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(''k''), the dualizing complex ''f''^!(''k'') on ''X'' is a line bundle (called the canonical bundle ''K''_''X''), viewed as a complex in degree −dim(''X''). If ''X'' is smooth of dimension ''n'' over ''k'', the canonical bundle ''K''_''X'' can be identified with the line bundle Ω^''n'' 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 ''K''_''U'' is a line bundle. The restriction from the divisor class group Cl(''X'') to Cl(''U'') is an isomorphism, and (since ''U'' is smooth) Cl(''U'') can be identified with the Picard group Pic(''U''). As a result, ''K''_''U'' defines a linear equivalence class of Weil divisors on ''X''. Any such divisor is called the canonical divisor ''K''_''X''. For a normal scheme ''X'', the canonical divisor ''K''_''X'' is said to be Q-Cartier if some positive multiple of the Weil divisor ''K''_''X'' is Cartier. (This property does not depend on the choice of Weil divisor in its linear equivalence class.) Alternatively, normal schemes ''X'' with ''K''_''X'' Q-Cartier are sometimes said to be Q-Gorenstein.

It is also useful to consider the normal schemes ''X'' for which the canonical divisor ''K''_''X'' is Cartier. Such a scheme is sometimes said to be Q-Gorenstein of index 1. (Some authors use "Gorenstein" for this property, but that can lead to confusion.) A normal scheme ''X'' is Gorenstein (as defined above) if and only if ''K''_''X'' is Cartier and ''X'' is Cohen–Macaulay.

Examples

*An algebraic variety with local complete intersection singularities, for example any hypersurface in a smooth variety, is Gorenstein.
*A variety ''X'' with quotient singularities over a field of characteristic zero is Cohen–Macaulay, and ''K''_''X'' is Q-Cartier. The quotient variety of a vector space ''V'' by a linear action of a finite group ''G'' is Gorenstein if ''G'' maps into the subgroup SL(''V'') of linear transformations of determinant 1. By contrast, if ''X'' is the quotient of C² by the cyclic group of order ''n'' acting by scalars, then ''K''_''X'' is not Cartier (and so ''X'' is not Gorenstein) for ''n'' ≥ 3.
*Generalizing the previous example, every variety ''X'' with klt (Kawamata log terminal) singularities over a field of characteristic zero is Cohen–Macaulay, and ''K''_''X'' is Q-Cartier.
*If a variety ''X'' has log canonical singularities, then ''K''_''X'' is Q-Cartier, but ''X'' need not be Cohen–Macaulay. For example, any affine cone ''X'' over an abelian variety ''Y'' is log canonical, and ''K''_''X'' is Cartier, but ''X'' is not Cohen–Macaulay when ''Y'' has dimension at least 2.

Notes

References

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External links

*{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/

Algebraic geometry
Algebraic varieties
Scheme theory