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
It can be proved by defining a homomorphism formula_2 with formula_3 and formula_4 in degree 1, surjective in degrees formula_5 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 to the sequence
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 
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 formula_8 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 formula_9 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 formula_9 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 formula_11. 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 formulaformula_12for any short exact sequence of the form formula_13.
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 sheavesformula_13we can compute the total chern class of formula_15 with the formula formula_16. For example, on formula_17 we findformula_18where formula_19 represents the hyperplane class in the chow ring formula_20. Using the exact sequenceformula_21we can again use the total chern class formula to findformula_22since we need to invert the polynomial in the denominator, this is equivalent to finding a power series formula_23 such that formula_24.