Morel's Purity Conjecture

Morel's purity conjecture asserts that the -homotopy sheaves of a pure -spectrum over a separated Noetherian base scheme of finite Krull dimension are pure (Morel 2005). The pure spectra are generated by suspension spectra of smooth projective schemes over the base under shifts, cofibers, retracts, and arbitrary coproducts. One requirement of purity for a sheaf is that its restriction to a dense open subscheme of a smooth scheme be injective on sections.

Feld (2026b) reports a failure of this requirement for the degree-2 sheaf of algebraic K-theory with coefficients over , where and is the ring of p-adic integers for . The restriction to has a nonzero kernel, by a finite-coefficient Gersten counterexample (Feld 2026a). Feld (2026b) also reports a counterexample over the regular affine scheme . Feld credits GPT-6 Astra with deriving the purity counterexample from the finite-coefficient construction and Claude Opus 5.5 with proposing a characterization via henselian local rings and the extension of that characterization to arbitrary coefficients.