Fp-spectra and Real-oriented homotopy theory

Speaker: Christian Carrick - Utrecht University

Abstract: If $X$ is a spectrum, then the cohomology groups $H^*(X;\mathbb{F}_ 2)$ form a module over the $\text{mod }2$ Steenrod algebra $\mathcal{A}$. Via the Adams spectral sequence, this algebraic structure almost completely determines the homotopy type of $X$ in many cases. However, $\mathcal{A}$ is a non-Noetherian algebra, so it is often convenient to study $H^*(X;\mathbb{F}_2)$ instead as a module over finite subalgebras of $\mathcal{A}$. When $H^*(X;\mathbb{F}_2)$ is a finitely presented $\mathcal{A}$-module, this results in no loss of information, and Mahowald-Rezk demonstrated that such spectra $X$ - “fp spectra” - admit a host of interesting categorical properties, including a chromatic version of Brown-Comenetz duality. I will discuss some of these properties and go through a number of easy examples of fp spectra, then finish with some new examples that come from Real-oriented homotopy theory.