Ramification and descent in homotopy theory and derived algebraic geometry

Abstract

We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[\Omega S^3]\otimes_R\Omega^1_{R/\mathbb{Z}}\to R[\Omega S^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(\Omega S^3;\Omega^1_{R/\mathbb{Z}})\to H_\ast(\Omega S^3\langle 3\rangle;R)\to THH_\ast(R)\to\cdots$. Any time an extension Y/X is a composite of unramified and totally ramified extensions, our results allow for the study of THH(X) in terms of THH(Y) by a kind of weak etale descent (ramified descent).