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.

Statement

For A a ring, there is an exact sequence of sheaves

$0\to \Omega _{\mathbb {P} _{A}^{n}/A}^{1}\to {\mathcal {O}}_{\mathbb {P} _{A}^{n}}(-1)^{\oplus n+1}\to {\mathcal {O}}_{\mathbb {P} _{A}^{n}}\to 0.$

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

Corollary

By taking the highest exterior power, one gets that the canonical sheaf $\omega _{\mathbb {P} _{A}^{n}/A}={\mathcal {O}}_{\mathbb {P} _{A}^{n}}(-(n+1))$. This has no non-zero global sections, so the geometric genus is 0.

References

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157