Gorenstein scheme

In algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein.^[1] 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).^[2] If X is smooth of dimension n over k, the canonical bundle K_X can be identified with the line bundle Ωⁿ of top-degree differential forms.^[3]

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.^[4]

Examples

• An algebraic variety with local complete intersection singularities, for example any hypersurface in a smooth variety, is Gorenstein.^[5]
• 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.^[6]
• 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.^[7]

Notes

1. Kollár (2013), section 2.5; Stacks Project, Tag 0AWV.
2. Hartshorne (1966), Proposition V.9.3.
3. Hartshorne (1966), section III.1.
4. Kollár & Mori (1998), Corollary 5.69.
5. Eisenbud (1995), Corollary 21.19.
6. Kollár & Mori (1998), Theorems 5.20 and 5.22.
7. Kollár (2013), Example 3.6.

References

• Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
• Hartshorne, Robin (1966), Residues and Duality, Lecture Notes in Mathematics, 20, Berlin, New York: Springer-Verlag, ISBN 978-3-540-03603-6, MR 0222093
• Kollár, János (2013), Singularities of the Minimal Model Program, Cambridge University Press, ISBN 978-1-107-03534-8, MR 3057950
• Kollár, János; Mori, Shigefumi (1998), Birational Geometry of Algebraic Varieties, Cambridge University Press, ISBN 0-521-63277-3, MR 1658959

External links

• The Stacks Project Authors, The Stacks Project