in this question <https://mathoverflow.net/a/55528/61732> it is stated that a normal variety is CM outside a set of codim at least 3. That would imply that normal surfaces are CM. [edited:] I wanted to ask if under the assumption that the variety is CM, the canonical divisor being Cartier is equivalent to the local rings being Gorenstein.


I would love to have a good book or paper on these questions with focus on algebraic geometry, cause I did not find any.


Many thanks,
Lukas



Normal surface is Cohen-Macaulay - reference
Serre's theorem tells you that normal implies $S\_2$, that is, Cohen-Macaulay in codimension 2. 
I am not sure I understand your second question, but of course Gorenstein implies Cohen-Macaulay. Finally I am sure that Eisenbud's "Commutative Algebra: with a View Toward Algebraic Geometry" contains all of these well-known facts.


**Answer to the edited question** : yes. This follows from the fact that a Cohen-Macaulay local ring is Gorenstein if and only if its dualizing module is free.


