# nLab jet scheme

## Idea and definition

For finite $m$ this is a version of $m$-jet space in algebraic geometry.

Let $k$ be the algebraically closed field, $Sch/k$ the category of schemes over $k$ and $X$ an object in $Sch/k$. The presheaf

$(Sch/k)^{op}\to Set\,\,\,\,\,\,\,\,\,\, Y\mapsto (Sch/k) (Y\times_k k[t]/t^{m+1},X)$

is representable by a $k$-scheme of finite type $X_m$ the $m$-jet scheme.

An analogue of the $\infty$-jet space is the arc space.