In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (n + 1)-fold sum of the dual of the Serre twisting sheaf.
Contents
The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.)
Statement
For A a ring, there is an exact sequence of sheaves
It can be proved by defining a homomorphism
Geometric interpretation
We assume that A is a field k.
The exact sequence above is equivalent to the sequence
where the last nonzero term is the tangent sheaf.
We consider V a n+1 dimensional vector space over k , and explain the exact sequence
This sequence is most easily understood by interpreting the central term as the sheaf of 1-homogeneous vector fields on the vector space V. There exists a remarkable section of this sheaf, the Euler vector field, tautologically defined by associating to a point of the vector space the identically associated tangent vector (ie. itself : it is the identity map seen as a vector field).
This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are invariant by homothetic rescaling, or "independent of the radial coordinate".
A function (defined on some open set) on
The second map is related to the notion of derivation, equivalent to that of vector field. Recall that a vector field on an open set U of the projective space
We see therefore that the kernel of the second morphism identifies with the range of the first one.
The canonical line bundle of projective spaces
By taking the highest exterior power, one sees that the canonical sheaf of a projective space is given by
In particular, projective spaces are Fano varieties, because the canonical bundle is anti-ample and this line bundle has no non-zero global sections, so the geometric genus is 0.