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 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 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.
Given a proper finitely presented morphism of schemes , (Kleiman 1980) defines the relative dualizing sheaf or as[3] the sheaf such that for each open subset and a quasi-coherent sheaf on , there is a canonical isomorphism
,
which is functorial in and commutes with open restrictions.
For a smooth curve C, its dualizing sheaf can be given by the canonical sheaf.
For a nodal curve C with a node p, we may consider the normalization with two points x, y identified. Let be the sheaf of rational 1-forms on with possible simple poles at x and y, and let be the subsheaf consisting of rational 1-forms with the sum of residues at x and y equal to zero. Then the direct image defines a dualizing sheaf for the nodal curve C. The construction can be easily generalized to nodal curves with multiple nodes.
This is used in the construction of the Hodge bundle on the compactified moduli space of curves: it allows us to extend the relative canonical sheaf over the boundary which parametrizes nodal curves. The Hodge bundle is then defined as the direct image of a relative dualizing sheaf.
Dualizing sheaf of projective schemes
As mentioned above, the dualizing sheaf exists for all projective schemes. For X a closed subscheme of Pn of codimension r, its dualizing sheaf can be given as . In other words, one uses the dualizing sheaf on the ambient Pn to construct the dualizing sheaf on X.[1]
dualizing, sheaf, algebraic, geometry, dualizing, sheaf, proper, scheme, dimension, over, field, coherent, sheaf, displaystyle, omega, together, with, linear, functional, displaystyle, operatorname, omega, that, induces, natural, isomorphism, vector, spaces, d. In algebraic geometry the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf w X displaystyle omega X together with a linear functional t X H n X w X k displaystyle t X operatorname H n X omega X to k that induces a natural isomorphism of vector spaces Hom X F w X H n X F f t X f 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 t X displaystyle t X is called a trace morphism A pair w X t X displaystyle omega X t X if it is exists is unique up to a natural isomorphism In fact in the language of category theory w X displaystyle omega X is an object representing the contravariant functor F H n X F 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 w X O X K X displaystyle omega X mathcal O X K X where K X 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 Supp F displaystyle operatorname Supp F is of pure dimension n there is a natural isomorphism 2 H i X F H n i X H o m F w X 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 Contents 1 Relative dualizing sheaf 2 Examples 2 1 Dualizing sheaf of a nodal curve 2 2 Dualizing sheaf of projective schemes 3 See also 4 Note 5 References 6 External linksRelative dualizing sheaf EditGiven a proper finitely presented morphism of schemes f X Y displaystyle f X to Y Kleiman 1980 defines the relative dualizing sheaf w f displaystyle omega f or w X Y displaystyle omega X Y as 3 the sheaf such that for each open subset U Y displaystyle U subset Y and a quasi coherent sheaf F displaystyle F on U displaystyle U there is a canonical isomorphism f U F w f O Y F displaystyle f U F omega f otimes mathcal O Y F which is functorial in F displaystyle F and commutes with open restrictions Example 4 If f displaystyle f is a local complete intersection morphism between schemes of finite type over a field then by definition each point of X displaystyle X has an open neighborhood U displaystyle U and a factorization f U U i Z p Y displaystyle f U U overset i to Z overset pi to Y a regular embedding of codimension k displaystyle k followed by a smooth morphism of relative dimension r displaystyle r Then w f U r i W p 1 k N U Z displaystyle omega f U simeq wedge r i Omega pi 1 otimes wedge k N U Z where W p 1 displaystyle Omega pi 1 is the sheaf of relative Kahler differentials and N U Z displaystyle N U Z is the normal bundle to i displaystyle i Examples EditDualizing sheaf of a nodal curve Edit For a smooth curve C its dualizing sheaf w C displaystyle omega C can be given by the canonical sheaf W C 1 displaystyle Omega C 1 For a nodal curve C with a node p we may consider the normalization p C C displaystyle pi tilde C to C with two points x y identified Let W C x y displaystyle Omega tilde C x y be the sheaf of rational 1 forms on C displaystyle tilde C with possible simple poles at x and y and let W C x y 0 displaystyle Omega tilde C x y 0 be the subsheaf consisting of rational 1 forms with the sum of residues at x and y equal to zero Then the direct image p W C x y 0 displaystyle pi Omega tilde C x y 0 defines a dualizing sheaf for the nodal curve C The construction can be easily generalized to nodal curves with multiple nodes This is used in the construction of the Hodge bundle on the compactified moduli space of curves it allows us to extend the relative canonical sheaf over the boundary which parametrizes nodal curves The Hodge bundle is then defined as the direct image of a relative dualizing sheaf Dualizing sheaf of projective schemes Edit As mentioned above the dualizing sheaf exists for all projective schemes For X a closed subscheme of Pn of codimension r its dualizing sheaf can be given as E x t P n r O X w P n displaystyle mathcal Ext mathbf P n r mathcal O X omega mathbf P n In other words one uses the dualizing sheaf on the ambient Pn to construct the dualizing sheaf on X 1 See also Editcoherent duality reflexive sheaf Gorenstein ring Dualizing moduleNote Edit a b Hartshorne 1977 Ch III 7 Kollar amp Mori 1998 Theorem 5 71 Kleiman 1980 Definition 6 Arbarello Cornalba amp Griffiths 2011 Ch X near the end of 2 References EditArbarello E Cornalba M Griffiths P A 2011 Geometry of Algebraic Curves Grundlehren der mathematischen Wissenschaften Vol 268 doi 10 1007 978 3 540 69392 5 ISBN 978 3 540 42688 2 MR 2807457 Kleiman Steven L 1980 Relative duality for quasi coherent sheaves PDF Compositio Mathematica 41 1 39 60 MR 0578050 Kollar Janos 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 0463157External links Edithttp math stanford edu vakil 0506 216 216class5354 pdf Relative dualizing sheaf reference behavior This algebraic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Dualizing sheaf amp oldid 1111740123, wikipedia, wiki, book, books, library,