= Cartier divisor exact sequence for an integral domain
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{title2=$1\to A^\times\to K^\times\to\operatorname{Cart}(A)\to\operatorname{Pic}(A)\to0$}
For an <integral domain> $A$ with <fraction field> $K$, realize <Cartier divisors> as <invertible fractional ideals>, with local divisor equation $f$ corresponding to the ideal $fA$. This is the $\mathcal O(-D)$ convention. The maps in the displayed <exact sequence> are unit inclusion, $f\mapsto fA$, and $I\mapsto[I]$. A principal fractional ideal is trivial as a divisor exactly when its generator is a unit. An invertible fractional ideal is trivial in the <Picard group of a ring> exactly when it is principal. Every rank-one projective module embeds in $K$ and produces an invertible fractional ideal, proving surjectivity. Local multiplication identifies $I\otimes_AJ$ with $IJ$, so these are group homomorphisms.
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