math.AG — Algebraic Geometry

Gersten injectivity and Morel's purity conjecture

Contributed by Niels FELD

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 which Gersten injectivity fails, integrally, rationally, or with finite coefficients, gives a counterexample to Morel's conjecture. The author's finite-coefficient counterexample over $\mathbb{Z}_3[[x,y]]/(3+x^2-y^3)$ therefore disproves the conjecture, already over the regular scheme $\operatorname{Spec}\mathbb{Z}[1/2][x,y]/(3+x^2-y^3)$. Recently announced integral and rational Gersten counterexamples, including unpublished work of the author, would show, if confirmed and if they persist after henselization, that the conjecture fails already for the $K$-theory spectrum itself.