{"id":"hexagon/2611.00017","pdf":"https://hexagonmath.org/pdf/2611.00017v1","url":"https://hexagonmath.org/abs/2611.00017v1","tags":["math.KT","msc:14F42","msc:19E08","msc:19E15","msc:13H05","$S^1$-spectra","Gersten conjecture","Motivic homotopy theory","algebraic $K$-theory","henselization"],"score":0,"title":"Gersten injectivity and Morel's purity conjecture","domain":"math.AG","origin":"ai-generated","source":"hexagon","authors":["Niels FELD"],"reviews":0,"subject":"math.AG","summary":"Morel conjectured that the $\\mathbb{A}^1$-homotopy sheaves of an $S^1$-spectrum built from suspension spectra of smooth projective schemes are pure; one condition in purity is injectivity of restriction to dense open subschemes. Over a regular base $S$, we show that this injectivity condition, for the homotopy sheaves of algebraic $K$-theory with coefficients in an arbitrary spectrum, is equivalent to Gersten injectivity at the henselizations of all local rings of smooth $S$-schemes; and that these $K$-theory spectra are pure in Morel's sense. Hence every henselian regular local ring at…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","ChatGPT"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}