In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents (or classifies) S-derivations[1] in the sense: for any -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 such that any S-derivation factors as with some .
In the case X and S are affine schemes, the above definition means that 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 .[2]
There are two important exact sequences:
If S →T is a morphism of schemes, then
If Z is a closed subscheme of X with ideal sheaf I, then
[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]
^"Section 17.27 (08RL): Modules of differentials—The Stacks project".
^In concise terms, this means:
^Hartshorne 1977, Ch. II, Proposition 8.12.
^https://mathoverflow.net/q/79956 as well as (Hartshorne 1977, Ch. II, Theorem 8.17.)
algebraic geometry, given a morphism f: X → S of schemes, the cotangentsheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules Ω X / S...
than smooth manifolds, such as complex manifolds, or (in the form of cotangentsheaf) algebraic varieties or schemes. In the smooth case, any Riemannian...
In mathematics, the cotangent complex is a common generalisation of the cotangentsheaf, normal bundle and virtual tangent bundle of a map of geometric...
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. A quasi-coherent sheaf on a ringed...
The cotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the cotangent bundle...
dimension n. Then the tangent sheaf of X is the dual of the cotangentsheaf Ω X {\displaystyle \Omega _{X}} and the canonical sheaf ω X {\displaystyle \omega...
Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous...
_{{\mathcal {O}}_{P}}(\Omega _{X/Y})P} is the divisor of the relative cotangentsheaf Ω X / Y {\displaystyle \Omega _{X/Y}} (called the different). Riemann–Roch...
{\displaystyle \,\!\Omega ^{n}=\omega } , which is the nth exterior power of the cotangent bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex...
\delta _{f}:X\to X\times _{Y}X} is an open immersion. The relative cotangentsheaf Ω X / Y {\displaystyle \Omega _{X/Y}} is zero. Finite extensions of...
tangent space. If x is a point of X and mx is ideal sheaf of all functions vanishing at x, then the cotangent space at x is mx / mx2. The tangent space is (mx...
theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications...
natural morphism from S to the grassmannian G(2,5) defined by the cotangentsheaf of S generated by its 5-dimensional space of global sections. Let F'...
I / I 2 {\displaystyle I/I^{2}} can be shown to be isomorphic to the cotangent space T x ∗ M {\displaystyle T_{x}^{*}M} through the use of Taylor's theorem...
holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle...
spectra. From the functor of points point-of-view, a derived scheme is a sheaf X on the category of simplicial commutative rings which admits an open affine...
the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle. More abstractly, given an immersion i : N → M {\displaystyle i:N\to...
geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic...
is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the normal sheaf, the dual of I / I 2 {\displaystyle I/I^{2}} , is locally...
of spheres to spheres. In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle...
structure known as a sheaf of real vector spaces on M {\displaystyle M} . The above construction applies equally well to the cotangent bundle – the differential...
X / Y {\displaystyle C_{X/Y}} of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec Spec X ( ⨁ n = 0 ∞ I n / I...
Global Information
