Is the following definition for local on the source and target correct?
def of local on both the source and target?
- (a) if a morphism $\pi: X \rightarrow Y$ has $P$, then for any open subset $U \subseteq X$ and $V \subseteq Y$ such that $\pi(U) \subseteq V$, the morphism $\pi|_U: U \rightarrow V$ has $P$.
- (b) If there exists open covers $\{U_i\}$ of $X$ and $\{V_i\}$ of $Y$, with $\pi(U_i) \subseteq V_i$, such that $\pi|_{U_i}: U_i\rightarrow V_i$ has $P$, then $\pi: X \rightarrow Y$ has $P$.
I checked "This property is local on" : properties of morphisms of $S$-schemes, but I cannot find an explicit statement about this.
Mainly, I want the definition for characterizing smooth morphism (Vakil's FOAG, Exercise 13.6.B smooth morphisms are local on the source and target, page 382). Since I am unsure about the definition, I am uncertain if I proved the correct statement.
As supplements, the definitions of local on source and local on target I believe should be:
A property $P$ of morphisms to local on the source, if the following two conditions hold:
- (a) if a morphism $\pi: X \rightarrow Y$ has $P$, then for any open subset $U \subseteq X$, the morphism $\pi|_U: U \rightarrow Y$ has $P$.
- (b) If there exists open covers $\{U_i\}$ of $X$, such that $\pi|_{U_i}: U_i\rightarrow Y$ has $P$, then $\pi: X \rightarrow Y$ has $P$.
and
A property $P$ of morphisms between schemes is local on the target if the following two statement holds:
- (a) If $\pi: X \rightarrow Y$ has property $P$, then for any open subset $V$ of $Y$, the restricted morphism $\pi^{-1}(V) \rightarrow$ $V$ has property $P$;
- and (b) for a morphism $\pi: X \rightarrow Y$, if there is an open cover $\left\{V_i\right\}$ of $Y$ for which each restricted morphism $\pi^{-1}\left(V_i\right) \rightarrow V_i$ has property $P$, then $\pi$ has property $P$.