Zariski tangent space

See dual numbers, closed point

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Equivalent definitions:

For manifolds:
- \({\mathbf{T}}_p M = \left\{{ \gamma\in C^\infty(I, M) {~\mathrel{\Big\vert}~}\gamma(0) = p}\right\}/\sim\) where \(\gamma \sim \eta \iff \gamma'(0) = \eta'(0)\).
- \({\mathbf{T}}_p M = ({\mathfrak{m}}_p/{\mathfrak{m}}_p^2) {}^{ \vee }\coloneqq {}_{k}{\mathsf{Mod}} ({\mathfrak{m}}_p/{\mathfrak{m}}_p^2, k)\) where \({\mathfrak{m}}_p = \left\{{f\in C^\infty(M; k) {~\mathrel{\Big\vert}~}f(p) = 0}\right\} \in \operatorname{mSpec}C^\infty(M; k)\)
- \({\mathbf{T}}_p M = \mathop{\mathrm{Der}}_{ {}_{k}{\mathsf{Mod}} }(C^\infty(M; k); k)\)
For affine schemes \(M = \operatorname{Spec}R\) and \({\mathfrak{m}}\in \operatorname{mSpec}R\) a closed point:
- \({\mathbf{T}}_{\mathfrak{m}}M = {}_{k} \mathsf{Alg} (R, k[{\varepsilon}])\)
- \({\mathbf{T}}_{\mathfrak{m}}M = \mathop{\mathrm{Der}}_{ {}_{k}{\mathsf{Mod}} }(R; k)\) – letting \(\mathop{\mathrm{proj}}_{\mathfrak{m}}: R\to R/{\mathfrak{m}}\), these must satisfy \(D(ab) = \mathop{\mathrm{proj}}_{\mathfrak{m}}(a) D(b) + \mathop{\mathrm{proj}}_{\mathfrak{m}}(b) D(a)\).
For general schemes: todo. Use the stalk \(R_{\mathfrak{m}}\) and embed \({\mathfrak{m}}\hookrightarrow R_{\mathfrak{m}}\).
Can be defined by taking defining equations \(\left\{{f_i}\right\}_{i=1}^r \subseteq k[x_1, \cdots, x_{n}]\), taking the matrix of partials \({\left[ {{\frac{\partial f_i}{\partial x_j}\,}} \right]}\), which evaluating at a point \(p\) yields a map \(Df: { \mathbf{F} }^n \to { \mathbf{F} }^r\) where \({ \mathbf{F} }\) is the residue field at \(p\).
- Then define the Zariski tangent space as \(\ker D_f\).