Jet Schemes

Author: Eiko

Time: 2025-04-12 04:11:30 - 2025-05-11 06:45:18 (UTC)

Jet Schemes

Let $X$ be a scheme of finite type over $k$ , the jet scheme $J_{m} (X)$ is characterized by the following identification of functor of points

$Hom (Spec (A), J_{m} (X)) ≅ Hom (Spec (A [t] / (t^{m + 1})), X)$

One can view this as an adjoint pair of functors, and the ordinary points of the jet scheme corresponds to thickened points of the original scheme

$J_{m} (X) (Spec (k)) ≅ X (Spec (k [t] / (t^{m + 1}))) .$

For example one can imagine $J_{1} (X)$ as the set of tangent vectors, elements in the tangent bundle $T X$ . When $m = 0$ , the jet scheme reduces to the original scheme $J_{0} (X) = X$ .

There is a canonical projection map when $n > m$ ,

$π_{n, m} : J_{n} (X) \to J_{m} (X)$

induced by the inclusion map $k [t] / (t^{m + 1}) \to k [t] / (t^{n + 1})$ (which sets higher order terms zero). This is a direct system, so $J_{n} (X)$ forms a projective system and we can take the limit

$J_{\infty} (X) = \lim_{\leftarrow} J_{n} (X)$

which is thought as arc space of $X$ , whose functor of points is obviously the limit

$Hom (Spec (A), J_{\infty} (X)) = Hom (Spec (A [[t]]), X)$

where $A [[t]]$ is the formal power series ring over $A$ .

The $J_{n}$ is a covariant endofunctor in the category of schemes of finite type over $k$ , if $f : X \to Y$ we would have $f_{m} := J_{m} (f) : J_{m} (X) \to J_{m} (Y)$ , defined as ‘pushforward’

$γ : Spec (k [t] / (t^{m + 1})) \to X \mapsto f \circ γ : Spec (k [t] / (t^{m + 1})) \to Y .$

The functor maps are compatible with the projection maps.

Examples

Jet Schemes of Affine Spaces

Let $X = A_{k}^{n}$ , we have

$\begin{aligned} J_{m} (A_{k}^{n}) & = Hom (k [x_{1}, \dots, x_{n}], k [t] / (t^{m + 1})) \\ = (k [t] / (t^{m + 1}))^{n} \\ ≅ A_{k}^{n (m + 1)} . \end{aligned}$

The map $k [x_{1}, \dots, x_{n}] \to k [t] / (t^{m + 1})$ specifies a $O (t^{m + 1})$ formula as a function of $t$ for each of the coordinate $x_{i}$

$x_{i} = a_{i 0} + a_{i 1} t + \dots + a_{i m} t^{m} + O (t^{m + 1})$

and this defines a local order $m$ curve, this local curve is a point in the jet scheme $J_{m} (A_{k}^{n})$ .

(Random thought: hey this looks like a point in a Hilbert scheme, for example at a smooth point of our variety, when two points collide, it remembers a tangent direction (but without direction, it’s a projective direction) we seem to have a point in the first jet space. Maybe if we define a $C^{*}$ action on the jet space, and take quotient, we will arrive at some projective jet scheme that is similar to Hilbert scheme?)

Easy Properties

If $J_{m} (X)$ exists and $U \subset X$ is open, then $J_{m} (U)$ exists as well given by $J_{m} (U) = π_{m}^{- 1} (U)$ .
If $X ↪ Y$ is an embedding, then $J_{m} (X) ↪ J_{m} (Y)$ is also an embedding.
Jet functor commutes with finite products,

$J_{m} (X \times Y) ≅ J_{m} (X) \times J_{m} (Y) .$
For a group scheme $G$ over $k$ , the jet scheme $J_{m} (G)$ is also a group scheme over $k$ . (imagine it as multiplication of two local curves)
For $f : Y \to X$ a morphism and $Z \subset X$ closed embedding,

$J_{m} (f^{- 1} Z) = f_{m}^{- 1} J_{m} (Z)$
For any etale morphism $f : X \to Y$ , the diagram

$rendering math failed o.o$

is cartesian. Intuitively this means you can lift a local curve uniquely over an etale morphism. (Does this look like the Homotopy lifting property in homotopy theory?)