Orientability and complete intersections for rings and ring spectra

以鲁棒形式表述性质和证明有双重好处。首先，它确定了它们在轻微扰动下是稳定的，因此记录了一些重要的东西。其次，它允许它们被更普遍地应用。例如，交换环的性质通常是非常刚性的，并且依赖于元素操作。用同调术语来表述这个是一个进步，然后人们可能希望用结构术语来表述派生范畴中的东西。最后，如果这个公式的形式合适，它也可以适用于环谱。例如，如果性质是局部环的正则性，则经典地将其表述为正则的元素序列，但用有限投影维数的Serre来表征，并在派生范畴中重新表述为能够在有限多步中从环构建剩余域。然后将其应用于环谱，并包括Dwyer-Wilkerson意义上的p紧群的分类空间。同样，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。该项目旨在进一步研究这一过程，特别是完全相交的概念。事实上，环的性质通常（例如在上面的例子中）是由代数几何引起的。因此，上面描述的过程给出了代数几何的同伦不变形式的一个成分，现在正从不同的方向进行研究。

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.