Connected components of qcqs schemes X are exactly f^{-1}(C) for the canonical f:X to Spec(R) with R global sections and C a connected component of Spec(R); connected components of P^n_S are P^n_C for connected components C of S equipped with closed subscheme structure.
Engelking, General Topology, Revised and completed edition
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Connected components of qcqs schemes and projective spaces
Connected components of qcqs schemes X are exactly f^{-1}(C) for the canonical f:X to Spec(R) with R global sections and C a connected component of Spec(R); connected components of P^n_S are P^n_C for connected components C of S equipped with closed subscheme structure.