Dualizing sheaf

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

$t_{X}:\operatorname {H} ^{n}(X,\omega _{X})\to k$

that induces a natural isomorphism of vector spaces

$\operatorname {Hom} _{X}(F,\omega _{X})\simeq \operatorname {H} ^{n}(X,F)^{*},\,\varphi \mapsto t_{X}\circ \varphi$

for each coherent sheaf F on X (the superscript * refers to a dual vector space). The linear functional $t_{X}$ is called a trace morphism.

A pair $(\omega _{X},t_{X})$, if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, $\omega _{X}$ is an object representing the contravariant functor $F\mapsto \operatorname {H} ^{n}(X,F)^{*}$ from the category of coherent sheaves on X to the category of k-vector spaces.

For a normal projective variety X, the dualizing sheaf exists and it is in fact the canonical sheaf: $\omega _{X}={\mathcal {O}}_{X}(K_{X})$ where $K_{X}$ is a canonical divisor. More generally, the dualuzing sheaf exists for any projective scheme.

There is the following variant of Serre's duality theorem: for a projective scheme X of pure dimension n and a Cohen–Macaulay sheaf F on X such that $\operatorname {Supp} (F)$ is of pure dimension n, there is a natural isomorphism

$\operatorname {H} ^{i}(X,F)\simeq \operatorname {H} ^{n-i}(X,\operatorname {{\mathcal {H}}om} (F,\omega _{X}))^{*}$.

In particular, if X itself is a Cohen–Macaulay scheme, then the above duality holds for any locally free sheaf.

See also

• coherent duality
• reflexive sheaf
• Gorenstein ring
• Dualizing module
• Hodge bundle

References

• Kleiman, Steven L. Relative duality for quasicoherent sheaves. Compositio Math. 41 (1980), no. 1, 39–60.
• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

External links

• http://math.stanford.edu/~vakil/0506-216/216class5354.pdf
• https://mathoverflow.net/questions/211158/relative-dualizing-sheaf-reference-behavior