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 (n + 1)-fold sum of the dual of the Serre twisting sheaf. 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. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) For A a ring, there is an exact sequence of sheaves It can be proved by defining a homomorphism S ( − 1 ) ⊕ n + 1 → S , e i ↦ x i {displaystyle S(-1)^{oplus n+1} o S,e_{i}mapsto x_{i}} with S = A [ x 0 , … , x n ] {displaystyle S=A} and e i = 1 {displaystyle e_{i}=1} in degree 1, surjective in degrees ≥ 1 {displaystyle geq 1} and checking that locally on the n + 1 standard charts the kernel is isomorphic to the relative differential module. We assume that A is a field k.