External canonical top-weight motivic-to-syntomic comparison

PDF p. 5: Proposition 2.6: torsion freeness

Let $k$ be algebraically closed, $X$ a finite-type $k$-scheme, $j\ge \dim X$, and $\mathrm{\ell }$ any prime. The canonical map of Elmanto–Morrow complexes $\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X){\otimes }^{\mathbf{L}}\mathbb{Z}\mathrm{/}\mathrm{\ell }⟶{\mathbb{F}}_{\mathrm{\ell }}(j{)}_{\mathrm{s}\mathrm{y}\mathrm{n}}(X)$ is an equivalence. If $\mathrm{\ell }$ is invertible in $k$, the target is $R{\mathrm{\Gamma }}_{\acute{e}t}(X,{\mu }_{\mathrm{\ell }}^{\otimes j})$. If $\mathrm{\ell }=\mathrm{char}k$, the target is the derived syntomic theory specified in External EM affine syntomic filtration and actual obstruction map.

Import the published report Theorem4, not its application to torsion-freeness. Its proof uses Artin vanishing at invertible primes. In characteristic $p$, it kills the cdh Artin–Schreier obstruction by henselian rigidity and the Kato–Kuzumaki bound for residue fields of transcendence degree at most $j$ over an algebraically closed field. Interpret that obstruction through the precise fiber sequence in External EM affine syntomic filtration and actual obstruction map. Thus this input concerns the actual map and specified EM theory, not merely abstractly isomorphic groups. The report's proof, not a purported reconstruction of the deeper field/rigidity results, is imported. Theorem4 has no reducedness hypothesis.

In the published report, Theorem 3(2), printed p.1994, has an étale lower-right corner; EM v1 Theorem 4.28, printed p.29, has the éh corner, identified there with the cdh localization of the étale presheaf. We retain this as a qualified discrepancy and do not import the report's printed square on arbitrary singular schemes. The precise comparison is supplied by External EM affine syntomic filtration and actual obstruction map.

The report's characteristic-$p$ proof of Theorem 4, printed p.1995, describes the cofiber without its shift. With its cited Artin–Schreier obstruction convention, EM v1 Corollary 4.32 gives the cofiber $R{\mathrm{\Gamma }}_{\mathrm{c}\mathrm{d}\mathrm{h}}(X,{\tilde{\nu }}_1(j))[-j-1]$. This is a confirmed nonfatal source-display omission: vanishing of the obstruction also kills its shift. It does not refute Theorem 4, whose full finite-type scope is retained. The reducedness premise of the separate Theorem 1 is not a premise of Theorem 4.