The Picard group of motivic \(\mathcal {A}_\mathbb {C}(1)\)

Abstract

We show that the Picard group \({{\mathrm{Pic}}}(\mathcal {A}_\mathbb {C}(1))\) of the stable category of modules over \(\mathbb {C}\)-motivic \(\mathcal {A}_\mathbb {C}(1)\) is isomorphic to \(\mathbb {Z}^4\). By comparison, the Picard group of classical \(\mathcal {A}(1)\) is \(\mathbb {Z}^2 \oplus \mathbb {Z}/2\). One extra copy of \(\mathbb {Z}\) arises from the motivic bigrading. The joker is a well-known exotic element of order 2 in the Picard group of classical \(\mathcal {A}(1)\). The \(\mathbb {C}\)-motivic joker has infinite order.