Conjectural realization of subgroup ideals by pure-factor ideals

Determine whether, under the product-of-pure-subgroups hypotheses $(*)$, every nontrivial subgroup ramification ideal is equal to one of the ideals $I_{\gamma_{ij}-\mathbf D_i}$ arising from a pure factor.

Background

Theorem 4.1 proves that the ideals from pure factors provide a lower bound for arbitrary subgroup ideals and proves equality when one component gives a strict minimum. The conjecture proposed equality without the strict-minimum condition. Example 4.1 subsequently gives a defectless Cp×CpC_p\times C_p counterexample, so the explicitly stated conjecture is resolved negatively rather than remaining open.

References

The following natural conjecture, which appeared in an earlier version of this manuscript, is false. Suppose that we are in situation $(*)$. Then for every nontrivial subgroup $H$ of $\mathcal G$ there exist $i,j$ such that

I_H=I_{\gamma_{ij}-\mathbf D_i}.

— Ramification ideals for products of pure subgroups  (2608.21210 - Novacoski, 21 Aug 2026) in Conjecture 1 in Section 4, immediately before Example 4.1

It is therefore natural to ask for conditions under which these defectless factors can also be chosen pure.

— Ramification ideals for products of pure subgroups  (2608.21210 - Novacoski, 21 Aug 2026) in Section 5, “Final remarks and open problems”

It would also be interesting to extend the adapted-decomposition statement to more general $p$-groups which are products of pure subgroups. A natural problem is to identify a group- and filtration-theoretic condition on a family of pure subgroups $H_1,\ldots,H_r$ ensuring that every ideal-indexed higher ramification group is generated by an appropriate subfamily and that every ramification ideal is realized by one factor.

— Ramification ideals for products of pure subgroups  (2608.21210 - Novacoski, 21 Aug 2026) in Section 5, “Final remarks and open problems”

A second problem is to understand when the lower bound of Theorem \ref{mainresultsofar} is sharp without assuming a strict minimum. The counterexample shows that this question is controlled by cancellation in the associated graded ring and is reflected in the position of the chosen factors relative to the ramification filtration. This suggests studying a refined invariant recording the initial forms of the contributions in veryimport.

veryimport:

σbb−1=σ‾1bb(σ1b‾1b‾1−1)+⋯+(σrbb−1).\frac{\sigma b}{b}-1= \frac{\overline\sigma_1b}{b} \left(\frac{\sigma_1\overline b_1}{\overline b_1}-1\right) +\cdots+ \left(\frac{\sigma_rb}{b}-1\right).

— Ramification ideals for products of pure subgroups  (2608.21210 - Novacoski, 21 Aug 2026) in Section 5, “Final remarks and open problems”