The Stacks project

Remark 45.4.5 (Lefschetz and Tate motive). Let $X = \mathbf{P}^1_ k$ and $c_2$ be as in Example 45.3.7. In the literature the motive $(X, c_2, 0)$ is sometimes called the Lefschetz motive and depending on the reference the notation $L$, $\mathbf{L}$, $\mathbf{Q}(-1)$, or $h^2(\mathbf{P}^1_ k)$ may be used to denote it. By Lemma 45.4.4 the Lefschetz motive is isomorphic to $\mathbf{1}(-1)$. Hence the Lefschetz motive is invertible (Categories, Definition 4.43.4) with inverse $\mathbf{1}(1)$. The motive $\mathbf{1}(1)$ is sometimes called the Tate motive and depending on the reference the notation $L^{-1}$, $\mathbf{L}^{-1}$, $\mathbf{T}$, or $\mathbf{Q}(1)$ may be used to denote it.