Sheaf of differential ring over manifold

Preliminary

Math Essays: Differential Ring (1): Some general result and interesting application (wuyulanliulongblog.blogspot.com)

Math Essays: Differential operator polynomial as a functor (wuyulanliulongblog.blogspot.com)

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Let $M$$M$ be a n-dimension smooth manifold, $f_1,...,f_n$$f_1,...,f_n$ be a family of smooth function.

Then

$$\begin{array}{}\text{(1)}& \delta =f_1\frac{\partial }{\partial x_1}+...+f_n\frac{\partial }{\partial x_n}\end{array}$$

is a derivation over $C^{\mathrm{\infty }}(M)$$C^{\infty}(M)$.

Hence the sheaf $\mathcal{F}(U)=(C^{\mathrm{\infty }}(U),\delta (U))$$\mathcal F(U)=(C^{\infty}(U),\delta(U))$ is a sheaf of differential ring.

The restriction map

$$\begin{array}{}\text{(2)}& {\mathrm{res}}_U^M:(C^{\mathrm{\infty }}(M),\delta (M))⟶(C^{\mathrm{\infty }}(U),\delta (U))\end{array}$$

is a differential ring homomorphism. Since ${\mathrm{res}}_U^M\circ \delta (M)=\delta (U)\circ {\mathrm{res}}_U^M$$\mathrm{res}_U^M\circ\delta(M)=\delta(U)\circ\mathrm{res}_U^M$.

Consider the kernel of $\delta (M):C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$$\delta(M):C^{\infty}(M)\to C^{\infty}(M)$, i.e. $\mathrm{Ker}\delta (M)$$\mathrm{Ker}\delta(M)$

by the functorial property of differential operator polynomial, ${\mathrm{res}}_U^M$$\mathrm{res}^M_U$ induced a ring homomorphism $\mathrm{Ker}(\delta (M))\to \mathrm{Ker}(\delta (U))$$\mathrm{Ker}(\delta(M))\to\mathrm{Ker}(\delta(U))$.

Hence we could consider embedding $i:\mathrm{Ker}\delta (M)⟶C^{\mathrm{\infty }}(U)$$i:\mathrm{Ker}\delta(M)\longrightarrow C^{\infty}(U)$, turns $\mathcal{F}(U)$$\mathcal F(U)$ to a sheaf of $\mathrm{Ker}\delta (M)$$\mathrm{Ker}\delta(M)$- Algebra.