(Affine communication lemma)
Let \(X\) be a scheme and \(P\) a property of affine open subschemes of \(X\) such that
if \(\Spec A \subseteq X\) has \(P\) and \(f \in A\), then \(\Spec A_f \subseteq X\) has \(P\), and
if \(\Spec A \subseteq X\) is an affine open, \(f_1, \ldots, f_n \in A\) generate the unit ideal, and every \(\Spec A_{f_i}\) has \(P\), then \(\Spec A\) has \(P\).
If \(X\) has a cover by affine opens \(\Spec A_i\) with \(P\), then every affine open of \(X\) has \(P\) [@Vak25, Lemma 5.3.2].
Being Noetherian, and for a scheme over a ring \(R\) being of finite type over \(R\), satisfy both hypotheses, so they can be checked on one affine cover; so do properties defined on stalks, such as being reduced.
Integrality does not satisfy hypothesis 2. For nonzero integral domains \(A\) and \(B\), the elements \(e_1 = (1, 0)\) and \(e_2 = (0, 1)\) of \(A \times B\) generate the unit ideal, and \((A \times B)_{e_1} \cong A\) and \((A \times B)_{e_2} \cong B\) are integral domains, but \(A \times B\) is not, since \(e_1 e_2 = 0\). Geometrically, \(\Spec(A \times B) = \Spec A \sqcup \Spec B\) is covered by integral affine opens and is not irreducible.
Dependencies
D-VKR54$\Spec A$ as a locally ringed spaceusesPR-SCHUNITCOVERDistinguished opens cover exactly when the functions generate the unit idealuses