A Gersten–Witt complex for hermitian Witt groups of coherent algebras over schemes, I: Involution of the first kind

摘要

Let $X$ be a noetherian scheme with dualizing complex and ${mathcal{A}}$ a coherent ${mathcal O}_{X}$-algebra with involution of the first kind $ au$. We develop in this work a coherent hermitian Witt theory for $({mathcal{A}}, au)$. As an application we construct a Gersten–Witt spectral sequence which converges to the coherent hermitian Witt theory of $({mathcal{A}}, au)$. We show then that the associated Gersten–Witt complex is exact if $X$ is the spectrum of a smooth semilocal ring and ${mathcal{A}}$ is locally free as ${mathcal O}_{X}$-module.