Two Cycle Class Maps on Torsion Cycles

We compare two cycle class maps on torsion cycles and show that they agree up to a minus sign. The first one goes back to Bloch (1979), with recent generalizations to non-closed fields. The second is the étale motivic cycle class map ${\alpha }_X^i\text{\hspace{0.17em}⁣}:{\text{CH}}^i(X{)}_{{\mathbb{Z}}_{\mathrm{\ell }}}\to H_L^{2i}(X,{\mathbb{Z}}_{\mathrm{\ell }}(i))$ restricted to torsion cycles.