math.AG — Algebraic Geometry

Torsion in $K_1$ and $p$-adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime $p$, we construct a regular strictly henselian local ring $A$ whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order $p$ in $SK_1(A[1/p])$, detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues $4$ and $8$ of $\psi^2$ are needed.