AI 中文总结
研究全局域阿贝尔扩张\(K/k\)中广义施蒂克尔贝格尔模秩一成分，在\(S\)的显式分裂条件下计算其广义指标并得\(p\)-主精细化，进而推导\(K\)和\(k\)的\(S\)-类数之间的整除关系。
AI 中文摘要
考虑全局域的有限阿贝尔扩张\(K/k\)，其伽罗瓦群为\(G\)。研究与\(K/k\)及\(k\)的有限位集合\(S\)相关的广义施蒂克尔贝格尔模的秩一成分。在\(S\)的显式分裂条件下，计算该模关于\(K\)的\(S\)-单位群无挠部分的广义指标。还得到\(p\)-主精细化，包括\(p\mid |G|\)的非半单情形。作为应用，推导了\(K\)和\(k\)的\(S\)-类数之间的整除关系，适用于数域和函数域。
英文摘要
Consider a finite abelian extension $K/k$ of global fields with Galois group $G$. We study the rank-one component of the generalized Stickelberger module associated with $K/k$ and a finite set $S$ of places of $k$. Under explicit splitting conditions on $S$, we compute generalized indices of this module with respect to the torsion-free part of the $S$-unit group of $K$. We also obtain $p$-primary refinements which include the non-semisimple case $p\mid |G|$. As applications, we derive divisibility relations between the $S$-class numbers of $K$ and $k$, both for number fields and for function fields.
Comments33 pages
