Gorenstein scheme

For a Gorenstein scheme X of finite type over a field, f: X → Spec(k), the dualizing complex f!

Let X be a normal scheme of finite type over a field k. Then X is regular outside a closed subset of codimension at least 2.

As a result, KU defines a linear equivalence class of Weil divisors on X.

(This property does not depend on the choice of Weil divisor in its linear equivalence class.)

A normal scheme X is Gorenstein (as defined above) if and only if KX is Cartier and X is Cohen–Macaulay.