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).
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.14568
 arXiv:
 arXiv:2112.14568
 Bibcode:
 2021arXiv211214568B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology
 EPrint:
 22 pages, comments welcome