Definition 55.4.1. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type $T$. The *Picard group of $T$* is the cokernel of the matrix $(a_{ij}/w_ i)$, more precisely

\[ \mathop{\mathrm{Pic}}\nolimits (T) = \mathop{\mathrm{Coker}}\left( \mathbf{Z}^{\oplus n} \to \mathbf{Z}^{\oplus n},\quad e_ i \mapsto \sum \frac{a_{ij}}{w_ j}e_ j \right) \]

where $e_ i$ denotes the $i$th standard basis vector for $\mathbf{Z}^{\oplus n}$.

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