Jets

Let $X$ be a separated scheme over $S$ . That means, we are given a separated morphism $\varphi:X\to S$ . Let $\mathcal I_\Delta$ be the defining ideal sheaf of the diagonal $\Delta$ in $X\times_S X$ . For any positive integer $n$ , we define $\Delta^{(n)}$ to be the closed subscheme of $X\times_S X$ defined by $\mathcal I_\Delta^{(n+1)}$ .

The sheaf $\mathcal J_n=p_{1 *} \mathcal{O}_\Delta^{(n+1)}$ on $X$ is called the sheaf of $n$ -jets on $X$ relative to $S$ . There is another description of this sheaf. Let

$$p_1^{(n)}: \Delta^{(n+1)} \to X,\quad p_2^{(n)}: \Delta^{(n+1)} \to X$$

be restrictions of the projections $p_1,p_2$ . Then we have

$$\mathcal J_n=(p_1^{(n)})_* (p_2^{(n)})^* \mathcal O.$$

For a local section $f$ of $\mathcal{O}_X$ , we define the jet (``the Taylor expansion'') of $f$ (of order $n$ ) by

$$Jet(f)=p_{1 *} p_2^* f$$

EXAMPLE 9.7   Let $A$ be any ring in which $n!$ is invertible. Let $x$ be an indeterminate. We put

$$X=\operatorname{Spec}(A[x]), \quad S=\operatorname{Spec}(A),$$

$\varphi:X\to S$ being the canonical projection.

Then we have $X\times_S X=\operatorname{Spec}(A[x, \bar{x}])$ . The sheaf of $n$ -jets on $X$ relative to $S$ is

$$A[x,\bar{x}]/(\bar{x}-x)^{n+1}$$

Let us put $h=\bar{x}-x$ . Then for any $p=p(x)\in A[x]$ , we have

$$Jet(f)=f(\bar{x})=\sum_{s=0}^n \frac{1}{s!}f^{(s)}(x) h^s.$$

When $n!$ is not invertible in $A$ , a similar formula is still valid. The thing is that the operator $1/s! (d/dx)^s$ is defined over $\mathbb{Z}$ .

$$(1/s! (d/dx)^s).( \sum_{t=0}^n a_t x^t)= \sum_{t=s}^n a_t\binom{t}{s} x^{t-s}$$

Like wise, for any quasi coherent sheaf $\mathcal{F}$ on $X$ , we may define the sheaf $\mathcal J_n(\mathcal{F})$ of $n$ -jets of $\mathcal{F}$ on $X$ relative to $S$ as

$$\mathcal J_n(\mathcal{F})=p_{1 *}p_2^* \mathcal{F}.$$

For any local section $f$ of $\mathcal{F}$ , we may define the $n$ -jet of it in the same way as above.