Cotangent sheaf

In algebraic geometry, given a morphism "f": "X" → "S" of schemes, the cotangent sheaf on "X" is the sheaf of formula_1-modules formula_2 that represents (or classifies) "S"-derivations in the sense: for any formula_1-modules "F", there is an isomorphism
that depends naturally on "F". In other words, the cotangent sheaf is characterized by the universal property: there is the differential formula_5 such that any "S"-derivation formula_6 factors as formula_7 with some formula_8.
In the case "X" and "S" are affine schemes, the above definition means that formula_2 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 formula_10.
There are two important exact sequences:
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 Ω"X" is a locally free sheaf of rank "n".
Construction through a diagonal morphism.
Let formula_13 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:
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".
See also: bundle of principal parts.
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 formula_15 for the projective space over a ring "R",
Cotangent stack.
For this notion, see § 1 of
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, formula_17 is the algebraic vector bundle corresponding to "E".)
See also: Hitchin fibration (the cotangent stack of formula_18 is the total space of the Hitchin fibration.)