Orientability and complete intersections for rings and ring spectra

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.