Motivic equivalence of quadratic forms

Abstract

Let $X_{\varphi }$ and $X_{\psi }$ be projective quadrics corresponding to quadratic forms $\varphi$ and $\psi$ over a field $F$. If $X_{\varphi }$ is isomorphic to $X_{\psi }$ in the category of Chow motives, we say that $\varphi$ and $\psi$ are motivic isomorphic and write $\varphi \stackrel{m}{\sim }\psi$. We show that in the case of odd-dimensional forms the condition $\varphi \stackrel{m}{\sim }\psi$ is equivalent to the similarity of $\varphi$ 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 $q_1\stackrel{m}{\sim }{q}_2$ and $q_1\nsim q_2$.