Gorenstein duality for topological Hochschild homology and its real form.

最好的交换环是正则局部环（离多项式环不远），然后通过分解一个正则序列得到这些环。然而，戈伦斯坦环类更为普遍，而且（在巴斯之后）无处不在。对于直到同伦的环也是如此，Greenlees证明了当R是特征为p的残馀域k的正则局部环时，THH（R;k）是Gorenstein。本课题是将这一结果在两个方向上进行推广。首先，当R是Gorenstein（而不仅仅是正则）时，人们期望它是正确的，这将极大地丰富示例的范围。其次，当R具有反对合性时，似乎更丰富的等变版本适用于实THH；这包括将上述工作与Dotto和Patchkoria的工作结合起来，以及Greenlees与Meier关于BPR的Gorenstein二元性的工作。在这样做的过程中，一些有趣的光谱序列将被开发出来，这将对显式计算有用。

The nicest sort of commutative rings are the regular local rings (not far from polynomial rings), and then those obtained from these by factoring out a regular sequence. However, the class of Gorenstein rings is much more general and famously (after Bass) ubiquitous. It turns out that this is also true for rings up to homotopy, and Greenlees shown that THH(R;k) is Gorenstein when R is regular local ring with residue field k of characteristic p. The project is to extend this result in two directions. Firstly, one expects it is true when R is Gorenstein (and not just regular), which would enormously enrich the range of examples. Secondly, it appears that a richer equivariant version applies to real THH when R has an anti-involution; this involves combining the above work with work of Dotto and Patchkoria and also work Greenlees with Meier on Gorenstein duality for BPR. In the course of doing this, a number of interesting spectral sequences will be developed which will be useful for explicit calculation.