Orientability and complete intersections for rings and ring spectra

以健壮的形式表达性质和证明有双重好处。首先，它确立了它们在微小扰动下是稳定的，从而记录了一些重要的东西。其次，它允许它们被更广泛地应用。例如，交换环的性质通常是非常严格的，并且依赖于元操作。用同源术语来表述这一点是向前迈出的一步，然后人们可能希望用结构术语来描述派生范畴中的事物。最后，如果这个公式是一种适当的形式，它也可以适用于环谱。例如，如果性质是局部环的正则性，则它经典地用正则元素序列来表示，但用有限投射维度来刻画Serre，并且在派生范畴中重新表示为能够在有限多个步骤中从环建立剩余域。然后，这适用于环谱，并包括在Dwyer-Wilkerson意义下的p-紧群的分类空间。类似地，Gorenstein交换环的概念扩展到环谱，其中包括流形的Poincar‘e对偶和群上同调的Benson-Carlson对偶。本项目旨在进一步研究这一过程，特别是完全交的概念。事实上，环的性质通常(例如在上述情况下)是由代数几何激发的。因此，上述过程给出了代数几何的同伦不变形式的成分之一，目前正从不同的角度进行研究。

Formulating properties and proofs in a robust form has double benefits. Firstly it establishes that they are stable under minor perturbations, and thus record something significant. Secondly, it allows them to be applied more generally.For example, properties of commutative rings are often very rigid and depend on elementwise manipulations. Formulating this in homological terms is a step forward, and then one may hope to state things in the derived category in structural terms. Finally, if this formulation is of an appropriate form, it may also apply to ring spectra. For instance if the property was regularity of a local ring, it is classically formulated in terms of a regular sequence of elements, but characterized by Serre in terms of finite projective dimension, and reformulated in the derived category as being able to build the residue field from the ring in finitely many steps. This then applies to ring spectra, and includes the classifying spaces of p-compact groups in the sense of Dwyer-Wilkerson. Similarly, the notion of Gorenstein commutative rings extends to ring spectra, where it includes Poincar\'e duality for manifolds and Benson-Carlson duality for group cohomology.The project aims to investigate this process further, and in particular for the notion of complete intersection. In fact the properties of rings are often (for example in the above cases) motivated by algebraic geometry. Thus the process described above gives one of the ingredients in a homotopy invariant form of algebraic geometry, now under investigation from various different directions.