## 33.45 Embedding dimension

There are several ways to define the embedding dimension, but for closed points on algebraic schemes over algebraically closed fields all definitions are equivalent to the following.

Definition 33.45.1. Let $k$ be an algebraically closed field. Let $X$ be a locally algebraic $k$-scheme and let $x \in X$ be a closed point. The embedding dimension of $X$ at $x$ is $\dim _ k \mathfrak m_ x/\mathfrak m_ x^2$.

Facts about embedding dimension. Let $k, X, x$ be as in Definition 33.45.1.

1. The embedding dimension of $X$ at $x$ is the dimension of the tangent space $T_{X/k, x}$ (Definition 33.16.3) as a $k$-vector space.

2. The embedding dimension of $X$ at $x$ is the smallest integer $d \geq 0$ such that there exists a surjection

$k[[x_1, \ldots , x_ d]] \longrightarrow \mathcal{O}_{X, x}^\wedge$

of $k$-algebras.

3. The embedding dimension of $X$ at $x$ is the smallest integer $d \geq 0$ such that there exists an open neighbourhood $U \subset X$ of $x$ and a closed immersion $U \to Y$ where $Y$ is a smooth variety of dimension $d$ over $k$.

4. The embedding dimension of $X$ at $x$ is the smallest integer $d \geq 0$ such that there exists an open neighbourhood $U \subset X$ of $x$ and an unramified morphism $U \to \mathbf{A}^ d_ k$.

5. If we are given a closed embedding $X \to Y$ with $Y$ smooth over $k$, then the embedding dimension of $X$ at $x$ is the smallest integer $d \geq 0$ such that there exists a closed subscheme $Z \subset Y$ with $X \subset Z$, with $Z \to \mathop{\mathrm{Spec}}(k)$ smooth at $x$, and with $\dim _ x(Z) = d$.

If we ever need these, we will formulate a precise result and provide a proof.

Non-algebraically closed ground fields or non-closed points. Let $k$ be a field and let $X$ be a locally algebraic $k$-scheme. If $x \in X$ is a point, then we have several options for the embedding dimension of $X$ at $x$. Namely, we could use

1. $\dim _{\kappa (x)}(\mathfrak m_ x/\mathfrak m_ x^2)$,

2. $\dim _{\kappa (x)}(T_{X/k, x}) = \dim _{\kappa (x)}(\Omega _{X/k, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x))$ (Lemma 33.16.4),

3. the smallest integer $d \geq 0$ such that there exists an open neighbourhood $U \subset X$ of $x$ and a closed immersion $U \to Y$ where $Y$ is a smooth variety of dimension $d$ over $k$.

In characteristic zero (1) $=$ (2) if $x$ is a closed point; more generally this holds if $\kappa (x)$ is separable algebraic over $k$, see Lemma 33.16.5. It seems that the geometric definition (3) corresponds most closely to the geometric intuition the phrase “embedding dimension” invokes. Since one can show that (3) and (2) define the same number (this follows from Lemma 33.18.5) this is what we will use. In our terminology we will make clear that we are taking the embedding dimension relative to the ground field.

Definition 33.45.2. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme. Let $x \in X$ be a point. The embedding dimension of $X/k$ at $x$ is $\dim _{\kappa (x)}(T_{X/k, x})$.

If $(A, \mathfrak m, \kappa )$ is a Noetherian local ring the embedding dimension of $A$ is sometimes defined as the dimension of $\mathfrak m/\mathfrak m^2$ over $\kappa$. Above we have seen that if $A$ is given as an algebra over a field $k$, it may be preferable to use $\dim _\kappa (\Omega _{A/k} \otimes _ A \kappa )$. Let us call this quantity the embedding dimension of $A/k$. With this terminology in place we have

$\text{embed dim of }X/k\text{ at }x = \text{embed dim of }\mathcal{O}_{X, x}/k = \text{embed dim of }\mathcal{O}_{X, x}^\wedge /k$

if $k, X, x$ are as in Definition 33.45.2.

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