Free Incomplete Tambara Functors are Almost Never Flat
Free Incomplete Tambara Functors are Almost Never Flat
复制标题
自由不完全 Tambara 函子几乎从不平坦
DOI：
10.1093/imrn/rnab361
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发表时间：
2022
影响因子：
1
通讯作者：
Quigley, James D.
中科院分区：
文献类型：
--
作者：
Hill, Michael A.;Mehrle, David;Quigley, James D.
在经典代数中，自由代数作为基环上的模总是自由的。在等变代数中，自由不完全Tambara函子扮演自由代数的角色，Mackey函子扮演模的角色。令人惊讶的是，自由的不完全Tambara函子往往不能像Mackey函子那样自由。在本文中，我们确定了所有有限群的条件下，一个自由的不完全Tambara函子是自由的麦基函子。对于可解群，我们表明，一个自由的不完整的Tambara函子是平坦的麦基函子正是当这些条件成立。我们的结果意味着自由不完全Tambara函子几乎从来没有平坦的Mackey函子。然而，我们表明，适当的本地化后，自由不完整的Tambara函子总是免费的麦基函子。
Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.