I am attempting to determine whether the following statements are correct. In all of these statement, I need to consider both the finite and the infinite case:
Let $A$ be an $R$-algebra, $X$ denote some affine scheme, $P$ denote some closed subscheme and $Q$ denote some open subscheme. The statements are:
By finite case, I mean taking the union of finitely many open/closed subschemes. Same for the so-called infinite case
(1)$(P_1 ∩ P_2)(A) = P_1(A) ∩ P_2(A)$
True.
(2)$(\bigcap _{\alpha\in I}P_{\alpha})(A)=\bigcap_{\alpha\in I} (P_{\alpha}(A))$
Ture.
(3)$(P_1 ∪ P_2)(A) = P_1(A) ∪ P_2(A)$
It holds for $C$ to be a field, but is not true in general.
(4)$(\bigcup _{\alpha\in I}P_{\alpha})(A)=\bigcup_{\alpha\in I} (P_{\alpha}(A))$
Does not make sense.
(5)$(Q_1 ∩ Q_2)(A) = Q_1(A) ∩ Q_2(A)$
True.
(6)$(\bigcap _{\alpha\in I}Q_{\alpha})(A)=\bigcap_{\alpha\in I} (Q_{\alpha}(A))$
True
(7)$(Q_1 ∪ Q_2)(A) = Q_1(A) ∪ Q_2(A)$(This is the main part that I am not sure)
It is true for field, but I can neither prove it is true in general nor find some counter examples yet. So could someone please prove it or disprove it by finding some counter examples?
(8)$(\bigcup _{\alpha\in I}Q_{\alpha})(A)=\bigcup_{\alpha\in I} (Q_{\alpha}(A))$
$ (X \setminus P)(A) = X(A) \setminus P(A)$
It is true when $A$ is a field, but doesn't hold in general, a quick example could be taking $R=\Bbb C, A=\Bbb C[t], X$ is the affine line, and $P= Spec_R(R)$ is the vanishing locus of the polynomial $x\in R[x]$ Then $P(A)=\{0\}$, $(X\setminus P)(A)=Spec_R(\Bbb C[x,y]/\langle xy-1\rangle)(\Bbb C[t])=\text{units in $\Bbb C$}$, which is not the whole $\Bbb C[t]\setminus \{0\}$.
For all of these parts, could someone please tell me if I am correct? And for the 4th statement, I have stuck for some time, so I would be very appreciate if someone could prove it or give some examples. Reference or ideas are also appreciate. Thanks in advance!