Cotangent sheaf

Let 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: :\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 \mathbf{P}^n_R for the projective space over a ring R, :0 \to \Omega_{\mathbf{P}^n_R/R} \to \mathcal{O}_{\mathbf{P}^n_R}(-1)^{\oplus(n+1)} \to \mathcal{O}_{\mathbf{P}^n_R} \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 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, \mathbf{Spec}(\operatorname{Sym}(\check{E})) is the algebraic vector bundle corresponding to E.) See also: Hitchin fibration (the cotangent stack of \operatorname{Bun}_G(X) is the total space of the Hitchin fibration.) == Notes ==