# 2 Smoothness and differentials

Oct. 16,

2023

General reference: [ GW1 ] Ch. 6.

## The Zariski tangent space

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

**Definition 2.1**

*(Zariski) tangent space of \(X\) in \(x\)*and is denoted by \(T_xX\).

**Definition 2.2**

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

**Remark 2.3**

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

The tangent space construction is functorial in the following sense: Given a scheme morphism \(f\colon X\to Y\) and \(x\in X\) such that \(\dim _{\kappa (f(x))}T_{f(x)}Y\) 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 2.4**

**Proposition 2.5**

**Remark 2.6**

It is possible to define the \(k\)-vector space structure on \(X(k[\varepsilon ]/(\varepsilon ^2))_x\) “directly”.

Similarly, one can define the relative tangent space of an \(S\)-scheme \(X\) in a \(K\)-valued point \(\xi \) for any field \(K\) and without restrictions on the residue class field of the image point of \(\xi \), as the set of \(S\)-morphisms \(f\colon \operatorname{Spec}K[\varepsilon ]/(\varepsilon ^2)\to X\) with \(\mathop{\rm Im}(f) = \mathop{\rm Im}(\xi )\) (and again, this set can be made into a \(K\)-vector space). This concept is sometimes useful, but the result is in general different from the Zariski tangent space.

## Smooth morphisms

**(2.2) Dimension of schemes**

Reference: [ GW1 ] Sections (5.3) ff.

Oct. 18,

2023

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 2.7**

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 2.8**

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

**Corollary 2.9**

**(2.3) Definition of smooth morphisms**

Reference: [ GW1 ] Section (6.8).

**Definition 2.10**

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 \(x\) has 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.)

**Remark 2.11**

(The Jacobian Conjecture) Let \(k\) be a field, \(n\ge 1\), and let \(f_1,\dots , f_n\in k[X_1,\dots , X_n]\). The \(f_i\) define a \(k\)-scheme morphism \(\mathbb {A}^n_k\to \mathbb {A}^n_k\), given on \(R\)-valued points by \((x_1,\dots , x_n)\mapsto (f_1(x_1,\dots , x_n),\dots , f_n(x_1, \dots x_n))\).

Assume that \(f\) is an isomorphism of \(k\)-schemes. It then follows easily, by similar computations as above (or expressed differently by the “multi-variable chain rule”), that the Jacobian matrix of the \(f_i\) is invertible in \({\rm Mat}_{n\times n}(k[X_\bullet ])\). Equivalently, the determinant of the Jacobian matrix lies in \(k^\times \).

*Jacobian conjecture* (O. Keller, 1939) Let \(k\) be a field of characteristic \(0\), \(n \ge 1\), and let \(f_1,\dots , f_n\in k[X_1,\dots , X_n]\). The morphism \(\mathbb {A}^n_k\to \mathbb {A}^n_k\) induced by the \(f_i\) is an isomorphism if and only if the Jacobian matrix \(\left(\frac{\partial f_i}{\partial X_j}\right)_{i,j}\in {\rm Mat}_{n\times n}(k[X_\bullet ])\) is invertible.

For \(n=1\) the statement is easy to prove, but the conjecture is open even for \(n = 2\) and is particularly well-known for the number of incorrect attempts of proving it.

It is not very hard to see that the condition that \(k\) has characteristic \(0\) cannot be dropped. Can you find an example for this?

With a bit of effort, one can show that equivalently, one can formulate the conjecture as follows: Let \(k\) be a field of characteristic \(0\), and let \(f\colon \mathbb {A}^n_k\to \mathbb {A}^n_k\) be an étale morphism. Then \(f\) is an isomorphism.

**(2.4) Existence of smooth points**

Reference: [ GW1 ] Section (6.9).

Let \(k\) be a field.

**Lemma 2.12**

^{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 2.13**

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 the case, 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 \(\operatorname{Spec}k[T_1,\dots T_d, T]/(g)\).

Oct. 23,

2023

**Theorem 2.14**

**(2.5) Regular rings**

**Definition 2.15**

*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 2.16**

*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. A key input for Part (4) is a version of Krull’s Principal Ideal Theorem.

**Theorem 2.17**

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 a unique factorization domain.

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 \(A/{\mathfrak m}\).

Note that Part (3) implies in particular that every regular local ring is a domain. The UFD property also implies that this domain is integrally closed in its field of fractions.

**(2.6) Smoothness and regularity**

Reference: [ GW1 ] Section (6.12).

Let \(k\) be a field.

**Lemma 2.18**

First, reduce to the case that (1) \(x\) is a closed point in \(X\). By the definition of smooth morphisms, it is then enough to consider the case of a closed point \(x\in \operatorname{Spec}k[X_\bullet ](f_\bullet )\) where the Jacobian matrix has full rank. By Theorem 2.17 (2) and (4) it is enough to show that the images of the \(f_i\) in \(\mathfrak m_x/\mathfrak m_x^2\) are linearly independent. This is clear (cf. Example 2.4) if \(\kappa (x) = k\), and the general case can be reduced to this one, using that in the base change \(X\otimes _k\kappa (x)\) there exists a point \(\overline{x}\) with residue class field \(\kappa (x)\) projecting to \(x\in X\) and that we have an inclusion \(\mathfrak m_x/\mathfrak m_x^2 \to \mathfrak m_{\overline{x}}/\mathfrak m_{\overline{x}}^2\) of \(\kappa (x)\)-vector spaces.

Oct. 25,

2023

**Lemma 2.19**

Write \(d = \dim {\mathscr O}_{X, x}\). After renumbering the \(g_i\), if necessary, we may assume that the first \(n-d\) columns of \(J_{g_\bullet }(x)\) are linearly independent. We then have

and \(x\) is smooth over \(k\) as a point of \(Y\). By the previous lemma, \(\dim {\mathscr O}_{Y, x} = d\). It follows that \({\mathscr O}_{X,x} = {\mathscr O}_{Y,x}\), and together with Lemma 2.12 we obtain the claim.

**Lemma 2.20**

For \(\ge \) choose some affine open neighborhood of \(x\), apply Noether normalization, and use that the properties *finite* and *injective* of a ring homomorphism are preserved under the base change \(-\otimes _kK\).

For \(\le \), use that the morphism \(X_K\to X\) (being obtained by base change from \(\operatorname{Spec}(K)\to \operatorname{Spec}(k)\)) is flat, and that flat ring homomorphisms satisfy a *going down* theorem. The key fact for the going down property is that for every flat local ring homomorphism \(A\to B\) between local rings, the map \(\operatorname{Spec}(B)\to \operatorname{Spec}(A)\) is surjective. Cf.
[
GW1
]
Lemma 14.9 or
[
M2
]
Theorem 7.3, Theorem 9.5. (In
[
GW1
]
, the proof of \(\le \) is given using the more difficult Proposition 5.44/Theorem 14.38 there, which is required in the book anyway; but at this point the above, related but simpler method works.)

**Theorem 2.21**

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 \(d\).

The implications (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii) and (iv) \(\Rightarrow \) (v) are easy.

Furthermore (iii) \(\Rightarrow \) (iv) and the regularity of \({\mathscr O}_{X,x}\) for a smooth point \(x\) follow from Lemma 2.18.

Next we show that the regularity of \({\mathscr O}_{X,x}\) implies that \(x\) is a smooth point if \(\kappa (x) = k\). Write \(d = \dim {\mathscr O}_{X, x} = \dim _k T_xX\). We embed an affine open neighborhood \(U\) into affine space, say as an open subscheme of \(V(g_1,\dots , g_s)\subseteq \mathbb {A}^n_k\). We are then in the situation of Lemma 2.19, and the lemma shows that \(x\) is a smooth point. This also shows (v) \(\Rightarrow \) (iii).

It remains to prove that (iii) \(\Rightarrow \) (i). It is enough to consider the case where \(x\) is a closed point of \(V(g_1,\dots , g_s)\subset \mathbb {A}^n_k\) for some polynomials \(g_i\). By Lemma 2.19, it is enough to show that \(\operatorname{rk}J_{g_\bullet }(x) = n-\dim {\mathscr O}_{X,x}\). But the rank of the Jacobian matrix does not change when we replace \(x\) by \(\overline{x}\) (and consider the polynomials \(g_i\) in \(K[X_\bullet ]\)), and \(\dim {\mathscr O}_{X,x} = \dim {\mathscr O}_{X_K, \overline{x}}\). Since \(\overline{x}\) is a regular point of \(X_K\) by (iii), which we now assume to hold, we are done.

Oct. 30,

2023

**Corollary 2.22**

**Corollary 2.23**

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 2.24**

\(X\) 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: [ GW2 ] Ch. 17, [ M2 ] §25, [ Bo ] Ch. 8, [ H ] II.8.

We now introduce the “module of differentials” of a ring homomorphism (and its sheaf version \(\Omega _{X/S}\) for a scheme morphism \(X\to S\)). This allows us to study how the (co-)tangent space varies in a family; as we will see, under suitable assumptions the fiber \(\Omega _{X/S}(x)\) at \(x\in X\) can be identified with the dual \(\mathfrak m_x/\mathfrak m_x^2\) of \(T_xX\), see Proposition 2.44. The theory we will set up is also closely related to the so-called infinitesimal lifting criterion for smooth morphisms, see Theorem 2.54.

**(2.7) Modules of differentials**

Let \(A\) be a ring.

**Definition 2.25**

*\(A\)-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 2.26**

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, Kähler) 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 2.27**

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 2.28**

**Corollary 2.29**

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 left 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 2.30**

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.

**Construction 2.31**

Let \(B\) be a ring and let \(M\) be a \(B\)-module. We construct a \(B\)-algebra \(D_B(M)\) as follows. As additive groups, we set \(D_B(M) = B\times M\). The multiplication is defined by

Then \(M = \{ 0\} \times M \subseteq D_B(M)\) is an ideal with \(M^2 = 0\).

For example, taking \(M=B\), we have \(D_B(B) \cong B[\varepsilon ](\varepsilon ^2)\), the ring of dual numbers over \(B\).

The projection \(\pi \colon D_B(M) \to B\) is a \(B\)-algebra homomorphism, i.e., the composition \(B\to D_B(M)\to B\) is the identity.

Now suppose that \(B\) is an \(A\)-algebra. One then checks that the map

is a \(B\)-module isomorphism.

**Proposition 2.32**

Assume that \(\Omega _{B/A} = 0\), and consider \(I\subset C\) and a commutative diagram as above. We need to show that there is at most one ring homomorphism \(B\to C\) making the diagram commutative. Assume that \(\varphi _1, \varphi _2\colon B\to C\) have this property. The \(C\)-module structure on \(I\) factors through a \(C/I\)-module structure since \(I^2=0\), so that we can view \(I\) as a \(B\)-module via the map \(B\to C/I\). Then the difference \(\varphi _1 - \varphi _2\) is an \(A\)-derivation \(B\to I\), and is hence zero by our assumption.

For the converse it is enough that every \(A\)-derivation \(B\to M\) vanishes. Let \(C = D_B(M)\) and \(I = M\). Then \(I^2 = 0\), and the assumption that \(B\) is formally unramified over \(A\) implies \(\operatorname{Der}_A(B, M)=0\).

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).

Nov. 6,

2023

**Theorem 2.33**

Let \(f\colon A\to B\), \(g\colon B\to C\) be ring homomorphisms.

Then we obtain a natural sequence of \(C\)-modules

\[ \Omega _{B/A}\otimes _BC \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]which is exact.

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

\[ 0\to \Omega _{B/A}\otimes _BC \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]is a split short exact sequence.

Conversely, assume that \(g\circ f\) is formally smooth and that the sequence

\[ 0\to \Omega _{B/A}\otimes _BC \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]is a split short exact sequence. Then \(g\) is formally smooth.

To check the exactness in Part (1), it is enough to check that the sequence gives rise to an exact sequence whenever we apply the functor \(\operatorname{Hom}_C(-, M)\) for \(M\) a \(C\)-module. Note that \(\operatorname{Hom}_C(\Omega _{B/A}\otimes _BC, M) = \operatorname{Hom}_B(\Omega _{B/A}, M)\) (where on the right we view \(M\) as a \(B\)-module via \(g\)). See [ ALG2 ] Satz 3.14.

Thus the first part follows, once we check that

is exact (as a sequence of \(A\)-modules or just abelian groups) for any \(C\)-module \(M\). But this is obvious.

Part (2). Now assume that \(g\) is formally smooth. Let us construct a \(C\)-module homomorphism \(\Omega _{C/A}\to \Omega _{B/A}\otimes _BC\) as follows. Constructing a homomorphism like this amounts to constructing an \(A\)-derivation \(C\to \Omega _{B/A}\otimes _BC=:M\). Similarly as above, we consider \(C\times M\) as a ring (with \(M^2=0\)). Let \(B\to C\times M\) be given by \(b\mapsto (g(b), db\otimes 1)\). One checks that this is a ring homomorphism. Since \(g\) is formally smooth, for this \(B\)-algebra structure we find a homomorphism \(C\to C\times M\) of \(B\)-algebras. Composing it with the projection to \(M\) we obtain an \(A\)-derivation \(C\to M\). One checks that the composition \(\Omega _{B/A}\otimes _BC\to \Omega _{C/A}\to \Omega _{B/A}\otimes _BC\) is the identity, and this finishes the proof.

See also [ GW2 ] Proposition 18.18 (1) for a slightly different proof of the second part (which is more along the lines of our proof of the first part).

Part (3). This can be proved by similar arguments as for Parts (1) and (2). We omit the proof for the time being (see [ GW2 ] Proposition 18.18 (2)).

**Theorem 2.34**

There is a natural sequence of \(C\)-modules

\[ {\mathfrak b}/{\mathfrak b}^2 \to \Omega _{B/A}\otimes _BC \to \Omega _{C/A} \to 0, \]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

\[ 0\to {\mathfrak b}/{\mathfrak b}^2 \to \Omega _{B/A}\otimes _BC \to \Omega _{C/A} \to 0 \]is a split short exact sequence.

All assertions in Part (1) follow from Theorem 2.33 and Lemma 2.28.

To prove Part (2), consider the short exact sequence

The assumption that \(g\circ f\) is formally smooth implies that \(p\) admits a section \(s\). Then \(s\circ p_{|{\mathfrak b}/{\mathfrak b}^2} = 0\), and \(p\circ (\operatorname{id}- s\circ p) = 0\). We obtain \(D := \operatorname{id}- s\circ p\colon B/{\mathfrak b}^2\to {\mathfrak b}/{\mathfrak b}^2\). This is an element of \(\operatorname{Der}_A(B/{\mathfrak b}^2, {\mathfrak b}/{\mathfrak b}^2) = \operatorname{Hom}_B(\Omega _{B/A}, {\mathfrak b}/{\mathfrak b}^2)\) and one checks that it gives rise to a retraction of the map \({\mathfrak b}/{\mathfrak b}^2 \to \Omega _{B/A}\otimes _BC\) in the sequence in Part (2).

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

**Remark 2.35**

Let again \(B\) an \(A\)-algebra. There is the following alternative construction of \(\Omega _{B/A}\): Let \(m\colon B\otimes _AB\to B\) be the multiplication map, and let \(I=\operatorname{Ker}(m)\). Then \(I/I^2\) is a \(B\)-module, and \(d\colon B\to I/I^2\), \(b\mapsto 1\otimes b - b\otimes 1\), is an \(A\)-derivation.

Let us show that \((I/I^2, d)\) satisfies the universal property defining \((\Omega _{B/A}, d_{B/A})\). Let \(M\) be a \(B\)-module. Composition with \(d\) gives a map \(\operatorname{Hom}_A(I/I^2, M)\to \operatorname{Der}_A(B, M)\). To show that it is injective, it is enough to show that \(I/I^2\) is generated by the image of \(d\) as a \(B\)-module. This follows from the following two computations (for \(b, b', b_i, b_i'\in B\)):

\(b\otimes b' = bb'\otimes 1 + (b\otimes 1)(1\otimes b' - b'\otimes 1)\),

if \(\sum b_ib_i' = 0\), then \(\sum b_i\otimes b_i' = \sum (b_i\otimes 1)(1\otimes b_i'-b_i'\otimes 1)\) by (1).

For the surjectivity, let \(D\in \operatorname{Der}_A(B, M)\) and let \(\psi \colon B\to D_B(M)\), \(b\mapsto (b, D(b))\), the corresponding map, cf. Construction 2.31. The diagram

(with exact rows) induces a map \(\delta \colon I/I^2\to M\) which makes the whole diagram commute, and \(\delta \circ d = D\).

Nov. 8,

2023

There are several ways of defining an \({\mathscr O}_X\)-module of differentials for a morphism \(f\colon X\to Y\) of schemes. One way is to proceed by gluing, using the following remark.

**Remark 2.36**

To pin down the sheaf of differentials we first define the notion of derivation in this context.

**Definition 2.37**

Let \(f\colon X\to Y\) 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 2.38**

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 \(f(U)\subseteq V\), \(\Omega _{X/Y|U} = \widetilde{\Omega _{B/A}}\) and \(d_{X/Y|U}\) is induced by \(d_{B/A}\).

Define \(\Omega _{X/Y} = \Delta ^*({\mathscr J}/{\mathscr J}^2)\), 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 \(\Delta (X) \subseteq W\). Define the derivation \(d_{X/Y}\) as the one 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. In all statements here, equality means that there is a unique isomorphism that is compatible with the universal derivations.

**Proposition 2.39**

Let \(g\colon Y'\to Y\) be a morphism of schemes, and let \(X'=X\times _Y Y'\). Denote by \(g'\colon X'\to X\) the base change of \(g\). There is a natural isomorphism \(\Omega _{X'/Y'} = (g')^*\Omega _{X/Y}\).

Let \(U\subseteq X\) and \(V\subseteq V\) be open subsets with \(f(U) \subseteq V\). There is a natural identification \(\Omega _{X/Y|U} = \Omega _{U/V}\).

Let \(x\in X\). Then \(\Omega _{X/Y,x} = \Omega _{{\mathscr O}_{x,x}/{\mathscr O}_{Y,y}}\).

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 2.40**

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.

**Proposition 2.41**

**Proposition 2.42**

**Remark 2.43**

*splits locally on a scheme \(X\)*, this means that there exists an open cover \(X = \bigcup _i U_i\) such that for each \(i\) the sequence splits after restricting it to \(U_i\). (It follows from the short exact sequence attached to the local-to-global spectral sequence for Ext sheaves and the vanishing of higher cohomology of quasi-coherent sheaves on affines that a short exact sequence of quasi-coherent \({\mathscr O}_X\)-modules that splits locally on \(X\) and where the term on the right hand side is of finite presentation, splits on every affine open of \(X\).)

Applying Proposition 2.42 to \(X\) a scheme of finite type over \(Y=\operatorname{Spec}(k)\), \(k\) a field, and \(Z=\operatorname{Spec}(k)\) so that \(i\) is a \(k\)-valued point, we obtain the following description of the fiber of the sheaf of differentials at \(x\).

**Proposition 2.44**

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and call this the *tangent sheaf* of \(X\) over \(S\).

**Proposition 2.45**

First note that \(\operatorname{Hom}^g(Y[\varepsilon ], X)\) can be identified with \(\operatorname{Der}_S({\mathscr O}_X, g_*{\mathscr O}_Y)\). In fact, this can be checked on an affine open cover, and in the affine case we have seen this in Construction 2.31. Now we conclude by the following chain of isomorphisms:

Using this description, we “compute” the sheaf of differentials of projective space.

**Proposition 2.46**

*Euler sequence*.

Write \(X = \mathbb {P}^n_R\). We have the “universal” surjection \({\mathscr O}_X^{n+1} \to {\mathscr O}_X(1)\) and denote by \({\mathscr K}\) its kernel. We want to show that \({\mathscr K}(-1):={\mathscr K}\otimes _{{\mathscr O}_X}{\mathscr O}_X(-1) \cong \Omega _{X/R}\). All the \({\mathscr O}_X\)-modules involved here are locally free of finite rank, so it is enough to prove that \({\mathscr T}_{X/R}\cong \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr K}, {\mathscr O}(1)) = \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr K}(-1), {\mathscr O}) = {\mathscr K}(-1)^\vee \).

Let \(U=\operatorname{Spec}(A)\subseteq X\) be open affine and denote by \(g\colon U\to X\) the inclusion. The morphism \(g\) corresponds to a surjection \(A^{n+1}\to L\) onto a locally free \(A\)-module \(L\) of rank \(1\) whose kernel we denote by \(K\). Note that \(K = {\mathscr K}_{|U}\). We now use the notation of Proposition 2.45.

*Claim.* There is a natural identification \(\operatorname{Hom}_A(K, A^{n+1}/K)\xrightarrow {\cong }\operatorname{Hom}^g(U[\varepsilon ], X)\).

*Proof of claim.* An element of \(\operatorname{Hom}^g(U[\varepsilon ], X) \subseteq X(U(\varepsilon ))\) is given by a surjection \((A[\varepsilon ]/(\varepsilon ^2))^{n+1} \to L'\), where \(L'\) is locally free over \(A[\varepsilon ]/(\varepsilon ^2)\) of rank \(1\), or equivalently its kernel \(K'\subset (A[\varepsilon ]/(\varepsilon ^2))^{n+1}\), such that \(K'\otimes _{A[\varepsilon ]/(\varepsilon ^2)} A = K\).

Now take an \(A\)-module homomorphism \(\alpha \colon K\to A^{n+1}/K\). We define \(K'\) as the \(A[\varepsilon ]/(\varepsilon ^2)\)-module generated by the image of the map \(K\to (A[\varepsilon ]/(\varepsilon ^2))^{n+1}\), \(x\to x+\varepsilon \alpha (x)\). (We define \(\varepsilon \alpha (x)\) by choosing a lift of \(\alpha (x)\in A^{n+1}/K\) in \(A^{n+1}\). The resulting \(K'\) is independent of the choice of lift.)

Note that \(A[\varepsilon ]^{n+1}/K'\) is locally free over \(A[\varepsilon ]\). To check this, we may localize and thus assume that \(K\) and \(A^{n+1}/K\) are free \(A\)-modules. Now choosing lifts of bases of \(K\) and \(A^{n+1}/K\) to \(A[\varepsilon ]^{n+1}\) gives us a family of \(n+1\) vectors. Write them as the columns of a matrix \(M\) over \(A[\varepsilon ]\). By construction, \(\det (M)\) maps to a unit in \(A\) and hence is a unit in \(A[\varepsilon ]\). Thus the lifts form a basis of \(A[\varepsilon ]^{n+1}\) and in particular \(A[\varepsilon ]^{n+1}/K'\) (and \(K'\)) are free.

This defines the desired bijection.

With the claim and Proposition 2.45 we can identify \(\Gamma (U, {\mathscr T}_{X/R})\) with

This identification is compatible with restrictions to smaller subsets and therefore defines the isomorphism of \({\mathscr O}_X\)-modules we are looking for.

**Remark 2.47**

In the course of the proof we have established a canonical identification of the tangent space \(T_x\mathbb {P}^n_k\) of projective space over a field \(k\) in a \(k\)-valued point \(x\) with the vector space \(\operatorname{Hom}_k(K, k^{n+1}/K)\), where \(K = \operatorname{Ker}(k^{n+1}\to L)\) is the kernel of the quotient of \(k^{n+1}\) corresponding to \(x\) via the functorial description of \(\mathbb {P}^n_k\). At this point we use the point of view that \(\mathbb {P}^n_k(k)\) is the set of all \(1\)-dimensional quotients of \(k^{n+1}\), or equivalently – passing to the kernel of the projection – of all hyperplanes in \(k^{n+1}\).

Passing to the dual (and classical) point of view, \(K\) gives us a line \(K^\perp = (k^{n+1}/K)^\vee \) in the dual vector space \((k^{n+1})^\vee \) (which we could identify with \(k^{n+1}\) via the standard basis). Then the tangent space is identified with \(\operatorname{Hom}_k(K^\perp , k^{n+1,\vee }/K^\perp )\), which is isomorphic to \(k^{n+1, \vee }/K^\perp \) since \(K^\perp \cong k\). This is “the same” description as using the natural surjection \(\mathbb {A}^{n+1}_k\setminus \{ 0\} \to \mathbb {P}^n_k\) which induces surjections on tangent spaces, cf. [ GW1 ] Prop. 6.10.

As for every short exact sequence of locally free modules of finite rank, we obtain an identification for the top exterior powers,

\[ \bigwedge \nolimits ^n\Omega _{\mathbb {P}^n_R/R} \cong \bigwedge \nolimits ^n\Omega _{\mathbb {P}^n_R/R} \otimes \bigwedge \nolimits ^1{\mathscr O}_{\mathbb {P}^n_R/R} \cong \bigwedge \nolimits ^{n+1} {\mathscr O}(-1)^{n+1} \cong {\mathscr O}(-n-1). \]

**Example 2.48**

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

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

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**Definition 2.49**

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 \(U \to \operatorname{Spec}R[T_1, \dots , T_n](f_1, \dots , f_{n-d}) \to \operatorname{Spec}R[T_1, \dots , T_n] = \mathbb {A}^n_R\).)

To see the equivalence, use that \(df = \sum _i \frac{\partial f}{\partial X_i} dX_i\).

**Proposition 2.50**

Since the assertion is local on \(X\), we may assume that \(Y = \operatorname{Spec}R\) and \(X = \operatorname{Spec}R[T_1, \dots , T_n]/(f_1, \dots , f_{n-d})\) with the \(df_i(x)\in \Omega _{\mathbb {A}^n_R/R}(x)\) linearly independent, as in Definition 2.49. We write \(\mathfrak a = (f_1, \dots , f_{n-d})\) and \(A = R[T_1,\dots , T_n]/\mathfrak a\). We have the exact sequence (Theorem 2.34)

Renumber the \(T_i\) (if necessary) so that the images of \(dT_1,\dots , dT_d, df_1,\dots , df_{n-d}\) are a basis of the fiber \((\Omega _{R[T_\bullet ]/R}\otimes _{R[T_\bullet ]}A)(x)\) over \(x\). By the lemma of Nakayama, these elements give us also a basis of the stalk, and hence even a basis on an open neighborhood \(U\) of \(x\). The image of \({\mathfrak a}/{\mathfrak a}^2\) is exactly the submodule generated by the \(df_i\), so this implies that \(\Omega _{A/R}\) is free over such a neighborhood.

**Remark 2.51**

**Theorem 2.52**

If \(\Omega _{X/k}\) is locally free of rank \(\dim X\), then \(X\) is regular and hence, since \(k\) is algebraically closed, also smooth over \(k\) (Theorem 2.21). Conversely, the smoothness of \(f\) implies that \(\Omega _{X/k}\) is locally free by Proposition 2.50. Again using Theorem 2.21, we also obtain that \(X\) is regular, and it follows that \(f\) must be smooth of relative dimension \(\dim X\).

**Proposition 2.53**

As in the proof of Proposition 2.50, it is enough to consider the local situation, and we again use the notation set up in the beginning of the proof of that proposition.

Consider a ring \(C\), an ideal \(I\) of \(C\) with \(I^2 = 0\) and a commutative diagram

We need to show that there exists a homomorphism \(\varphi \colon A\to C\) making the diagram commutative. We start by choosing arbitarily an \(R\)-algebra homomorphism \(\psi \colon R[T_1,\dots , T_n]\to C\) such that the diagram

is commutative. Then \(\psi ({\mathfrak a})\subseteq I\) (but of course there is no reason to expect that \(\psi \) will factor through \(A\); we will now change it appropriately to achieve that). In Remark 2.51 we have seen that the sequence

is split exact, at least after replacing \(A\) by a suitable localization. Since the proposition makes only a local statement the localization is harmless and we suppress it from the notation. The restriction of \(\psi \) to \({\mathfrak a}\) induces a map \({\mathfrak a}/{\mathfrak a}^2 \to I/I^2=I\), as we have already noted, and since the sequence is split, we can extend that map to a map \(\xi \colon \Omega _{R[T_\bullet ]/R}\otimes _{R[T_\bullet ]}A \to I\). We define \(D\) as the composition

an \(R\)-derivation with the property that \(\psi _{|{\mathfrak a}} = D_{|{\mathfrak a}}\). Setting \(\varphi = \psi -D\), we obtain a map that maps \({\mathfrak a}\) to \(0\) and (since \(D\) is a derivation) is a ring homomorphism. Thus \(\varphi \) factors through a homomorphism \(\varphi \colon A\to C\). This makes the above diagram commutative, so we are done.

**Theorem 2.54**

Let \(f\) be formally smooth and locally of finite presentation. To show that \(f\) is smooth, we may work locally on \(X\) and \(Y\) and therefore pass to an affine situation, i.e., assume that \(f\) is given by a ring homomorphism \(R\to R[T_\bullet ]\to A\) with \(R[T_\bullet ]\to A\) surjective with kernel \({\mathfrak a}\). Then Theorem 2.34 shows that the sequence

is split exact. Choosing a basis of \({\mathfrak a}/{\mathfrak a}^2\) and lifting its elements to polynomials \(f_1,\dots , f_{n-d}\in {\mathfrak a}\), we see that the conditions of Definition 2.49 are satisfied and the morphism \(\operatorname{Spec}A\to \operatorname{Spec}R\) is smooth.

For the converse, note that the previous proposition shows already that a smooth morphism is at least “locally formally smooth”. We only give some very sketchy indications on how to get a global version. See [ Bo ] Ch. 8.5 for more details. See also [ GW2 ] Section (18.10) for a slightly different approach.

Consider a diagram

where, as usual, \(I\subseteq C\) is an ideal with \(I^2=0\). We have seen that there exists an open cover \(X=\bigcup _i U_i\) such that each \(U_i\) is formally smooth over \(Y\). In particular after restricting \(a_0\) to \(U_i\) and the inverse image of \(U_i\) in \(\operatorname{Spec}C/I\), we can find the desired diagonal morphism that extends \(a_0\) to \(\operatorname{Spec}C\). In other words, we find an open cover \((V_i)_i\) of \(\operatorname{Spec}C\) (which topologically is \(=\operatorname{Spec}C/I\)) and morphisms \(\varphi _i\colon V_i\to U_i\subseteq X\) making the above diagram commutative. The idea is to replace the \(\varphi _i\) by \(\varphi '_i\) such that \(\varphi '_i\) and \(\varphi '_j\) coincide on \(V_i\cap V_j\). By gluing one obtains the desired map \(\operatorname{Spec}C\to X\).

Here, we want to set \(\varphi '_i = \varphi _i - D_i\) for some derivation \(D_i\) (cf. Construction 2.31 where we have seen this principle). Writing this out one sees that there exists a family \((D_i)_i\) with the desired properties if and only if a certain class (depending on the \(\varphi _i\)) in \(\check{H}^1(\operatorname{Spec}C/I, \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_{\operatorname{Spec}C/I}}(a_0^*\Omega _{X/Y}, \tilde{I})\) vanishes. But this cohomology group vanishes entirely since \(\operatorname{Spec}C/I\) is affine and \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_{\operatorname{Spec}C/I}}(a_0^*\Omega _{X/Y}, \tilde{I})\), \(\Omega _{X/Y}\) being of finite presentation by our assumptions, is quasi-coherent.

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**Remark 2.55**

A morphism of schemes is smooth if and only if it is locally of finite presentation, flat and has regular geometric fibers. (Here the geometric fibers of a morphism \(X\to Y\) are the schemes \(X\times _Y\operatorname{Spec}(K)\) where \(K\) is an algebraically closed field and the fiber product is taken with respect to \(f\) and a \(K\)-valued point of \(Y\).) This also gives us a “fiber criterion for smoothness”, cf. [ GW2 ] Corollary 18.77.

A morphism of schemes is étale (which we have defined as smooth of relative dimension \(0\)) if and only if it is locally of finite presentation and formally étale if and only if it is flat and unramified.

An étale morphism is locally standard-étale ( [ GW2 ] Theorem 18.42): For \(f\colon X\to Y\) locally of finite presentation and \(x\in X\), \(y= f(x)\), \(f\) is étale at \(x\) if and only if there exist affine open neighborhoods \(U\subseteq X\) of \(x\) and \(V =\operatorname{Spec}R \subseteq Y\) of \(y\) where \(U \cong \operatorname{Spec}(R[T]/(f))_g\) with \(f,g\in R[T]\) and \(f'\) a unit in the localization \(R[T]_g\).

If \(f\colon X\to Y\) is smooth at \(x\in X\), then there exists an open neighborhood \(U\) of \(x\) such that \(f_{|U}\) can be factorized as \(U\to \mathbb {A}^n_Y\to Y\) with \(U\to \mathbb {A}^n_Y\) étale.