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.)
Best Answer
The answer is yes.
The Hartshorne definition means that the generic point of any irreducible closed subset of codimension one is a regular point.
Since the local ring at the generic point can be obtained from the local ring at any other point by localizing (and localizing a regular local ring gives you a regular local ring again) this is equivalent to say that any irreducible closed subset of codimension one admits at least one regular point.
The latter is of course equivalent of saying that the singular locus (assuming we have already shown it is closed) has at least codimension two.