Euler sequence

On projective space, the Euler sequence is

0 \to O \to O (1)^{\oplus (m + 1)} \to T P^{m} \to 0.

Table of contents
- Cotangent Euler sequence in homogeneous coordinates Euler sequence
  - Vanishing of global sections of the once-twisted projective cotangent bundle Cotangent Euler sequence in homogeneous coordinates

Cotangent Euler sequence in homogeneous coordinates ( $0 \to Ω_{P^{n}}^{1} \to O (- 1)^{\oplus (n + 1)} \to O \to 0$ )

For a degree-zero rational function regular on an open subset of projective space, each derivative

\partial f / \partial X_{i}

is a regular homogeneous rational function of degree

- 1

. These derivatives define a derivation with values in

O (- 1)^{\oplus (n + 1)}

, hence a map from the Kähler differential sheaf. The last map is

(g_{i}) \mapsto \sum_{i} X_{i} g_{i}

. On

X_{i} \neq = 0

, put

y_{j} = X_{j} / X_{i}

and use frame

X_{i}^{- 1}

. The derivative of

y_{j}

has coordinate vector with

1

at

j

,

- y_{j}

at

i

and zero elsewhere. Those vectors form a free basis of the kernel of

(b_{i}) \mapsto b_{i} + \sum_{j \neq = i} y_{j} b_{j}

, proving exactness directly in every characteristic.

Vanishing of global sections of the once-twisted projective cotangent bundle ( $H^{0} (P^{n}, Ω^{1} (1)) = 0$ )

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain

0 \to Ω^{1} (1) \to O^{n + 1} \to O (1) \to 0

. On global sections the last map sends

(c_{i})

to

\sum_{i} c_{i} X_{i}

, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.

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Euler sequence by

Wikipedia Bot  1

The term "Euler sequence" can refer to different concepts depending on the context, but one of the most common uses is related to the Euler numbers or the sequence of Euler's totient function. 1. **Euler Numbers**: In combinatorial mathematics, Euler numbers (not to be confused with Eulerian numbers) are a sequence of integers that occur in the expansion of certain generating functions. They can be defined recursively and are used in various areas of mathematics, such as topology and number theory.

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