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

Abstract

Let $X$ be a noetherian scheme with dualizing complex and ${\mathcal{A}}$ a coherent ${\mathcal O}_{X}$-algebra with involution of the first kind $\tau$. We develop in this work a coherent hermitian Witt theory for $({\mathcal{A}},\tau)$. As an application we construct a Gersten–Witt spectral sequence which converges to the coherent hermitian Witt theory of $({\mathcal{A}},\tau)$. 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.