AI 中文总结
本文研究乘法上同调Tambara函子，证明相关猜想及该类函子的Nakayama引理，确定局部Tambara函子上有限生成投射模必为自由模的适用条件，该条件适用于常值$C_{p^n}$-Tambara函子$\u005cunderline{\u005cmathbb{Z}}_p$。
AI 中文摘要
我们研究一类性质良好的Tambara函子，称为乘法上同调Tambara函子。首个结果是证明了Balchin、Quigley和Spitz的一个猜想，表明每个乘法上同调Tambara函子也是加法上同调的。在此基础上，我们证明了乘法上同调Tambara函子的Nakayama引理。作为应用，我们确定了一个条件，在此条件下局部Tambara函子上有限生成投射模必为自由模，该条件尤其适用于常值$C_{p^n}$-Tambara函子$\u005cunderline{\u005cmathbb{Z}}_p$。
英文摘要
We study a nice class of Tambara functors, called multiplicatively cohomological Tambara functors. The first result is a proof of a conjecture of Balchin, Quigley, and Spitz showing that every multiplicatively cohomological Tambara functor is also additively cohomological. Building on this, we prove Nakayama's lemma for multiplicatively cohomological Tambara functors. As an application, we identify a condition under which finitely generated projective modules over local Tambara functors are always free. This applies, in particular, to the constant $C_{p^n}$-Tambara functor $\underline{\mathbb{Z}}_p$.
Comments19 pages. Comments welcome!
