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).[1] 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 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 $\operatorname {Supp} (F)$ is of pure dimension n, there is a natural isomorphism[2]

\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 $f:X\to Y$ , (Kleiman 1980) defines the relative dualizing sheaf $\omega _{f}$ or $\omega _{X/Y}$ as[3] the sheaf such that for each open subset $U\subset Y$ and a quasi-coherent sheaf $F$ on $U$ , there is a canonical isomorphism

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

,

which is functorial in $F$ and commutes with open restrictions.

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

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

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

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

References

Hartshorne, Ch. III, § 7.
Kollár–Mori, Theorem 5.71.
Kleiman 1980, Definition 6
Arbarello–Cornalba–Griffiths 2011, Ch. X., near the end of § 2.

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

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[1] Hartshorne, Ch. III, § 7.

[2] Kollár–Mori, Theorem 5.71.

[3] Kleiman 1980, Definition 6

[4] Arbarello–Cornalba–Griffiths 2011, Ch. X., near the end of § 2.