Dualizing sheaf

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

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

that induces a natural isomorphism of vector spaces

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

for each coherent sheaf F on X (the superscript * refers to a dual vector space).[1] The linear functional ${\displaystyle t_{X}}$ is called a trace morphism.

A pair ${\displaystyle (\omega _{X},t_{X})}$, if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, ${\displaystyle \omega _{X}}$ is an object representing the contravariant functor ${\displaystyle F\mapsto \operatorname {H} ^{n}(X,F)^{*}}$ 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: ${\displaystyle \omega _{X}={\mathcal {O}}_{X}(K_{X})}$ where ${\displaystyle K_{X}}$ is a canonical divisor. More generally, the dualizing 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 ${\displaystyle \operatorname {Supp} (F)}$ is of pure dimension n, there is a natural isomorphism[2]

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

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

Relative dualizing sheaf

Given a proper finitely presented morphism of schemes ${\displaystyle f:X\to Y}$, (Kleiman 1980) harv error: no target: CITEREFKleiman1980 (help) defines the relative dualizing sheaf ${\displaystyle \omega _{f}}$ or ${\displaystyle \omega _{X/Y}}$ as[3] the sheaf such that for each open subset ${\displaystyle U\subset Y}$ and a quasi-coherent sheaf ${\displaystyle F}$ on ${\displaystyle U}$, there is a canonical isomorphism

${\displaystyle (f|_{U})^{!}F=\omega _{f}\otimes _{{\mathcal {O}}_{Y}}F}$,

which is functorial in ${\displaystyle F}$ and commutes with open restrictions.

Example:[4] If ${\displaystyle f}$ is a local complete intersection morphism between schemes of finite type over a field, then (by definition) each point of ${\displaystyle X}$ has an open neighborhood ${\displaystyle U}$ and a factorization ${\displaystyle f|_{U}:U{\overset {i}{\to }}Z{\overset {\pi }{\to }}Y}$, a regular embedding of codimension ${\displaystyle k}$ followed by a smooth morphism of relative dimension ${\displaystyle r}$. Then

${\displaystyle \omega _{f}|_{U}\simeq \wedge ^{r}i^{*}\Omega _{\pi }^{1}\otimes \wedge ^{k}N_{U/Z}}$

where ${\displaystyle \Omega _{\pi }^{1}}$ is the sheaf of relative Kähler differentials and ${\displaystyle N_{U/Z}}$ is the normal bundle to ${\displaystyle i}$.

See also: Hodge bundle (which is the direct image of a relative dualizing sheaf).

See also

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

References

1. Hartshorne, Ch. III, § 7. harvnb error: no target: CITEREFHartshorne (help)
2. Kollár–Mori, Theorem 5.71. harvnb error: no target: CITEREFKollár–Mori (help)
3. Kleiman 1980, Definition 6 harvnb error: no target: CITEREFKleiman1980 (help)
4. Arbarello–Cornalba–Griffiths 2011, Ch. X., near the end of § 2. harvnb error: no target: CITEREFArbarello–Cornalba–Griffiths2011 (help)

• E. Arbarello, M. Cornalba, and P.A. Griffiths, Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, vol. 268, Springer, Heidelberg, 2011. MR-2807457
• 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, 134, Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

External links

• http://math.stanford.edu/~vakil/0506-216/216class5354.pdf
• Relative dualizing sheaf (reference, behavior)