# 4 Smoothness and differentials *

General reference: [ GW1 ] Ch. 6.

## The Zariski tangent space

**(4.1) Definition of the Zariski tangent space**

**Definition 4.1**

*(Zariski) tangent space of \(X\) in \(x\)*.

**Definition 4.2**

*Jacobian matrix*of the polynomials \(f_i\). Here the partial derivatives are to be understood in a formal sense.

**Remark 4.3**

If in the above setting the ideal \({\mathfrak m}\) is finitely generated, then \(\dim _{\kappa (x)} T_xX\) is the minial number of elements needed to generate \({\mathfrak m}\) and in particular is finite.

The tangent space construction if functorial in the following sense: Given a scheme morphism \(f\colon X\to Y\) and \(x\in X\) such that \(\dim _{\kappa (x)}T_xX\) is finite or \([\kappa (x) : \kappa (f(x)) ]\) is finite, then we obtain a map

\[ df_x \colon T_xX \to T_{f(x)} Y \otimes _{\kappa (f(x))} \kappa (x). \]

**Example 4.4**

**Proposition 4.5**

## Smooth morphisms

**(4.2) Definition of smooth morphisms**

**Definition 4.6**

A morphism \(f\colon X\to Y\) of schemes is called *smooth of relative dimension \(d\ge 0\) in \(x\in X\)*, if there exist affine open neighborhoods \(U \subseteq X\) of \(x\) and \(V=\operatorname{Spec}R\subseteq Y\) of \(f(x)\) such that \(f(U) \subseteq V\) and an open immersion \(j \colon U \to \operatorname{Spec}R[T_1, \dots , T_n](f_1, \dots , f_{n-d})\) such that the triangle

is commutative, and that the Jacobian matrix \(J_{f_1, \dots , f_{n-d}}(x)\) has rank \(n-d\).

We say that \(f\colon X\to Y\) is *smooth of relative dimension \(d\)* if \(f\) is smooth of relative dimension \(d\) at every point of \(X\). Instead of *smooth of relative dimension \(0\)*, we also use the term *étale*.

With notation as above, if \(f\) is smooth at \(x\in X\), then \(x\) as an open neighborhood such that \(f\) is smooth at all points of this open neighborhood. Clearly, \(\mathbb {A}^n_S\) and \(\mathbb {P}^n_S\) are smooth of relative dimension \(n\) for every scheme \(S\). (It is harder to give examples of non-smooth schemes directly from the definition; we will come back to this later.)

**(4.3) Dimension of schemes**

Recall from commutative algebra that for a ring \(R\) we define the (Krull) dimension \(\dim R\) of \(R\) as the supremum over all lengths of chains of prime ideals, or equivalently as the dimension of the topological space \(\operatorname{Spec}R\) in the sense of the following definition.

**Definition 4.7**

We will use this notion of dimension for non-affine schemes, as well. Recall the following theorem about the dimension of finitely generated algebras over a field from commutative algebra:

**Theorem 4.8**

By passing to an affine cover, we obtain the following corollary:

**Corollary 4.9**

**(4.4) Existence of smooth points**

Let \(k\) be a field.

**Lemma 4.10**

^{1}] \(k\)-schemes which are locally of finite type over \(k\). Let \(x\in X\), \(y\in Y\), and let \(\varphi \colon {\mathscr O}_{Y,y}\to {\mathscr O}_{X,x}\) be an isomorphism of \(k\)-algebras. Then there exist open neighborhoods \(U\) of \(x\) and \(V\) of \(y\) and an isomorphism \(h\colon U\to V\) of \(k\)-schemes with \(h^\sharp _x = \varphi \).

**Proposition 4.11**

Let \(X\) be an integral \(k\)-scheme of finite type. Assume that \(K(X)\cong k(T_1, \dots , T_d)[\alpha ]\) with \(\alpha \) separable algebraic over \(k(T_1, \dots , T_d)\). (This is always possible if \(k\) is perfect.) (Then \(\dim X=d\) by the above.)

Then there exists a dense open subset \(U\subseteq X\) and a separable irreducible polynomial \(g \in k(T_1,\dots , T_d)[T]\) with coefficients in \(k[T_1,\dots , T_d]\), such that \(U\) is isomorphic to a dense open subset of \(\operatorname{Spec}k[T_1,\dots T_d]/(g)\).

**Theorem 4.12**

**(4.5) Regular rings**

For references to the literature, see [ GW1 ] App. B, in particular B.73, B.74, B.75

**Definition 4.13**

*regular*, if \(\dim A = \dim _\kappa {\mathfrak m}/{\mathfrak m}^2\).

One can show that the inequality \(\dim A \le \dim _\kappa {\mathfrak m}/{\mathfrak m}^2\) always holds. Therefore we can rephrase the definition as saying that \(A\) is regular if \({\mathfrak m}\) has a generating system consisting of \(\dim A\) elements.

**Definition 4.14**

*regular*, if \(A_{\mathfrak m}\) is regular for every maximal ideal \({\mathfrak m}\subset A\).

We quote the following (mostly non-trivial) results about regular rings:

**Theorem 4.15**

Every localization of a regular ring is regular.

If \(A\) is regular, then the polynomial ring \(A[T]\) is regular.

(Theorem of Auslander–Buchsbaum) Every regular local ring is factorial.

Let \(A\) be a regular local ring with maximal ideal \({\mathfrak m}\) and of dimension \(d\), and let \(f_1,\dots , f_r\in {\mathfrak m}\). Then \(A/(f_1, \dots , f_r)\) is regular of dimension \(d-r\) if and only if the images of the \(f_i\) in \({\mathfrak m}/{\mathfrak m}^2\) are linearly independent over \(A/{\mathfrak m}\).

**(4.6) Smoothness and regularity**

Let \(k\) be a field.

**Lemma 4.16**

**Lemma 4.17**

**Theorem 4.18**

The morphism \(X\to \operatorname{Spec}k\) is smooth of relative dimension \(d\) at \(x\).

For all points \(\overline{x}\in X_K\) lying over \(x\), \(X_K\) is smooth over \(K\) of relative dimension \(d\) at \(\overline{x}\).

There exists a point \(\overline{x}\in X_K\) lying over \(x\), such that \(X_K\) is smooth over \(K\) of relative dimension \(d\) at \(\overline{x}\).

For all points \(\overline{x}\in X_K\) lying over \(x\), the local ring \({\mathscr O}_{X_K,\overline{x}}\) is regular of dimension \(d\).

There exists a point \(\overline{x}\in X_K\) lying over \(x\), such that the local ring \({\mathscr O}_{X_K,\overline{x}}\) is regular of dimension \(d\).

**Corollary 4.19**

**Corollary 4.20**

Let \(X = V(g_1, \dots , g_s)\subseteq \mathbb {A}^n_k\) and let \(x\in X\) be a smooth closed point. Let \(d=\dim {\mathscr O}_{X,x}\). Then \(J_{g_1, \dots , g_s}(x)\) has rank \(n-d\). In particular, \(s\ge n-d\).

After renumbering the \(g_i\), if necessary, there exists an open neighborhood \(U\) of \(x\) and an open immersion \(U \subseteq V(g_1, \dots , g_{n-d})\), i.e., locally around \(x\), “\(X\) is cut out in affine space by the expected number of equations”.

**Corollary 4.21**

\(X\) is smooth over \(k\).

\(X\otimes _kL\) is regular for every field extension \(L/k\).

There exists an algebraically closed extension field \(K\) of \(k\) such that \(X\otimes _kK\) is regular.

## The sheaf of differentials

General references: [ M2 ] §25, [ Bo ] Ch. 8, [ H ] II.8.

**(4.7) Modules of differentials**

Let \(A\) be a ring.

**Definition 4.22**

*\(A\)-derivation*from \(B\) to \(M\) is a homomorphism \(D\colon B\to M\) of abelian groups such that

(Leibniz rule) \(D(bb') = bD(b') + b'D(b)\) for all \(b, b'\in B\),

\(d(a) = 0\) for all \(a\in A\).

Assuming property (a), property (b) is equivalent to saying that \(D\) is a homomorphism of \(A\)-modules. We denote the set of \(A\)-derivations \(B\to M\) by \(\operatorname{Der}_A(B, M)\); it is naturally a \(B\)-module.

**Definition 4.23**

Let \(B\) be an \(A\)-algebra. We call a \(B\)-module \(\Omega _{B/A}\) together with an \(A\)-derivation \(d_{B/A}\colon B\to \Omega _{B/A}\) a module of (relative, KÃ¤hler) differentials of \(B\) over \(A\) if it satisfies the following universal property:

For every \(B\)-module \(M\) and every \(A\)-derivation \(D\colon B\to M\), there exists a unique \(B\)-module homomorphism \(\psi \colon \Omega _{B/A}\to M\) such that \(D = \psi \circ d_{B/A}\).

In other words, the map \(\operatorname{Hom}_B(\Omega _{B/A}, M) \to \operatorname{Der}_A(B, M)\), \(\psi \mapsto \psi \circ d_{B/A}\) is a bijection.

**Lemma 4.24**

Let \(I\) be a set, \(B= A[T_i, i\in I]\) the polynomial ring. Then \(\Omega _{B/A} := B^{(I)}\) with \(d_{B/A}(T_i) = e_i\), the “\(i\)-th standard basis vector” is a module of differentials of \(B/A\).

So we can write \(\Omega _{B/A} = \bigoplus _{i\in I} Bd_{B/A}(T_i)\).

**Lemma 4.25**

**Corollary 4.26**

We will see later that for a scheme morphism \(X\to Y\), one can construct an \({\mathscr O}_X\)-module \(\Omega _{X/Y}\) together with a “derivation” \({\mathscr O}_X\to \Omega _{X/Y}\) by gluing sheaves associated to modules of differentials attached to the coordinate rings of suitable affine open subschemes of \(X\) and \(Y\).

Let \(\varphi \colon A\to B\) be a ring homomorphism. For the next definition, we will consider the following situation: Let \(C\) be a ring, \(I\subseteq C\) an ideal with \(I^2 = 0\), and let

be a commutative diagram (where the right vertical arrow is the canonical projection). We will consider the question whether for these data, there exists a homomorphism \(B\to C\) (dashed in the following diagram) making the whole diagram commutative:

**Definition 4.27**

We say that \(\varphi \) is

*formally unramified*, if in every situation as above, there exists at most one homomorphism \(B\to C\) making the diagram commutative.We say that \(\varphi \) is

*formally smooth*, if in every situation as above, there exists at least one homomorphism \(B\to C\) making the diagram commutative.We say that \(\varphi \) is

*formally étale*, if in every situation as above, there exists a unique homomorphism \(B\to C\) making the diagram commutative.

Passing to the spectra of these rings, we can interpret the situation in geometric terms: \(\operatorname{Spec}C/I\) is a closed subscheme of \(\operatorname{Spec}C\) with the same topological space, so we can view the latter as an “infinitesimal thickening” of the former. The question becomes the question whether we can extend the morphism from \(\operatorname{Spec}C/I\) to \(\operatorname{Spec}B\) to a morphism from this thickening.

**Proposition 4.28**

For an algebraic field extension \(L/K\) one can show that \(K\to L\) is formally unramified if and only if it is formally smooth if and only if \(L/K\) is separable. Cf. Problem 27 and [ M2 ] §25, §26 (where the discussion is extended to the general, not necessarily algebraic, case).

**Theorem 4.29**

Let \(f\colon A\to B\), \(g\colon B\to C\) be ring homomorphisms. Then we obtain a natural sequence of \(C\)-modules

which is exact.

If moreover \(g\) is formally smooth, then the sequence

is a split short exact sequence.

**Theorem 4.30**

Let \(f\colon A\to B\), \(g\colon B\to C\) be ring homomorphisms. Assume that \(g\) is surjective with kernel \({\mathfrak b}\). Then we obtain a natural sequence of \(C\)-modules

where the homomorphism \({\mathfrak b}/{\mathfrak b}^2 \to \Omega _{B/A}\otimes _BC\) is given by \(x\mapsto d_{B/A}(x)\otimes 1\).

If moreover \(g\circ f\) is formally smooth, then the sequence

is a split short exact sequence.

**(4.8) The sheaf of differentials of a scheme morphism**

**Remark 4.31**

We can use a similar definition as we used for ring homomorphisms above to define the notions of formally unramified, formally smooth and formally étale morphisms of schemes.

**Definition 4.32**

We say that \(f\) is

*formally unramified*, if for every ring \(C\), every ideal \(I\) with \(I^2=0\), and every morphism \(\operatorname{Spec}C\to Y\) (which we use to view \(\operatorname{Spec}C\) and \(\operatorname{Spec}C/I\) as \(Y\)-schemes), the composition with the natural closed embedding \(\operatorname{Spec}C/I\to \operatorname{Spec}C\) yields an injective map \(\operatorname{Hom}_Y(\operatorname{Spec}C, X) \to \operatorname{Hom}_Y(\operatorname{Spec}C/I, X)\).We say that \(f\) is

*formally smooth*, if for every ring \(C\), every ideal \(I\) with \(I^2=0\), and every morphism \(\operatorname{Spec}C\to Y\), the composition with the natural closed embedding \(\operatorname{Spec}C/I\to \operatorname{Spec}C\) yields a surjective map \(\operatorname{Hom}_Y(\operatorname{Spec}C, X) \to \operatorname{Hom}_Y(\operatorname{Spec}C/I, X)\).We say that \(f\) is

*formally étale*, if \(f\) is formally unramified and formally smooth.

If \(f\) is a morphism of affine schemes, then \(f\) has one of the properties of this definition if and only if the corresponding ring homomorphism has the same property in the sense of our previous definition.

**Lemma 4.33**

Every monomorphism of schemes (in particular: every immersion) is formally unramified.

Let \(A\to B\to C\) be ring homomorphisms such that \(A\to B\) is formally unramified. Then we can naturally identify \(\Omega _{C/A} = \Omega _{C/B}\).

**Definition 4.34**

Let \(X\to Y\) be a morphism of schemes, and let \({\mathscr M}\) be an \({\mathscr O}_X\)-module. A derivation \(D\colon {\mathscr O}_X\to {\mathscr M}\) is a homomorphism of abelian sheaves such that for all open subsets \(U\subseteq X\), \(V\subseteq Y\) with \(f(U)\subseteq V\), the map \({\mathscr O}(U)\to {\mathscr M}(U)\) is an \({\mathscr O}_Y(V)\)-derivation.

Equivalently, \(D\colon {\mathscr O}_X\to {\mathscr M}\) is a homomorphism of \(f^{-1}({\mathscr O}_Y)\)-modules such that for every open \(U\subseteq X\), the Leibniz rule

holds.

We denote the set of all these derivations by \(\operatorname{Der}_Y({\mathscr O}_X, {\mathscr M})\); it is a \(\Gamma (X, {\mathscr O}_X)\)-module.

**Definition/Proposition 4.35**

There exists a unique \({\mathscr O}_X\)-module \(\Omega _{X/Y}\) together with a derivation \(d_{X/Y}\colon {\mathscr O}_X\to \Omega _{X/Y}\) such that for all affine open subsets \(\operatorname{Spec}B = U\subseteq X\), \(\operatorname{Spec}A = V\subseteq Y\) with \(f(U)\subseteq V\), \(\Omega _{X/Y} = \widetilde{\Omega _{B/A}}\) and \(d_{X/Y|U}\) is induced by \(d_{B/A}\).

\(\Omega _{X/Y} = \Delta ^*({\mathscr J}/{\mathscr J}^2)\), where \(\Delta \colon X\to X\times _YX\) is the diagonal morphism, \(W\subseteq X\times _YX\) is open such that \(\mathop{\rm im}(\Delta )\subseteq W\) is closed (if \(f\) is separated we can take \(W=X\times _YX\)), and \({\mathscr J}\) is the quasi-coherent ideal defining the closed subscheme \(\Delta (X) \subseteq W\). The derivation \(d_{X/Y}\) is induced, on affine opens, by the map \(b\mapsto 1\otimes b-b\otimes 1\).

The quasi-coherent \({\mathscr O}_X\)-module \(\Omega _{X/Y}\) together with \(d_{X/Y}\) is characterized by the universal property that composition with \(d_{X/Y}\) induces bijections

\[ \operatorname{Hom}_{{\mathscr O}_X}(\Omega _{X/Y}, {\mathscr M}) \overset {\sim }{\to }\operatorname{Der}_{Y}({\mathscr O}_X, {\mathscr M}) \]for every quasi-coherent \({\mathscr O}_X\)-module \({\mathscr M}\), functorially in \({\mathscr M}\).

The properties we proved for modules of differentials can be translated into statements for sheaves of differentials:

**Proposition 4.36**

**Proposition 4.37**

**Proposition 4.38**

**Proposition 4.39**

**Proposition 4.40**

**(4.9) Sheaves of differentials and smoothness**

We start by slightly rephrasing the definition of a smooth morphism.

**Definition 4.41**

A morphism \(f\colon X\to Y\) of schemes is called *smooth of relative dimension \(d\ge 0\) in \(x\in X\)*, if there exist affine open neighborhoods \(U \subseteq X\) of \(x\) and \(V=\operatorname{Spec}R\subseteq Y\) of \(f(x)\) such that \(f(U) \subseteq V\) and an open immersion \(j \colon U \to \operatorname{Spec}R[T_1, \dots , T_n](f_1, \dots , f_{n-d})\) such that the triangle

is commutative, and that the images of \(df_1\), …, \(df_{n-d}\) in the fiber \(\Omega _{\mathbb {A}^n_R/R}^1\otimes \kappa (x)\) are linearly independent over \(\kappa (x)\). (We view \(x\) as a point of \(\mathbb {A}^n_R\) via the embedding \(U \to \operatorname{Spec}R[T_1, \dots , T_n](f_1, \dots , f_{n-d}) \to \operatorname{Spec}R[T_1, \dots , T_n] = \mathbb {A}^n_R\).)

**Proposition 4.42**

**Theorem 4.43**

**Proposition 4.44**

**Theorem 4.45**

We skip the proof that smoothness implies formal smoothness, see for instance [ Bo ] Ch. 8.5. (But cf. the previous proposition which shows that a smooth morphism is at least “locally formally smooth”.)