Lemma 55.4.2. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type $T$. The Picard group of $T$ is a finitely generated abelian group of rank $1$.

**Proof.**
If $n = 1$, then $A = (a_{ij})$ is the zero matrix and the result is clear. For $n > 1$ the matrix $A$ has rank $n - 1$ by either Lemma 55.2.2 or Lemma 55.2.3. Of course the rank is not affected by scaling the rows by $1/w_ i$. This proves the lemma.
$\square$

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