\(\Spec\) and \(\Proj\), gluing, fibre products, and the properties — reduced, irreducible, integral, Noetherian — read off the ring.
A scheme \(X\) is affine exactly when the canonical morphism \(X\to\Spec\OO_X(X)\) is an isomorphism.
What is a scheme, the definition with the reason for each clause, and the criterion \(X\cong\Spec\OO_X(X)\) for affineness.
Properties from the ring, reduced, irreducible, integral, and noetherian schemes in terms of rings, and generic points.
Proj and gluing, gluing affine schemes along open subschemes, and \(\Proj\) of a graded ring.
Fibre products and base change, intersections, fibres, and base change to field extensions as fibre products, and the functor of points.
Subschemes and points, the closed subscheme structures on one closed set, and values of functions in residue fields.