Motivic equivalence of quadratic forms

Abstract

Let $X_\phi$ and $X_\psi$ be projective quadrics corresponding to quadratic forms $\phi$ and $\psi$ over a field $F$. If $X_\phi$ is isomorphic to $X_\psi$ in the category of Chow motives, we say that $\phi$ and $\psi$ are motivic isomorphic and write $\phi\msim\psi$. We show that in the case of odd-dimensional forms the condition $\phi\msim\psi$ is equivalent to the similarity of $\phi$ and $\psi$. After this, we discuss the case of even-dimensional forms. In particular, we construct examples of generalized Albert forms $q_1$ and $q_2$ such that $q_1\msim q_2$ and $q_1\not\sim q_2$.