Editing Jet bundle (section)

=== Algebro-geometric perspective ===
An independently motivated construction of the sheaf of sections <math>\Gamma J^k\left(\pi_{TM}\right)</math>'' is given''.''

Consider a diagonal map <math display="inline">\Delta_n: M \to \prod_{i=1}^{n+1} M</math>, where the smooth manifold <math>M</math> is a [[locally ringed space]] by <math>C^k(U)</math> for each open <math>U</math>. Let <math>\mathcal{I}</math> be the [[ideal sheaf]] of <math>\Delta_n(M)</math>, equivalently let <math>\mathcal{I}</math> be the [[Sheaf (mathematics)|sheaf]] of smooth [[Germ (mathematics)|germs]] which vanish on <math>\Delta_n(M)</math> for all <math>0 < n \leq k</math>.  The [[Inverse image functor|pullback]] of the [[quotient sheaf]] <math>{\Delta_n}^*\left(\mathcal{I}/\mathcal{I}^{n+1}\right)</math> from <math display="inline">\prod_{i=1}^{n+1} M</math> to <math>M</math> by <math>\Delta_n</math> is the sheaf of k-jets.<ref>{{Cite web|url=http://math.stanford.edu/~vakil/files/jets.pdf|title=A beginner's guide to jet bundles from the point of view of algebraic geometry|last=Vakil|first=Ravi|date=August 25, 1998|access-date=June 25, 2017}}</ref>

The [[direct limit]] of the sequence of injections given by the canonical inclusions <math>\mathcal{I}^{n+1} \hookrightarrow \mathcal{I}^n</math> of sheaves, gives rise to the '''infinite jet sheaf''' <math>\mathcal{J}^\infty(TM)</math>. Observe that by the direct limit construction it is a filtered ring.