- See also
- References
- External links
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
that induces a natural isomorphism of vector spaces
for each coherent sheaf F on X (the superscript * refers to a dual vector space).[1] The linear functional 
is called a trace morphism.
A pair 
, if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, is an object representing the contravariant functor 
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: 
where 
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 
is of pure dimension n, there is a natural isomorphism[2]
.
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
2. ^{{harvnb|Kollár–Mori|loc=Theorem 5.71.}}
3. ^{{Cite arxiv|title = Vanishing of the higher direct images of the structure sheaf|arxiv = 1404.1827 |date = 2014-04-07|first = Andre|last = Chatzistamatiou|first2 = Kay|last2 = Rülling}}
- Kleiman, Steven L. Relative duality for quasicoherent sheaves. Compositio Math. 41 (1980), no. 1, 39–60.
- {{Citation | last1=Kollár | first1=János | last2=Mori | first2=Shigefumi | title=Birational geometry of algebraic varieties | publisher=Cambridge University Press | series=Cambridge Tracts in Mathematics | isbn=978-0-521-63277-5 |mr=1658959 | year=1998 | volume=134}}
- {{Hartshorne AG}}
External links
- http://math.stanford.edu/~vakil/0506-216/216class5354.pdf
- https://mathoverflow.net/questions/211158/relative-dualizing-sheaf-reference-behavior
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