Abstract

In 2006, Jeff Smith gave a talk explaining how to define ideals in the context of ring spectra, and laying out applications to algebraic K-theory computations. In 2014, Mark Hovey posted a preprint working out the homotopy theory of such ideals, in any stable model category. Since every map is homotopic to an inclusion, ideals must be defined diagrammatically rather than as subobjects. In this talk, I will explain the definition, and then discuss joint work with Donald Yau that extends Hovey’s work to develop a homotopy theory of ideals with additional structure encoded by an operad. I’ll unpack the definition in algebraic settings like chain complexes and the stable module category. If there’s time, I’ll discuss applications to algebraic K-theory and other localizing invariants, and open problems.