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In [[mathematics]], the '''Euler sequence''' is a particular [[exact sequence]] of [[Sheaf (mathematics)|sheaves]] on ''n''-dimensional [[projective space]] over a [[ring (mathematics)|ring]]. It shows that the [[sheaf of relative differentials]] is [[stably isomorphic]] to an <math>(''n''&nbsp;+&nbsp;1)</math>-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 ==
Let <math>\mathbb{P}^n_A</math> be the ''n''-dimensional projective space over a commutative ring ''A''. Let <math>\Omega^1 = \Omega^1_{\mathbb P^n_A/A}</math> be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on <math>\mathbb{P}^n_A</math>:
For ''A'' a ring, there is an exact sequence of sheaves
 
::<math display="block"> 0 \tolongrightarrow \Omega^1_{\mathbb P^n_A/A}1 \tolongrightarrow \mathcal{O}_{\mathbb{P}^n_A}(-1)^{\oplus (n+1)} \tolongrightarrow \mathcal{O}_{\mathbb{P}^n_A} \tolongrightarrow 0.</math>
 
ItThe sequence can be provedconstructed by defining a homomorphism <math>S(-1)^{\oplus n+1} \to S, e_i \mapsto x_i</math> with <math>S = A[x_0, \ldots, x_n]</math> and <math>e_i = 1</math> in degree 1, surjective in degrees <math>\geq 1</math>, and checking that locally on the ''<math>n''&nbsp;+&nbsp;1</math> standard charts, the kernel is isomorphic to the relative differential module.<ref>Theorem II.8.13 in {{Harvnb|Hartshorne|1977}}</ref>
 
==Geometric interpretation==
We assume that ''A'' is a [[field (mathematics)|field]] '''k'''.▼
 
The exact sequence above is equivalentdual to the sequence▼
▲We assume that ''A'' is a [[field (mathematics)|field]] '''k'''.
:<math> 0 \longrightarrow \mathcal O \longrightarrow \mathcal O (1)^{\oplus (n+1)} \longrightarrow \mathcal T \longrightarrow 0 </math>,
where <math>\mathcal T</math> is the [[tangent sheaf]] of <math>\mathbb{P}^n</math>.
 
Let us explain the coordinate-free version of this sequence, on <math>\mathbb{P} V</math> for an <math>(n+1)</math>-dimensional [[vector space]] ''V'' over ''k'':
▲The exact sequence above is equivalent to the sequence
:<math> 0\longrightarrow \to \mathcal O_{\mathbb P^{nP} V} \tolongrightarrow \mathcal O (1)^O_{\oplusmathbb{P} V}(n+1)}\otimes V \tolongrightarrow \mathcal T_{\mathbb {P^n} V} \tolongrightarrow 0. </math>,
where the last nonzero term is the [[tangent sheaf]].
 
This sequence is most easily understood by interpreting sections of the central term as 1-homogeneous [[vector field]]s on ''V''. One such section, the [[Euler vector field]], associates to each point <math>v</math> of the variety <math>V</math> the tangent vector <math>v</math>. 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''".▼
We consider ''V'' a ''n+1'' dimensional [[vector space]] over ''k '', and explain the exact sequence
:<math>0\to \mathcal O_{\mathbb P(V)} \to \mathcal O_{\mathbb P (V)}(1)\otimes V \to \mathcal T_{\mathbb P (V)} \to 0 </math>▼
 
A function (defined on some open set) on <math>\mathbb P (V)</math> 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.▼
This sequence is most easily understood by interpreting the central term as the sheaf of 1-homogeneous [[vector field]]s 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).
 
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 <math>\mathbb {P(} V)</math> 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 <math>\mathbb{P} V</math> can be thus obtained, and the defect of injectivity of this mapping consists precisely of the radial vector fields.▼
▲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''".
 
We see therefore thatTherefore the kernel of the second morphism identifies withequals the rangeimage of the first one.▼
▲A function (defined on some open set) on <math>\mathbb P (V)</math> 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 canonical line bundle of projective spaces==▼
The second map is related to the notion of derivation, equivalent to that of vector field.
By taking the highest [[exterior power]], one sees that the [[canonical sheaf]] of a [[Algebraic geometry of projective spaces|projective space]] is given by <math display="block">\omega_{\mathbb{P}^n_A/A} = \mathcal{O}_{\mathbb{P}^n_A}(-(n+1)).</math> In particular, projective spaces are [[Fano varieties]], because the canonical bundle is anti-[[ample line bundle|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<ref>{{Cite book|last=Vakil|first=Ravi|url=http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf| title=Rising Sea|location=386|archive-url=https://web.archive.org/web/20191130195401/http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf|archive-date=2019-11-30}}</ref> <math display="block">\det(\mathcal{E}) = \det(\mathcal{E}') \otimes \det(\mathcal{E}'')</math> for any short exact sequence of the form <math>0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}''\to 0</math>.
▲Recall that a vector field on an open set ''U'' of the projective space <math>\mathbb P(V)</math> 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 <math>\mathbb P(V)</math> can be thus obtained, and the defect of injectivity of this mapping consists precisely of the radial vector fields.
 
==Chern classes==
▲We see therefore that the kernel of the second morphism identifies with the range of the first one.
The Euler sequence can be used to compute the [[Chern class]]es of projective space. Recall that given a short exact sequence of coherent sheaves,
 
▲:<math display="block">0 \to \mathcal O_{\mathbb P(V)E}' \to \mathcal O_{\mathbb P (V)E}(1)\otimes V \to \mathcal T_{\mathbb P (V)E} ''\to 0 ,</math>
▲== The canonical line bundle of projective spaces==
we can compute the total Chern class of <math>\mathcal{E}</math> with the formula <math>c(\mathcal{E}) = c(\mathcal{E}')\cdot c(\mathcal{E}'')</math>.<ref>{{Cite web|url=https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf|title=3264 and all that|page=169}}</ref> For example, on <math>\mathbb{P}^2</math> we find<ref>Note that <math>[H]^3 = 0</math> in the Chow ring for dimension reasons.</ref> <math display="block">\begin{align}
By taking the highest [[exterior power]], one sees that the [[canonical sheaf]] of a [[Algebraic geometry of projective spaces|projective space]] is given by
:<math>c(\omega_Omega^1_{\mathbb{P}^n_A/A2}) &= \frac{c(\mathcal{O}_(-1)^{\mathbb{P}^n_A}(-oplus (n2+1)})</math>.}{c(\mathcal{O})} \\
&= (1 - [H])^3 \\
In particular, projective spaces are [[Fano varieties]], because the canonical bundle is anti-[[ample line bundle|ample]] and this line bundle has no non-zero global sections, so the [[geometric genus]] is 0.
&= 1 - 3[H] + 3[H]^2 - [H]^3 \\
&= 1 - 3[H] + 3[H]^2,
\end{align}</math>
where <math>[H]</math> represents the hyperplane class in the [[Chow group|Chow ring]] <math>A^\bullet(\mathbb{P}^2)</math>. Using the exact sequence<ref>{{Cite web|url=https://www.math.purdue.edu/~arapura/preprints/book-chap17.pdf|title=Computation of Some Hodge Numbers| last=Arapura|first=Donu| url-status=live|archive-url=https://web.archive.org/web/20200201234047/https://www.math.purdue.edu/~arapura/preprints/book-chap17.pdf|archive-date=1 February 2020}}</ref>
<math display="block">0 \to \Omega^2 \to \mathcal{O}(-2)^{\oplus 3} \to \Omega^1 \to 0,</math>
we can again use the total Chern class formula to find
<math display="block">\begin{align}
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{align}</math>
Since we need to invert the polynomial in the denominator, this is equivalent to finding a [[power series]] <math>a([H]) = a_0 + a_1[H] + a_2[H]^2 +a_3[H]^3 + \cdots</math> such that <math>a([H])c(\Omega^1) = 1</math>.
 
== Notes ==
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== References ==
* {{Hartshorne AG}}
* {{Citation | last1=Rubei | first1=Elena | title=Algebraic Geometry, a concise dictionary | publisher=[[De Gruyter|Walter De Gruyter]] | location=Berlin/Boston | isbn=978-3-11-031622-3 | year=2014|ref=none}}
 
 
 
[[Category:Algebraic geometry]]
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