A normal variety is said to have Gorenstein singularity iff its canonical divisor is a Cartier divisor (one can always define the canonical divisor on a normal variety and it can be proved to be a Weil divisor).
What is the relation between Gorenstein singularity and Gorenstein ring? More precisely, is it true that a normal variety has Gorenstein singularity iff the local ring of its structure sheaf at any point is Gorenstein?
One can assume the normal variety is moreover Cohen-Macaulay if that helps.