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Let $A$ be a local algebra over a field $k$ with maximal ideal $m,$ such that $A/m=k$. Suppose that $A$ is regular. Then it seems to me that we can think of the tangent space as being any of the equivalent notions given by $$\mathrm{Hom}_{local-k-alg}(A,k[t]/t^2)\cong \mathrm{Hom}_k(m/m^2,k) \cong \mathrm{Hom}_{k-alg}(gr(A),k),$$ where $gr(A) = k\oplus m/m^2 \oplus m^2/m^3 \oplus...$ is the graded ring of $A$. The last isomorphism holds because under these assumptions $gr(A)$ is the symmetric algebra of $m/m^2$.
In general, is there a (perhaps noncommutative) $k$-algebra $F_n(A)$, depending functorially on $A$, such that $$\mathrm{Hom}_{local-k-alg}(A,k[t]/t^n)\cong\mathrm{Hom}_{k-alg}(F_n(A),k)?$$

Thanks!

John Zimmerman
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xlord
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1 Answers1

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Yes, this is the so called Jet space or Jet scheme and the construction is pretty general, and you can deduce the general case by looking at what happens for $\mathbb{A}_k^n = \operatorname{Spec} k[x_1,\ldots,x_n]$. By definition we want a $k$-algebra $A_n$ such that $\operatorname{Hom}_k(A_m,k) = \operatorname{Hom}_k(k[x_1,\ldots,x_n],k[t]/t^m)$. A morphism in the latter is determined by $x_i \mapsto a_0+a_1t^1+\cdots+a_{n-1}t^{m-1}$. The assignment $x_i \mapsto(a_0,\ldots,a_{m-1})$ is equivalent to the assignment of $m$ coordinates $x_{i1}\mapsto a_0,\ldots, x_{im} \mapsto a_{m-1}$ we see that $A_m$ is represented by $\mathbb{A}_k^{nm}$. If $A=k[x_1,\ldots,x_n]/I$ and $I=(f_1,\ldots,f_t)$, then the $m$-th Jet ring $A_m$ is given by the quotient in $k[x_{11},\ldots,x_{nm}]$ by the functions $f(x_{11},\ldots,x_{1m},\ldots,x_{n1},\ldots,x_{nm})$ as constructed above, e.g. the ideal is generated by in some sense the vanishing of $f$ and along with it's up to $m$-th Taylor expansion.

KReiser
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wskrsk
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