Gorenstein scheme


In algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its properties are much the same as in the special case of smooth schemes.

Related properties

For a Gorenstein scheme X of finite type over a field, f: X → Spec, the dualizing complex f! on X is a line bundle, viewed as a complex in degree −dim. If X is smooth of dimension n over k, the canonical bundle KX can be identified with the line bundle Ωn of top-degree differential forms.
Using the canonical bundle, Serre duality takes the same form for Gorenstein schemes as it does for smooth schemes.
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. Let U be the open subset where X is regular; then the canonical bundle KU is a line bundle. The restriction from the divisor class group Cl to Cl is an isomorphism, and Cl can be identified with the Picard group Pic. As a result, KU defines a linear equivalence class of Weil divisors on X. Any such divisor is called the canonical divisor KX. For a normal scheme X, the canonical divisor KX is said to be Q-Cartier if some positive multiple of the Weil divisor KX is Cartier. Alternatively, normal schemes X with KX Q-Cartier are sometimes said to be Q-Gorenstein.
It is also useful to consider the normal schemes X for which the canonical divisor KX is Cartier. Such a scheme is sometimes said to be Q-Gorenstein of index 1. A normal scheme X is Gorenstein if and only if KX is Cartier and X is Cohen–Macaulay.

Examples