
- A 2023 result shows that the dual-Steenrod-algebra comodule category fails axiom for every , but does not establish nonexistence of nonzero projectives.
September 2026 claimed resolution
In a September 2026 preprint, Andrew Salch states that Ext-preservation results prove the old nonexistence conjecture and derive related mapping formulas. The claim is currently unverified.
Current status (as of September 2026): A new preprint claims the full nonexistence theorem, while independent confirmation is not recorded; connective-subcategory results remain distinct from the full-category claim.
Solutions 0
No solutions have been posted yet.