Dualizing sheaf (original) (raw)
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). 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. .
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| dbo:abstract | 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). 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 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 F on X such that is of pure dimension n, there is a natural isomorphism . In particular, if X itself is a Cohen–Macaulay scheme, then the above duality holds for any locally free sheaf. (en) |
| dbo:wikiPageExternalLink | https://mathoverflow.net/q/211158 http://www.numdam.org/item/CM_1980__41_1_39_0.pdf%7Ctitle=Relative http://math.stanford.edu/~vakil/0506-216/216class5354.pdf |
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| dct:subject | dbc:Algebraic_geometry |
| rdfs:comment | 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). 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. . (en) |
| rdfs:label | Dualizing sheaf (en) |
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