Lemma 55.4.3. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type $T$. Then $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Coker}}(A)$ where $A = (a_{ij})$.

**Proof.**
Since $\mathop{\mathrm{Pic}}\nolimits (T)$ is the cokernel of $(a_{ij}/w_ i)$ we see that there is a commutative diagram

with exact rows. By the snake lemma we conclude that $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Coker}}(A)$. $\square$

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