Euler sequence

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 -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.) Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on : The sequence can be constructed by defining a homomorphism with and in degree 1, surjective in degrees , and checking that locally on the standard charts, the kernel is isomorphic to the relative differential module. We assume that A is a field k. The exact sequence above is dual to the sequence where is the tangent sheaf of . Let us explain the coordinate-free version of this sequence, on for an -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, associates to each point of the variety the tangent vector . 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 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 of the projective space 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 that preserves 0-homogeneous functions.