Cotangent sheaf

 ( Algebraic Geometry ) 

In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of $\backslash mathcal\{O\}\_X$-modules $\backslash Omega\_\{X/S\}$ that represents (or classifies) S-derivations in the sense: for any $\backslash mathcal\{O\}\_X$-modules F, there is an isomorphism

$\backslash operatorname\{Hom\}\_\{\backslash mathcal\{O\}\_X\}(\backslash Omega\_\{X/S\},\; F)\; =\; \backslash operatorname\{Der\}\_S(\backslash mathcal\{O\}\_X,\; F)$

that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential $d:\; \backslash mathcal\{O\}\_X\; \backslash to\; \backslash Omega\_\{X/S\}$ such that any S-derivation $D:\; \backslash mathcal\{O\}\_X\; \backslash to\; F$ factors as $D\; =\; \backslash alpha\; \backslash circ\; d$ with some $\backslash alpha:\; \backslash Omega\_\{X/S\}\; \backslash to\; F$.

In the case X and S are affine schemes, the above definition means that $\backslash Omega\_\{X/S\}$ is the module of Kähler differentials. The standard way to construct a cotangent sheaf (e.g., Hartshorne, Ch II. § 8) is through a diagonal morphism (which amounts to gluing modules of Kähler differentials on affine charts to get the globally defined cotangent sheaf). The dual module of the cotangent sheaf on a scheme X is called the tangent sheaf on X and is sometimes denoted by $\backslash Theta\_X$.In concise terms, this means:

$\backslash Theta\_X\; \backslash overset\{\backslash mathrm\{def\}\}\; =\; \backslash mathcal\{H\}om\_\{\backslash mathcal\{O\}\_X\}(\backslash Omega\_X,\; \backslash mathcal\{O\}\_X)\; =\; \backslash mathcal\{D\}er(\backslash mathcal\{O\}\_X).$

There are two important :

1. If S → T is a morphism of schemes, then
2. :$f^*\; \backslash Omega\_\{S/T\}\; \backslash to\; \backslash Omega\_\{X/T\}\; \backslash to\; \backslash Omega\_\{X/S\}\; \backslash to\; 0.$
3. If Z is a closed subscheme of X with ideal sheaf I, then
4. :$I/I^$ as well as

The cotangent sheaf is closely related to smooth variety of a variety or scheme. For example, an algebraic variety is smooth variety of dimension n if and only if Ω _X is a locally free sheaf of rank n.

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Construction through a diagonal morphism Let $f:\; X\; \backslash to\; S$ be a morphism of schemes as in the introduction and Δ: X → X × _S X the diagonal morphism. Then the image of Δ is locally closed; i.e., closed in some open subset W of X × _S X (the image is closed if and only if f is separated). Let I be the ideal sheaf of Δ( X) in W. One then puts:

$\backslash Omega\_\{X/S\}\; =\; \backslash Delta^*\; (I/I^2)$

and checks this sheaf of modules satisfies the required universal property of a cotangent sheaf (Hartshorne, Ch II. Remark 8.9.2). The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if S is Noetherian and f is of finite type.

The above definition means that the cotangent sheaf on X is the restriction to X of the conormal sheaf to the diagonal embedding of X over S.

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Relation to a tautological line bundle The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing $\backslash mathbf\{P\}^n\_R$ for the projective space over a ring R,

$0\; \backslash to\; \backslash Omega\_\{\backslash mathbf\{P\}^n\_R/R\}\; \backslash to\; \backslash mathcal\{O\}\_\{\backslash mathbf\{P\}^n\_R\}(-1)^\{\backslash oplus(n+1)\}\; \backslash to\; \backslash mathcal\{O\}\_\{\backslash mathbf\{P\}^n\_R\}\; \backslash to\; 0.$

(See also Chern class#Complex projective space.)

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Cotangent stack For this notion, see § 1 of

A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves [1] see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf

There, the cotangent stack on an algebraic stack X is defined as the relative Spec of the symmetric algebra of the tangent sheaf on X. (Note: in general, if E is a locally free sheaf of finite rank, $\backslash mathbf\{Spec\}(\backslash operatorname\{Sym\}(\backslash check\{E\}))$ is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of $\backslash operatorname\{Bun\}\_G(X)$ is the total space of the Hitchin fibration.)

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Notes

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See also

• Canonical sheaf
• Cotangent complex

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External links

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