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Let $f:C\to B$ be a finite morphism of curves. Let $D$ be a divisor on $B$.

Does the equality of divisors $$(f^\ast D)_{red} = f^\ast (D_{red})$$ hold on $C$? (I'm asking for an equality of divisors, not divisor classes.)

If yes, then the equality $$ \deg_C (f^\ast D)_{red} = \deg(f) \deg_B (D_{red})$$ holds. Is this equality true even if the above equality of divisors fails?

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up vote 4 down vote accepted

No. Take a ramified map, e.g. the parabola $y^2 = x$ mapping to the $x$-axis, and suppose your divisor is the origin. Then $(f^*D)_\text{red}$ has degree 1, but $D = D_\text{red}$ and $f^*D$ has degree 2. (maybe I am misunderstanding what "red" means - do you mean take the reduced closed subscheme underlying the divisor?)

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