# Regular in codimension one VS Singular locus is codimension at least two

In Hartshorne, a scheme is regular in codimension one if the local ring at any (non-closed) point representing a codimension one subscheme is a regular local ring (of Krull dimension one).

For varieties, the most naive notion of being regular in codimension one (at least to me!) is just to say that the set of singular points is subvariety of codimension at least two.

Is my naive definition of "regular in codim one" equivalent to the definition in Hartshorne?

(I ask this because my naive definition is easy to verify: for instance, a surface with only ADE singularities is clearly regular in codimension one by my naive definition - there is no need to do any commutative algebra, which I'm terrible at. But I do need to know if this singular variety satisfies the condition in Hartshorne, because having DVRs in codimension one allows me to define Weil divisors.)

• Typo: "Weyl" should be "Weil". I would change it but the software won't allow me to. – Nefertiti Feb 17 '17 at 11:01
• Also note that the assumption 'regular in codimension one' is not needed to define the notion of a Weil Divisor. It is actually needed to associate to any rational function a Weil Divisor, so it is needed to define the notion of a principal Weil Divisor. – MooS Feb 17 '17 at 13:39
• @MooS I don't think you need regular in codimension 1 to associate a Weil divisor to a rational function: Fulton in section 1.2 of Intersection Theory gets around this by defining order using length. – Takumi Murayama Feb 17 '17 at 17:34
• Yes, you are of course correct. – MooS Feb 17 '17 at 20:00