# Normal surface is Cohen-Macaulay - reference

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

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.

• stacks.math.columbia.edu/tag/033P – rghthndsd Nov 13 '14 at 17:15
• thanks a lot, I'm sorry that my second question wasn't well formulated: 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. (there is sometimes a bit confusion about the notions of Gorenstein in the case of varieties, what is expressed in this question) – Lukas Braun Nov 13 '14 at 17:20
• Just as a complement to abx's answer, you could take a look at "Young Person's guide to Canonical Singularities" of Miles Reid. The appendix to section 3 is called "Cohen-Macaulay and all that", and precisely gives a short introduction to Cohen-Macaulayness from an geometric point of view. For a more detailed discussion you may refer to the book of Kollar and Mori "Birational Geometry of Algebraic Varieties", Chapter 5: Singularities of the Minimal Model Program. – Pedro Montero Nov 14 '14 at 16:45