论文标题
派生的Mackey函子的光谱
The spectrum of derived Mackey functors
论文作者
论文摘要
我们计算所有有限组的派生的Mackey函数类别（在Kaledin的意义上）的频谱。我们发现该空间可准确捕获稳定同型类别的光谱的顶部和底层（即高度无穷大和高度零部分）。由于色彩信息的这种截断，我们能够为所有有限组的频谱提供完整的描述，尽管我们对均衡稳定同型类别的光谱的拓扑不完整。从不同的角度来看，我们表明可以将派生的Mackey函数的光谱理解为从伯恩赛环频谱中获得的空间，这是通过“拆开”封闭点来理解的。为了计算频谱，我们提供了Kaledin类别的新描述，作为模棱两可的环频谱的派生类别，这可能具有独立的兴趣。实际上，我们阐明了几个不同类别之间的关系，建立了对称的单体等效性和Kaledin构造之间的比较，Barwick的频谱Mackey函数，Mackey函数的普通派生类别，以及在某些等价环形光谱上的模块类别。我们还说明了Mackey函子的普通派生类别的一个有趣功能，该特征将其与与其几何固定点的行为有关的其他模棱两可的类别区分开来。
We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by "ungluing" closed points. In order to compute the spectrum, we provide a new description of Kaledin's category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.
