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-fold sum of the dual of the Serre twisting sheaf.

The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.)

Statement

be the n-dimensional projective space over a commutative ring A:

standard charts, the kernel is isomorphic to the relative differential module.

Geometric interpretation

We assume that A is a field k.

The exact sequence above is dual to the sequence

1. ,

is the tangent sheaf.

-dimensional vector space V over k:

This sequence is most easily understood by interpreting sections of the central term as 1-homogeneous vector fields on V. One such section, the Euler 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 gives rise by pull-back to a 0-homogeneous function on V (again partially defined). We obtain 1-homogeneous vector fields by multiplying the Euler vector field by such functions. This is the definition of the first map, and its injectivity is immediate.

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 can be defined as a derivation of the functions defined on this open set. Pulled-back in V, this is equivalent to a derivation on the preimage of U can be thus obtained, and the defect of injectivity of this mapping consists precisely of the radial vector fields.

Therefore the kernel of the second morphism equals the image 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 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. This can be found by looking at the Euler sequence and plugging it into the determinant formula.

Chern classes

The Euler sequence can be used to compute the Chern classes. we find represents the hyperplane class in the Chow ring. Using the exact sequence Since we need to invert the polynomial in the denominator, this is equivalent to finding a power series.

Notes

References