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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

$\displaystyle 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

$\displaystyle \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

$\displaystyle 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

$\displaystyle 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

$\displaystyle 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

$\displaystyle 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}$ .

$\displaystyle (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

$\displaystyle \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.
next up previous
Next: definition of linear differential Up: Linear differential operators Previous: Linear differential operators
2007-12-11