تقرير
Towards a refinement of the Bloch-Kato conjecture
| العنوان: | Towards a refinement of the Bloch-Kato conjecture |
|---|---|
| المؤلفون: | Chebolu, Sunil K., Mináč, Ján, Okay, Cihan, Schultz, Andrew, Ure, Charlotte |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, Mathematics - K-Theory and Homology, Mathematics - Rings and Algebras, 12F10, 12G10, 20E18 |
| الوصف: | Rost and Voevodsky proved the Bloch-Kato conjecture relating Milnor $k$-theory and Galois cohomology. Their result implies that for a field $F$ containing a primitive $p$th root of unity, the Galois cohomology ring of $F$ with $\mathbb{F}_p$ coefficients is generated by elements of degree 1 as an $\mathbb{F}_p$-algebra. Therefore, for a given Galois extension $K/F$ and an element $\alpha$ in $H^*(\text{Gal}(K/F), \mathbb{F}_p)$ there exits a Galois extension $L/F$ containing $K/F$ such that the inflation of $\alpha$ in $H^*(\text{Gal}(L/F), \mathbb{F}_p)$ belongs to an $\mathbb{F}_p$-subalgebra of $H^*(\text{Gal}(L/F), \mathbb{F}_p)$ generated by $1$-dimensional classes. It is interesting to find relatively small explicit Galois extensions $L/F$ with the above property for a given $\alpha$ in $H^*(\text{Gal}(K/F), \mathbb{F}_p)$ as above. In this paper, we provide some answers to this question for cohomology classes in degree two, thus setting the first step toward refining the Bloch-Kato conjecture. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean and $p$-rigid fields. Additionally, as a byproduct of our work, we characterize elementary abelian $2$-groups as the only finite $p$-groups whose mod-$p$ cohomology ring is generated by degree-one elements. This provides additional motivation for studying refinements of the Bloch-Kato conjecture and exploring the connections between group cohomology and Galois theory. Comment: 24 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.13223 |
| رقم الأكسشن: | edsarx.2405.13223 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |