Tangent sheaf

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In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} {\displaystyle {\mathcal {O}}_{X}}-modules Ω X / S {\displaystyle \Omega _{X/S}} {\displaystyle \Omega _{X/S}} that represents (or classifies) S-derivations[1] in the sense: for any O X {\displaystyle {\mathcal {O}}_{X}} {\displaystyle {\mathcal {O}}_{X}}-modules F, there is an isomorphism

Hom O X ⁡ ( Ω X / S , F ) = Der S ⁡ ( O X , F ) {\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{X}}(\Omega _{X/S},F)=\operatorname {Der} _{S}({\mathcal {O}}_{X},F)} {\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{X}}(\Omega _{X/S},F)=\operatorname {Der} _{S}({\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 : O X → Ω X / S {\displaystyle d:{\mathcal {O}}_{X}\to \Omega _{X/S}} {\displaystyle d:{\mathcal {O}}_{X}\to \Omega _{X/S}} such that any S-derivation D : O X → F {\displaystyle D:{\mathcal {O}}_{X}\to F} {\displaystyle D:{\mathcal {O}}_{X}\to F} factors as D = α ∘ d {\displaystyle D=\alpha \circ d} {\displaystyle D=\alpha \circ d} with some α : Ω X / S → F {\displaystyle \alpha :\Omega _{X/S}\to F} {\displaystyle \alpha :\Omega _{X/S}\to F}.

In the case X and S are affine schemes, the above definition means that Ω X / S {\displaystyle \Omega _{X/S}} {\displaystyle \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 Θ X {\displaystyle \Theta _{X}} {\displaystyle \Theta _{X}}.[2]

There are two important exact sequences:

  1. If S →T is a morphism of schemes, then
    f ∗ Ω S / T → Ω X / T → Ω X / S → 0. {\displaystyle f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.} {\displaystyle f^{*}\Omega _{S/T}\to \Omega _{X/T}\to \Omega _{X/S}\to 0.}
  2. If Z is a closed subscheme of X with ideal sheaf I, then
    I / I 2 → Ω X / S ⊗ O X O Z → Ω Z / S → 0. {\displaystyle I/I^{2}\to \Omega _{X/S}\otimes _{O_{X}}{\mathcal {O}}_{Z}\to \Omega _{Z/S}\to 0.} {\displaystyle I/I^{2}\to \Omega _{X/S}\otimes _{O_{X}}{\mathcal {O}}_{Z}\to \Omega _{Z/S}\to 0.}[3][4]

The cotangent sheaf is closely related to smoothness of a variety or scheme. For example, an algebraic variety is smooth of dimension n if and only if ΩX is a locally free sheaf of rank n.[5]

Construction through a diagonal morphism

Let f : X → S {\displaystyle f:X\to S} {\displaystyle f:X\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:

Ω X / S = Δ ∗ ( I / I 2 ) {\displaystyle \Omega _{X/S}=\Delta ^{*}(I/I^{2})} {\displaystyle \Omega _{X/S}=\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.

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 P R n {\displaystyle \mathbf {P} _{R}^{n}} {\displaystyle \mathbf {P} _{R}^{n}} for the projective space over a ring R,

0 → Ω P R n / R → O P R n ( − 1 ) ⊕ ( n + 1 ) → O P R n → 0. {\displaystyle 0\to \Omega _{\mathbf {P} _{R}^{n}/R}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}(-1)^{\oplus (n+1)}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}\to 0.} {\displaystyle 0\to \Omega _{\mathbf {P} _{R}^{n}/R}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}(-1)^{\oplus (n+1)}\to {\mathcal {O}}_{\mathbf {P} _{R}^{n}}\to 0.}

(See also Chern class#Complex projective space.)

Cotangent stack

For this notion, see § 1 of

A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves Archived 2015-01-05 at the Wayback Machine[6]

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, S p e c ( Sym ⁡ ( E ˇ ) ) {\displaystyle \mathbf {Spec} (\operatorname {Sym} ({\check {E}}))} {\displaystyle \mathbf {Spec} (\operatorname {Sym} ({\check {E}}))} is the algebraic vector bundle corresponding to E.)

See also: Hitchin fibration (the cotangent stack of Bun G ⁡ ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} {\displaystyle \operatorname {Bun} _{G}(X)} is the total space of the Hitchin fibration.)

Notes

  1. "Section 17.27 (08RL): Modules of differentials". The Stacks project.
  2. In concise terms, this means:
    Θ X = d e f H o m O X ( Ω X , O X ) = D e r ( O X ) . {\displaystyle \Theta _{X}{\overset {\mathrm {def} }{=}}{\mathcal {H}}om_{{\mathcal {O}}_{X}}(\Omega _{X},{\mathcal {O}}_{X})={\mathcal {D}}er({\mathcal {O}}_{X}).} {\displaystyle \Theta _{X}{\overset {\mathrm {def} }{=}}{\mathcal {H}}om_{{\mathcal {O}}_{X}}(\Omega _{X},{\mathcal {O}}_{X})={\mathcal {D}}er({\mathcal {O}}_{X}).}
  3. Hartshorne 1977, Ch. II, Proposition 8.12.
  4. https://mathoverflow.net/q/79956 as well as (Hartshorne 1977, Ch. II, Theorem 8.17.)
  5. Hartshorne 1977, Ch. II, Theorem 8.15.
  6. see also: § 3 of http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf

See also

References