Gorenstein scheme

Last updated January 15, 2024

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 bundleK_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 divisorK_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_XQ-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

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

Related Research Articles

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group $is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.$

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

<span class="mw-page-title-main">Birational geometry</span> Field of algebraic geometry

In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.

In mathematics, the canonical bundle of a non-singular algebraic variety $of dimension over a field is the line bundle, which is the n th exterior power of the cotangent bundle on .$

In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative". The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors, there is an equivalent notion of an ample divisor.

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are derived from the notion of divisibility in the integers and algebraic number fields.

In mathematics, more particularly in the field of algebraic geometry, a scheme $has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map$

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change.

In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors, there is an equivalent notion of a nef divisor.

In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by Reid (1980). Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.

The concept of a Projective space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective spaces.

In mathematics, an arithmetic surface over a Dedekind domain R with fraction field $is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When R is the ring of integers Z, this intuition depends on the prime ideal spectrum Spec(Z) being seen as analogous to a line. Arithmetic surfaces arise naturally in diophantine geometry, when an algebraic curve defined over K is thought of as having reductions over the fields R / P, where P is a prime ideal of R, for almost all P; and are helpful in specifying what should happen about the process of reducing to R / P when the most naive way fails to make sense.$

This is a glossary of algebraic geometry.

This is a glossary of commutative algebra.

In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf of finite rank and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry.

In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf $together with a linear functional$

References

Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
Hartshorne, Robin (1966), Residues and Duality, Lecture Notes in Mathematics, vol. 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

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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.)

[1]

[2]

[3]

[4]

[5]

[6]

[7]