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.

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 ${\displaystyle S(-1)^{\oplus n+1}\to S,e_{i}\mapsto x_{i}}$ with ${\displaystyle S=A[x_{0},\ldots ,x_{n}]}$ and ${\displaystyle e_{i}=1}$ in degree 1, surjective in degrees ${\displaystyle \geq 1}$ and checking that locally on the n + 1 standard charts the kernel is isomorphic to the relative differential module.

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