List of Local to Global principles What are some of the local to global principles in different areas of mathematics?

$C^*$-Algebras
Let $\mathfrak{A}$ be a unital $C^*$-algebra, $\mathfrak{C}$ a $C^*$-subalgebra of the center of $\mathfrak{A}$ which contains the unit of $\mathfrak{A}$ and for any maximal ideal $x$ of $\mathfrak{C}$ let $I_x$ be the smallest closed two-sided ideal of $\mathfrak{A}$ containing $x$.
Now the local principle by Allan and Douglas says, that $a\in\mathfrak{A}$ is invertible in $\mathfrak{A}$ if and only if $a+I_x$ is invertible in the quotient algebra $\mathfrak{A}/I_x$ for every maximal ideal $x$ of $\mathfrak{C}$.
One can use this result to tackle questions of invertibility and Fredholmness in operator algebras, e.g. one can characterize Fredholmness properties of Toeplitz operators with piecewise continuous symbols by means of easily checked properties of their respective symbols.
