---- > [!definition] Definition. ([[Zariski tangent space]]) > Consider $\frac{k[t]}{\langle t^{2} \rangle}$. We have $\text{Spec } \frac{k[t]}{\langle t^{2} \rangle}=\{ \langle t^{} \rangle \}$.[^1] The [[structure sheaf on a ring spectrum|structure sheaf]] [[structure sheaf on a ring spectrum#^basic-example|amounts to]] $\frac{k[t]}{\langle t^{2} \rangle}$ itself. So we have an [[affine scheme]] $D:= \big( \{ \langle t \rangle \}, \frac{k[t]}{\langle t^{2} \rangle } \big).$ > Let $X$ be any [[scheme]] [[scheme over a field|over]] $k$ (i.e. an object of $\mathsf{Sch} /\text{Spec } k$). Recall that this turns $\mathcal{O}_{X}$ into a [[sheaf]] of $k$-[[algebra|algebras]]. > > > [[base scheme|What is]] $X(D)=\text{Hom}_{\mathsf{Sch} / \text{Spec } k}(D \to k, X \to k)$? Given $f:D \to X$ a [[morphism of locally ringed spaces|morphism of schemes]] over $k$, we get a point $x_{0} \in X$ as the image of $f$ and a $k$-[[algebra]] [[algebra homomorphism|homomorphism]] $f_{x_{0}}^{\sharp}:\mathcal{O}_{X, x} \to \frac{k[t]}{\langle t^{2} \rangle}$ that is [[homomorphism of local rings|local]]: > with notation $\mathfrak{m}_{x_{0}}=\mathfrak{m}_{\mathcal{O}_{X, x_{0}}}$, we have $f_{x_{0}}^{\sharp}(\mathfrak{m}_{x_{0}}) \subset \langle t \rangle$ and thus a well-defined morphism of $k$-[[vector space|vector spaces]] $\mathfrak{m}_{x_{0}} \to \langle t \rangle$. Under this mapping, $\mathfrak{m}_{x_{0}}^{2} \mapsto (0)$, [[first isomorphism theorem for modules|inducing in turn]] a $k$-[[linear map|linear map]] $\frac{\mathfrak{m}}{\mathfrak{m}_{x_{0}}^{2}} \to \langle t \rangle=k $ > where we have used that $\langle t \rangle=k \cdot t \cong k$ as $k$-[[vector space|vector spaces]]. > > This is thus an element of the [[dual vector space|dual]] $k$-[[vector space]] $\left( \frac{\mathfrak{m}_{x_{0}}}{\mathfrak{m}_{x_{0}}^{2}} \right)^{*}$, called the **Zariski tangent space to $X$ at $x_{0}$**. > [^1]: Indeed, $\langle t \rangle$ is a [[prime ideal|prime]] (and [[prime iff maximal for nonzero ideals in PID|hence maximal]]) [[ideal]] of $k[t]$, since $t$ is [[irreducible element of an integral domain|irreducible]] (see [[prime iff maximal iff irreducible for nonzero ideals in a PID|see here, or something]]). Thus $\langle t \rangle / \langle t ^{2}\rangle$ is the unique [[maximal ideal]] of the [[quotient ring|quotient]] (see example in [[local ring]]). Using the usual abuse of notation, we write $\langle t \rangle$ for the image of $\langle t \rangle$ under the quotient map. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```