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.

Geometric interpretation

We assume that A is a field k.

The exact sequence above is equivalent the sequence

$0\to {\mathcal {O}}_{\mathbb {P} ^{n}}\to {\mathcal {O}}(1)^{\oplus (n+1)}\to {\mathcal {T}}_{\mathbb {P} ^{n}}\to 0$.

We consider V a n+1 dimensional vector space over k , and explain the exact sequence

$0\to {\mathcal {O}}_{\mathbb {P} (V)}\to {\mathcal {O}}_{\mathbb {P} (V)}(1)\otimes V\to {\mathcal {T}}_{\mathbb {P} (V)}\to 0$

This sequence is most easily understood by interpreting the central term as the sheaf of 1-homogeneous vector fields on the vector space V. There exists a remarkable section of this sheaf, the Euler vector field tautologically defined by associating to a point of the vector space the identically associated tangent vector (ie. itself : it is the identity map seen as a vector field).

This vector field is radial in the sense that it vanishes uniformly on 0-homogeneous functions, that is, the functions that are ΅independent of the radial coordinate".

A function (defined on some open set) on $\mathbb {P} (V)$ gives rise to a 0-homogeneous function 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. If a vector field on Ǘ transforms 0-homogeneous functions into 0-homogeneous functions by derivation, then it induces a vector field on the projective space $\mathbb {P} (V)$, and any vector field can be thus obtained, and the defect of injectivity of this mapping consists precisely of the radial vector fields.

We see therefore that the kernel of the second morphism identifies with the range of the first one.

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

\mathcal O_{\mathbb P^{n}} \to