Gersten injectivity and Morel's purity conjecture
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…
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