Equivariant Homotopy-Invariant Commutative Algebra

我的项目涉及从交换代数到稳定同伦理论的思想泛化。特别地，在整个代数和代数拓扑中，许多对偶现象都可以用Dwyer-Greenlees-Iyengar意义下的环谱的Gorenstein性质来表示。这已经得到了广泛的研究，我的一个目的是在一个等变的背景下探索这一点。对于具有G作用的环谱，一个很大的问题是看Gorenstein性质何时从环下降到同伦不动点谱。例子包括具有共轭作用的复理论谱(及其连通覆盖)，以及经典的代数中关于多项式不变环的渡边定理。考虑真正的环G-谱也很有趣，它的Gorenstein位移具有RO(G)分次位移。例子包括可定向流形的双重覆盖上的奇异余链，通过甲板变换等变，其中Poincaré对偶通过适当的扭曲被恢复。另一个可在稳定同伦理论中考虑的交换代数概念是奇点范畴，由Greenlees-Stevenson为环谱定义。一个重要的例子是紧李群的分类空间上的余链--我的一个项目是将模表示理论中的“核”的概念改编到更具拓扑性的背景下，在那里我已经证明了它与Benson-Greenlees在原始背景下猜测的余链的奇点范畴的支持相一致。我期望对环谱奇点范畴的这一支持理论的理解将成为我项目的一部分，特别是在使用它来对超曲面环谱的厚子范畴进行分类时--可能用离散环推广Stevenson和Takahashi的结果。

My project involves generalisation of ideas from commutative algebra to stable homotopy theory. In particular, many duality phenomena throughout algebra and algebraic topology can be expressed in terms of the Gorenstein property of ring spectra over a field in the sense of Dwyer-Greenlees-Iyengar. This has been studied extensively, and an aim of mine is to explore this in an equivariant context. For ring spectra with G-action, a big question is to see when the Gorenstein property descends from the ring to the homotopy-fixed point spectrum. Examples include the complex theory spectrum (and its connective cover) with the action of conjugation, and classically in algebra a theorem of Watanabe for polynomial invariant rings. It is also of interest to consider genuine ring G-spectra, whose Gorenstein shifts have an RO(G)-graded shift. Examples include singular cochains on the double cover of an orientable manifold, equivariant via deck transformations, where Poincaré duality is recovered with an appropriate twist.Another commutative-algebraic notion that can be considered in stable homotopy theory is the singularity category, defined for ring spectra by Greenlees-Stevenson. A crucial example is the cochains on the classifying space of a compact Lie group - a project of mine has been to adapt a notion of the ``nucleus'' in modular representation theory to a more topological context, where I have shown it to coincide with the support of the singularity category of the cochains, as conjectured in the original context by Benson-Greenlees. I expect understanding this support theory for singularity categories of ring spectra will form a part of my project, especially in using it to classify thick subcategories for hypersurface ring spectra - possibly extending results of Stevenson and Takahashi with discrete rings.