Commutative S-algebras and relative smash products
In the 1990's it was a major advance to construct good models of the category of spectra. It allowed for a more accurate adaptation of intuition from classical and homological algebra. One major avenue of study is E∞-ring spectra or commutative S-algebras which play the role of commutative rings. One point of divergence between the classical theory and homotopical theory is that being a commutative S-algebra is not a property but a extra structure. So one can then ask, how many commutative S-algebra structures are there on a given spectrum? How does one measure differences between two commutative S-algebra structures? This talk will focus on the utilization of relative smash products and power operations to examine these questions. We will see that the choices made in how one cones off 2 on a commutative S-algebra controls a lot. These choices are controlled by a universal object, a relative smash product, which we will compute using the Künneth spectral sequence. If time permits, we will also examine the relationship of relative smash products to derived algebraic geometry and Tannakian formalism.