The Picard group of motivic A(1)

TL;DR

This paper computes the Picard group of the motivic A(1) algebra, revealing it is isomorphic to Z^4, which differs from the classical case by an additional Z factor due to motivic bigrading and an infinite order joker element.

Contribution

The paper determines the Picard group of the stable category of modules over motivic A(1), showing it is Z^4, and highlights differences from the classical case including the nature of the joker element.

Findings

Picard group of motivic A(1) is Z^4

Motivic joker has infinite order

Extra Z factor from motivic bigrading

Abstract

We show that the Picard group $Pic(A(1))$ of the stable category of modules over $\mathbb{C}$-motivic $A(1)$ is isomorphic to $\mathbb{Z}^4$. By comparison, the Picard group of classical $A(1)$ 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 $A(1)$. The $\mathbb{C}$-motivic joker has infinite order.