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What does the notation $f_*$ mean in $\mathcal{O}_S$?
I am reading Artin’s Algebraic Geometry, and I’m coming across the notation $f_*$. It seems to mean that the set of connected components of the fiber is a subgroup of the set of connected components of $S$. Does anyone know where this notation comes from?
A:
$f_*$ is the pushforward. In the Zariski topology, $f$ is a proper morphism, $f^*$ is the pullback, and $f_*$ is the functor of points. A sheaf on $S$ is a commutative diagram of maps between sheaves of sets:
$$\begin{array}{ccc}
f_*\mathcal O_S &\to &\mathcal O_S \\
\uparrow & & \uparrow \\
f_*\mathcal G &\to &\mathcal G
\end{array}$$
If $f$ is generically constant, $f_*$ is a functor from connected components to connected components: If $f^{ -1}(s)=\{\sigma_1,\ldots,\sigma_n\}$, then $f_*\mathcal O_S(s)=\mathcal O_S(\sigma_1)\times\ldots\times\mathcal O_S(\sigma_n)$.
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