E0030 definition notation E0020 → E0001

Jets and Taylor realization

The jet space is the completion of the smooth functions at a point, dual to the point-supported distributions E0029. The Taylor map realizes jets as elements of the completed symmetric cotangent space.

(1) Definition (Maximal ideal, jets, completion).

  • \(\mathfrak{m}_x := \set{f \in \mathcal{E}(X) : f(x) = 0}\) is the maximal ideal of \(x\).
  • The \(k\)-jet space at \(x\) is the quotient algebra \(\mathcal{E}_k(X; x) := \mathcal{E}(X) / \mathfrak{m}_x^{k+1}\); the \(k\)-jet of \(f\) is its class \(j^k_x f\).
  • \(\hat{\mathcal{E}}(X; x) := \varprojlim_k \mathcal{E}_k(X; x)\) is the formal completion of the functions at \(x\).

(2) Proposition (Jet–distribution duality). The pairing \(\langle \hat{f}, u \rangle := u(f)\) induces an isomorphism \(\mathcal{E}'(X; x) \xrightarrow{\sim} \hat{\mathcal{E}}(X; x)'\).

Proof. A distribution of order \(\leq k\) supported at \(x\) vanishes on \(\mathfrak{m}_x^{k+1}\) (flat functions are approximable by functions vanishing near \(x\)), so it factors through the \(k\)-jet quotient. Conversely, a continuous functional on the projective limit factors through some finite stage, giving a distribution of finite order supported at \(x\).

(3) Definition (Taylor map).

  • The Taylor expansion of \(f \in C^\infty(X, \IR)\) at \(x\) is

    \[ t(x; f) := \sum_{r \geq 0} \frac{1}{r!}\, D^r(f; x) \in \STH^*(X; x). \]

  • The jet pairing is \(\langle t(x; f), \xi \rangle = D(f; x; \xi)\) for \(\xi \in ST_k(X; x)\).

  • The Taylor map is surjective: for any \(\alpha \in ST^k(X; x)\), the polynomial \(p_\alpha(x + v) := \sum_r \alpha_r(v, \dots, v)\) satisfies \(t(x; p_\alpha) = \alpha\) (jet realization).

(4) Proposition (Jets are multiplicative). \(t(x; fg) = \pi_{\leq k}(t(x; f) \cdot t(x; g))\) for \(f, g \in C^k(X, \IR)\).

Proof. The \(r\)-th component of \(t(x; fg)\) is \(D^r(fg; x) / r!\). Expanding by the Fréchet product rule E0023 and matching with the truncated product of Taylor expansions gives equality.

(5) Proposition (Formal functoriality). For smooth \(\phi: X \to Y\) with \(\phi(x) = y\), the pullback \(\phi^*\) preserves the maximal ideal (\(\phi^* \mathfrak{m}_y \subseteq \mathfrak{m}_x\)) and descends to algebra morphisms \(\hat{\phi}^*: \hat{\mathcal{E}}(Y; y) \to \hat{\mathcal{E}}(X; x)\) with

\[ \widehat{f \circ \phi} = \hat{\phi}^* \hat{f}, \qquad \widehat{(\psi \circ \phi)}^* = \hat{\phi}^* \circ \hat{\psi}^*. \]

Proof. \(f(y) = 0\) implies \(f(\phi(x)) = 0\), and \(\phi^*\) is multiplicative, so the ideal containments hold; functoriality is that of \(\phi^*\).

(6) Theorem (Jet pullback). For \(\phi\) \(C^k\) near \(x\) and \(\hat{f} \in \hat{\mathcal{E}}(Y; y)\),

\[ \hat{\phi}^* \hat{f} = \langle \hat{f},\, D_+(\phi; x)\, (-) \rangle, \]

that is, the jet pullback is the adjoint of the symmetric pushforward \(D_+(\phi; x)\) E0027 under the jet pairing.

Proof. Pair with \(\xi \in ST_k(X; x)\): by the jet pairing, \(\langle \hat{\phi}^* \hat{f}, \xi \rangle = D(f \circ \phi; x; \xi)\). By the smooth adjunction E0028, this equals \(D(f; y; D_+(\phi; x)\, \xi) = \langle \hat{f}, D_+(\phi; x)\, \xi \rangle\).