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.)

Statement

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

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

It can be proved by defining a homomorphism ${\displaystyle S(-1)^{\oplus n+1}\to S,e_{i}\mapsto x_{i}}$ 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.[1]

Geometric interpretation

We assume that A is a field k.

The exact sequence above is equivalent to the sequence

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

where the last nonzero term is the tangent sheaf.

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

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

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 invariant by homothetic rescaling, or "independent of the radial coordinate".

A function (defined on some open set) on ${\displaystyle \mathbb {P} (V)}$ 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 ${\displaystyle \mathbb {P} (V)}$ 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. Any vector field on ${\displaystyle \mathbb {P} (V)}$ 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.

The canonical line bundle of projective spaces

By taking the highest exterior power, one sees that the canonical sheaf of a projective space is given by

${\displaystyle \omega _{\mathbb {P} _{A}^{n}/A}={\mathcal {O}}_{\mathbb {P} _{A}^{n}}(-(n+1))}$.

In particular, projective spaces are Fano varieties, because the canonical bundle is anti-ample and this line bundle has no non-zero global sections, so the geometric genus is 0. This can be found by looking at the Euler sequence and plugging it into the determinant formula

${\displaystyle {\text{det}}({\mathcal {E}})={\text{det}}({\mathcal {E}}')\otimes {\text{det}}({\mathcal {E}}'')}$[2]

for any short exact sequence of the form ${\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0}$.

Chern Classes

The euler sequence can be used to compute the Chern classes of projective space. Recall that given a short exact sequence of coherent sheaves

${\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0}$

we can compute the total chern class of ${\displaystyle {\mathcal {E}}}$ with the formula ${\displaystyle c({\mathcal {E}})=c({\mathcal {E}}')\cdot c({\mathcal {E}}'')}$.[3] For example, on ${\displaystyle \mathbb {P} ^{2}}$ we find

${\displaystyle {\begin{aligned}c(\Omega _{\mathbb {P} ^{2}}^{1})&={\frac {c({\mathcal {O}}(-1)^{\oplus (2+1)})}{c({\mathcal {O}})}}\\&=(1-[H])^{3}\\&=1-3[H]+3[H]^{2}-[H]^{3}\\&=1-3[H]+3[H]^{2}\end{aligned}}}$[4]

where ${\displaystyle [H]}$ represents the hyperplane class in the chow ring ${\displaystyle A^{\bullet }(\mathbb {P} ^{2})}$. Using the exact sequence

${\displaystyle 0\to \Omega ^{2}\to {\mathcal {O}}(-2)^{\oplus 3}\to \Omega ^{1}\to 0}$[5]

we can again use the total chern class formula to find

${\displaystyle {\begin{aligned}c(\Omega ^{2})&={\frac {c({\mathcal {O}}(-2)^{\oplus 3})}{c(\Omega ^{1})}}\\&={\frac {(1-2[H])^{3}}{1-3[H]+3[H]^{2}}}\\\end{aligned}}}$

since we need to invert the polynomial in the denominator, this is equivalent to finding a power series ${\displaystyle a([H])=a_{0}+a_{1}[H]+a_{2}[H]^{2}+a_{3}[H]^{3}+\cdots }$ such that ${\displaystyle a([H])c(\Omega ^{1})=1}$.

Notes

1. Theorem II.8.13 in Hartshorne 1977
2. Vakil, Ravi. Rising Sea (PDF). 386. Archived from the original (PDF) on 2019-11-30.CS1 maint: location (link)
3. "3264 and all that" (PDF). p. 169.
4. Note that ${\displaystyle [H]^{3}=0}$ in the chow ring for dimension reasons.
5. Arapura, Donu. "Computation of Some Hodge Numbers" (PDF). Archived (PDF) from the original on 1 February 2020.

References

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3