Algebraic Geometry 2, SS 2019.
Lecture course notes.

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3 Smoothness and differentials

April 29,
2019

General reference: [ GW ] Ch. 6.

The Zariski tangent space

 

(3.1) Definition of the Zariski tangent space

Definition 3.1
Let $X$ be a scheme, $x\in X$, ${\mathfrak m}_x \subset {\mathscr O}_{X,x}$ the maximal ideal in the local ring at $x$, $\kappa (x)$ the residue class field of $X$ in $x$. The $\kappa (x)$-vector space ${({\mathfrak m}/{\mathfrak m}^2)}^*$ is called the (Zariski) tangent space of $X$ in $x$.

Definition 3.2
Let $R$ be a ring, $f_1, \dots , f_r\in R[T_1, \dots , T_n]$. We call the matrix
\[ J_{f_1, \dots , f_r} := {\left( \frac{\partial f_i}{\partial T_j}\right)}_{i, j} \in M_{r\times n}(R[T_\bullet ]) \]
the Jacobian matrix of the polynomials $f_i$. Here the partial derivatives are to be understood in a formal sense.

Remark 3.3

  1. 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.

  2. 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 3.4
Let $k$ be a field, $X = V(f_1, \dots , f_m)\subseteq \mathbb {A}^n_k$, $f_i\in k[T_1, \dots , T_n]$, $x = (x_i)_i\in k^n= \mathbb {A}^n(k)$. Then there is a natural identification $T_xX = \operatorname{Ker}(J_{f_1, \dots , f_m} (x))$, where $J_{f_1, \dots , f_m}(x)$ denotes the matrix with entries in $\kappa (x) = k$ obtained by mapping each entry of $J_{f_1, \dots , f_m}$ to $\kappa (x)$, which amounts to evaluating these polynomials at $x$.

Proposition 3.5
Let $k$ be a field, $X$ a $k$-scheme, $x\in X(k)$. There is an identification (functorial in $(X, x)$)
\[ X(k[\varepsilon ]/(\varepsilon ^2))_x := \{ f\in \operatorname{Hom}_k(\operatorname{Spec}k[\varepsilon ]/(\varepsilon ^2), X);\ \mathop{\rm im}(f) = \{ x\} \} = T_xX. \]

Smooth morphisms

 

(3.2) Definition of smooth morphisms

Definition 3.6

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

\begin{tikzcd} U \arrow[rd, "f"]\arrow[rr, "j"] & & \Spec R[T_1, \dots, T_n](f_1, \dots, f_{n-d}) \arrow[ld]\\ & V & \end{tikzcd}

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.)

May 6,
2019

(3.3) 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 3.7
Let $X$ be a topological space. We define the dimension of $X$ as \begin{align*} \dim X := \sup \{ \ell ;\ & \text{there exists a chain}\ Z_0 \supsetneq Z_1 \supsetneq \cdots \supsetneq Z_\ell \\ & \ \text{of closed irreducible subsets}\ Z_i\subseteq X\} . \end{align*}

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 3.8
Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra which isi a domain. Let ${\mathfrak m}\subset A$ be a maximal ideal. Then
\[ \dim A = \mathop{\rm trdeg}\nolimits _k(\operatorname{Frac}(A)) = \dim A_{{\mathfrak m}}. \]

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

Corollary 3.9
Let $k$ be a field, and let $X$ be an integral $k$-scheme which is of finite type over $k$. Denote by $K(X)$ its field of rational functions. Let $U\subseteq X$ be a non-empty open subset, and let $x\in X$ be a closed point. Then
\[ \dim X = \dim U = \mathop{\rm trdeg}\nolimits _k(K(X)) = \dim {\mathscr O}_{X,x}. \]

(3.4) Existence of smooth points

Let $k$ be a field.

Lemma 3.10
Let $X$, $Y$ be [integral 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 3.11

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 3.12
Let $k$ be a perfect field, and let $X$ be a nonempty reduced $k$-scheme locally of finite type over $k$. Then the smooth locus
\[ X_{\rm sm} := \{ x\in X;\ X\to \operatorname{Spec}k\ \text{is smooth in} x\} \]
of $X$ is open and dense.

May 8,
2019

(3.5) Regular rings

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

Definition 3.13
A noetherian local ring $A$ with maximal ideal ${\mathfrak m}$ and residue class field $\kappa $ is called 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 3.14
A noetherian ring $A$ is called 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 3.15

  1. Every localization of a regular ring is regular.

  2. If $A$ is regular, then the polynomial ring $A[T]$ is regular.

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

  4. 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 $A/{\mathfrak m}$.

(3.6) Smoothness and regularity

Let $k$ be a field.

Lemma 3.16
Let $X$ be a $k$-scheme locally of finite type. Let $x\in X$ such that $X\to \operatorname{Spec}k$ is smooth of relative dimension $d$ in $x$. Then ${\mathscr O}_{X,x}$ is regular of dimension $\le d$. If moreover $x$ is closed, then ${\mathscr O}_{X,x}$ is regular of dimension $d$.

Lemma 3.17
Let $X = V(g_1, \dots , g_s)\subseteq \mathbb {A}^n_k$, and let $x\in X$ be a closed point. If $\operatorname{rk}J_{g_1,\dots , g_s}(x) \ge n - \dim {\mathscr O}_{X,x}$, then $x$ is smooth in $X/k$, and $\operatorname{rk}J_{g_1,\dots , g_s}(x) = n - \dim {\mathscr O}_{X,x}$.

Theorem 3.18
Let $X$ be a $k$-scheme locally of finite type, $x\in X$ a closed point, $d\ge 0$. Fix an algebraically closed extension field $K$ of $k$ and write $X_K=X\otimes _kK$. The following are equivalent:

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

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

  3. 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 $\overline{x}$.

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

  5. 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 $d$.

If these conditions hold, then the local ring ${\mathscr O}_{X,{x}}$ is regular of dimension $d$, and if $\kappa (x) = k$, then this last condition is equivalent to the previous ones.

May 13,
2019

Corollary 3.19
Let $X$ be an irreducible scheme of finite type over $k$, and let $x\in X(k)$ be a $k$-valued point. Then $X\to \operatorname{Spec}k$ is smooth at $x$ if and only if $\dim X= \dim _k T_xX$.

Corollary 3.20

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 3.21
Let $X$ be locally of finite type over $k$. The following are equivalent:

  1. $X$ is smooth over $k$.

  2. $X\otimes _kL$ is regular for every field extension $L/k$.

  3. 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.

(3.7) Modules of differentials

Let $A$ be a ring.

Definition 3.22
Let $B$ be an $A$-algebra, and $M$ a $B$-module. An $A$-derivation from $B$ to $M$ is a homomorphism $D\colon B\to M$ of abelian groups such that

  1. (Leibniz rule) $D(bb’) = bD(b’) + b’D(b)$ for all $b, b’\in B$,

  2. $d(a) = 0$ 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 3.23

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, 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 3.24

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 3.25
Let $\varphi \colon B\to B’$ be a surjective homomorphism of $A$-algebras, and write ${\mathfrak b}= \operatorname{Ker}(\varphi )$. Assume that a module of differentials $(\Omega _{B/A}, d_{B/A})$ for $B/A$ exists. Then
\[ \Omega _{B/A}/({\mathfrak b}\Omega _{B/A} + B’ d({\mathfrak b})) \]
together with the derivation $d_{B'/A}$ induced by $d_{B/A}$ is a module of differentials for $B’/A$.

Corollary 3.26
For every $A$-algebra $B$, a module $\Omega _{B/A}$ of differentials exists. It is unique up to unique isomorphism.

May 15,
2019

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

\begin{tikzcd} B \arrow[r] & C/I \\ A\arrow[u, "\varphi"]\arrow[r] & C\arrow[u] \end{tikzcd}

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:

\begin{tikzcd} B \arrow[r]\arrow[rd, dashed] & C/I \\ A\arrow[u, "\varphi"]\arrow[r] & C\arrow[u] \end{tikzcd}

Definition 3.27
Let $\varphi \colon A\to B$ be a ring homomorphism.

  1. 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.

  2. 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.

  3. 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 3.28
Let $\varphi \colon A\to B$ be a ring homomorphism. Then $\varphi $ is formally unramified if and only if $\Omega _{B/A} = 0$.

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).

Theorem 3.29

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.

May 20,
2019

Theorem 3.30

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

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

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.

(3.8) The sheaf of differentials of a scheme morphism

Remark 3.31
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. One shows that $(I/I^2, d)$ satisfies the universal property defining $(\Omega _{B/A}, d_{B/A})$.

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 3.32
Let $f\colon X\to Y$ be a morphism of schemes.

  1. 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 $\operatorname{Hom}_Y(\operatorname{Spec}C, X) \to \operatorname{Hom}_Y(\operatorname{Spec}C/I, X)$.

  2. 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 $\operatorname{Hom}_Y(\operatorname{Spec}C, X) \to \operatorname{Hom}_Y(\operatorname{Spec}C/I, X)$.

  3. 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 3.33

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

  2. 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 3.34

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

\[ D(U)(bb’) = bD(U)(b’) + b’D(U)(b),\qquad \forall b, b’\in \Gamma (U, {\mathscr O}_X) \]

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.

May 22,
2019

Definition/Proposition 3.35
Let $f\colon X\to Y$ be a morphism of schemes. The following three definitions give the same result (up to unique isomorphism), called the sheaf of differentials of $f$ or of $X$ over $Y$, denoted $\Omega _{X/Y}$ — a quasi-coherent ${\mathscr O}_X$-module together with a derivation $d_{X/Y}\colon {\mathscr O}_X\to \Omega _{X/Y}$.

  1. 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 $d_{B/A}$.

  2. $\Omega _{X/Y} = \Delta ^*({\mathscr J}/{\mathscr J}^2)$, 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 $b\mapsto 1\otimes b-b\otimes 1$.

  3. 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 3.36
Let $f\colon X\to Y$, $g\colon Y’\to Y$ be morphisms 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 $\Omega _{X'/Y'} \cong (g’)^*\Omega _{X/Y}$, compatible with the universal derivations.

Proposition 3.37
Let $f\colon X\to Y$, $g\colon Y\to Z$ be morphisms of schemes. Then there is an exact sequence
\[ f^*\Omega _{Y/Z} \to \Omega _{X/Z} \to \Omega _{X/Y} \to 0 \]
of ${\mathscr O}_X$-modules. If $f$ is formally smooth, then the sequence
\[ 0 \to f^*\Omega _{Y/Z} \to \Omega _{X/Z} \to \Omega _{X/Y} \to 0 \]
is exact and splits locally on $X$.

Proposition 3.38
Let $i\colon Z\to X$ be a closed immersion with corresponding ideal sheaf ${\mathscr J}\subseteq {\mathscr O}_X$, and let $g\colon X\to Y$ be a scheme morphism. Then there is an exact sequence
\[ i^*({\mathscr J}/{\mathscr J}^2) \to i^*\Omega _{X/Y} \to \Omega _{Z/Y} \to 0 \]
of ${\mathscr O}_Z$-modules. If $Z$ is formally smooth over $Y$, then the sequence
\[ 0\to i^*({\mathscr J}/{\mathscr J}^2) \to i^*\Omega _{X/Y} \to \Omega _{Z/Y} \to 0 \]
is exact and splits locally on $Z$.

Proposition 3.39
Let $K$ be a field, and let $X$ be a $k$-scheme of finite type. Let $x\in X(k)$. Then we have an isomorphism $T_xX = \Omega _{X/k}(x)$ between the Zariski tangent space at $x$ and the fiber of the sheaf of differentials of $X/k$ at $x$.

(3.9) Sheaves of differentials and smoothness

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

Definition 3.40

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

\begin{tikzcd} U \arrow[rd, "f"]\arrow[rr, "j"] & & \Spec R[T_1, \dots, T_n](f_1, \dots, f_{n-d}) \arrow[ld]\\ & V & \end{tikzcd}

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 3.41
Let $f\colon X\to S$ be smooth of relative dimension at $x\in X$. Then there exists an open neighborhood $U$ of $x$ such that the restriction $\Omega _{X/Y|U} (=\Omega _{U/Y})$ is free of rank $d$.

May 27,
2019

Theorem 3.42
Let $k$ be an algebraically closed field, and let $X$ be an irreducible $k$-scheme of finite type. Let $d=\dim X$. Then $X$ is smooth over $k$ if and only if $\Omega _{X/k}$ is locally free of rank $d$.

Proposition 3.43
Let $f\colon X\to S$ be smooth of relative dimension $d$ at $x\in X$. Then there exists an open neighborhood $U$ of $x$ such that the restriction $U\to S$ of $f$ to $U$ is formally smooth.

Theorem 3.44
Let $f\colon X\to Y$ be a morphism locally of finite presentation (e.g., if $Y$ is noetherian and $f$ is locally of finite type). Then $f$ is smooth if and only if $f$ is formally smooth.

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”.)

May 29,
2019

Projective schemes

 

References: [ GW ] , Ch. 8, Ch. 11, in particular Example 11.43, (8.5); [ H ] II.6, II.7.

(3.10) Line bundles on $\mathbb {P}^n_k$

Let $R$ be a ring. We cover $\mathbb {P}^n_R$ by the standard charts $U_i := D_+(T_i)$, as usual, and write $U_{ij} := U_i\cap U_j$. For $d\in \mathbb {Z}$, the elements $(T_i/T_j)^d \in \Gamma (U_{ij}, {\mathscr O}_{\mathbb {P}^n_R})^\times $ define isomorphisms ${\mathscr O}_{U_i|U_{ij}}\to {\mathscr O}_{U_j|U_{ij}}$ which give rise to a gluing datum of the ${\mathscr O}_{U_i}$-modules ${\mathscr O}_{U_i}$. By gluing of sheaves, we obtain a line bundle ${\mathscr O}_{\mathbb {P}^n_R}(d)$. (Cf. Problems 9, 10.)

Lemma 3.45
We obtain a group homomorphism $\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_R)$, $d\mapsto {\mathscr O}(d)$.

Proposition 3.46
Writing $R[T_0, \dots , T_n]_d$ for the submodule of homogeneous polynomials of degree $d$ (with $R[T_0, \dots , T_n]_d=0$ for $d<0$), we have
\[ \Gamma (\mathbb {P}^n_R, {\mathscr O}(d)) \cong R[T_0, \dots , T_n]_d \]
for all $d\in \mathbb {Z}$.

Now let $R=k$ be a field.

The closed subscheme $V_+(T_0)$ is a Weil divisor on $\mathbb {P}^n_k$, and it corresponds to the Cartier divisor $(U_i, T_0/T_i)_i$. The corresponding line bundle is ${\mathscr O}(1)$. By passing to multiples/negatives of this divisor, we can describe all ${\mathscr O}(d)$ in a similar way.

June 3,
2019

Remark 3.47
One can show that every locally free ${\mathscr O}_{\mathbb {P}^1_k}$-module is isomorphic to a direct sum of line bundles. Note though that this statement is not true for $\mathbb {P}^n_k$, $n>1$.

Proposition 3.48
Let $A$ be a unique factorization domain, and let $Z = V({\mathfrak p})\subset \operatorname{Spec}A$ a closed irreducible subset of codimension $1$, i.e., ${\mathfrak p}\ne 0$ is a prime ideal which is minimal among all non-zero prime ideals. Then ${\mathfrak p}$ is a principal ideal, i.e., considering $Z$ as a Weil divisor, it is principal.

Corollary 3.49
Let $k$ be a field, and let $Z\subset \mathbb {A}^n_k$ be an integral closed subscheme of codimension $1$. Then $Z = V(f)$ for some polynomial $f$.

Proposition 3.50
Let $k$ be a field, and let $Z\subseteq \mathbb {P}^n_k$ be an integral closed subscheme of codimension $1$. Then $Z=V_+(f)$ for some homogeneous polynomial $f$.

Proposition 3.51
The above homomorphism $\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_k)$, $d\mapsto {\mathscr O}(d)$, is an isomorphism.

Proposition 3.52
Let $R$ be a ring. We have a short exact sequence
\[ 0 \to \Omega _{\mathbb {P}^n_R/R} \to {\mathscr O}(-1)^{n+1} \to {\mathscr O}^n \to 0 \]
of ${\mathscr O}_X$-modules.

(3.11) Functorial description of $\mathbb {P}^n$

June 5,
2019

As we have seen last term, every scheme $X$ defines a contravariant functor $T\mapsto X(T):=\operatorname{Hom}_{\textup{(Sch)}}(T, X)$ from the category of schemes to the category of sets. This functor determines $X$ up to unique isomorphism. In this section, we want to describe the functor attached in this way to projective space $\mathbb {P}^n_R$ for $R$ a ring.

Proposition 3.53
Let $R$ be a ring, and let $S$ be an $R$-scheme. There is a bijection, functorial in $S$, \begin{align*} \mathbb {P}^n_R(S) = \{ ({\mathscr L}, \alpha );\ & {\mathscr L}\ \text{a line bundle on}\ S,\\ & \alpha \colon {\mathscr O}_S^{n+1}\twoheadrightarrow {\mathscr L}\ \text{a surjective} {\mathscr O}_S\text{-module homom.} \} /\cong \end{align*} Here we consider pairs $({\mathscr L}, \alpha )$, $({\mathscr L}’, \alpha ’)$ as isomorphic, if there exists an ${\mathscr O}_S$-module isomorphism $\beta \colon {\mathscr L}\to {\mathscr L}’$ with $\alpha = \alpha ’\circ \beta $.

Note that a homomorphism $\alpha \colon {\mathscr O}_S^{n+1}\twoheadrightarrow {\mathscr L}$ corresponds to $n+1$ global sections in $\Gamma (S, {\mathscr L})$ (the “images of the standard basis vectors”). Thus $T_0, \dots , T_n\in \Gamma (\mathbb {P}^n_R, {\mathscr O}(1))$ give rise to a (surjective) homomorphism ${\mathscr O}_{\mathbb {P}^n_R}^{n+1}\to {\mathscr O}(1)$. Given a morphism $S\to \mathbb {P}^n_R$, we can pull this homomorphism back to $S$ and obtain an element of the right hand side in the statement of the proposition.

Conversely, given a pair $({\mathscr L}, \alpha )$ on $S$, we can think of the corresponding morphism $S\to \mathbb {P}^n_R$ in terms of homogeneous coordinates (i.e., for $K$-valued points for some field $K$), as follows: Denote by $f_0, \to f_n\in \Gamma (S, {\mathscr L})$ the global sections corresponding to $\alpha $. For a point $x\in S$, the fiber ${\mathscr L}(x)$ is a one-dimensional $\kappa (x)$-vector space generated by the elements $f_0(x), \dots , f_n(x)$ (i.e., at least one of them ins $\ne 0$ – this holds since $\alpha $ is surjective). We choose an isomorphism ${\mathscr L}(x) \cong \kappa (x)$, and hence can view the $f_i(x)$ as elements of $\kappa (x)$. Then the morphism $S\to \mathbb {P}^n_S$ maps $x$ to $(f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))$. While the individual $f_i(x)$, as elements of $\kappa (x)$, depend on the choice of isomorphism ${\mathscr L}(x)\cong \kappa (x)$, the point $(f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))$ is independent of this choice.

(3.12) The Proj construction

Reference: [ GW ] Ch. 13.

Definition 3.54

  1. A graded ring is a ring $A$ with a decomposition $A = \bigoplus _{d\ge 0}A_d$ as abelian groups such that $A_d\cdot A_e\subseteq A_{d+e}$ for all $d, e$. The elements of $A_d$ are called homogeneous of degree $d$.

  2. Let $R$ be a ring. A graded $R$-algebra is a graded ring $A$ together with a ring homomorphism $R\to A$.

  3. A homomorphism $A\to B$ of graded rings (or graded $R$-algebras) is a ring homomorphism (or $R$-algebra homomorphism, respectively) $f\colon A\to B$ such that $f(A_d)\subseteq B_d$ for all $d$.

  4. Let $A$ be a graded ring. A graded $A$-module is an $A$-module $M$ with a decomposition $M=\bigoplus _{d\in \mathbb {Z}} M_d$ such that $A_d\cdot M_e\subseteq M_{d+e}$ for all $d, e$. The elements of $M_d$ are called homogeneous of degree $d$.

  5. A homomorphism $M\to N$ of graded $A$-modules is an $A$-module homomorphism $f\colon M\to N$ such that $f(M_d)\subseteq N_d$ for all $d$.

  6. Let $A$ be a graded ring and let $M$ be a graded $A$-module. A homogeneous submodule of $M$ is a submodule $N\subseteq M$ such that $N = \bigoplus _{d\in \mathbb {Z}} (N\cap M_d)$. In this way, $N$ is itself a graded $A$-module and the inclusion $N\hookrightarrow M$ is a homomorphism of graded $A$-modules. (And conversely, every injective homomorphism of graded $A$-modules has a homogeneous submodule as its image.) A homogeneous submodule of $A$ is called a homogeneous ideal.

Example 3.55
Let $R$ be a ring. Then the polynomial ring $R[T_0, \dots , T_n]$ is a graded $R$-algebra if we set $R[T_0, \dots , T_n]_d$ to be the $R$-submodule of homogeneous polynomials of degree $d$.

We now fix a graded ring $A$.

For a homogeneous element $f\in A_e$, and a graded $A$-module $M$, the localization $M_f$ is a graded $A$-module via

\[ M_{f, d} = \{ \frac{m}{f^i};\ m\in M_{d+ei} \} . \]

Applying this to $A$ as an $A$-module, we obtain a grading on $A_f$ giving $A_f$ the structure of a graded ring. Then $M_f$ is a graded $A_f$-module.

We define

\[ M_{(f)} := M_{f, 0}, \]

the degree $0$ part of $M_f$. Then $A_{(f)}$ is a ring and $M_{(f)}$ is an $A_{(f)}$-module.

Example 3.56
Let $R$ be a ring. Then
\[ R[T_0, \dots , T_n]_{(T_i)} = R[\frac{T_0}{T_i}, \dots , \frac{T_n}{T_i}]. \]

Definition 3.57
We write $A_+ := \bigoplus _{d>0} A_d$, an ideal of $A$. A homogeneous prime ideal ${\mathfrak p}\subset A$ is called relevant if $A_+ \not\subseteq {\mathfrak p}$.

Definition 3.58
We denote by $\operatorname{Proj}(A)$ the set of all relevant homogeneous prime ideals of $A$. We equip $\operatorname{Proj}(A)$ with the Zariski topology, by saying that the closed subsets are the subsets of the form
\[ V_+(I) := \{ {\mathfrak p}\in \operatorname{Proj}(A);\ I \subseteq {\mathfrak p}\} . \]
for homogeneous ideals $I\subseteq A$.

For a homogeneous element $f$, we write $D_+(f) := \operatorname{Proj}(A)\setminus V_+(f)$.

Lemma 3.59
Let $f\in A$ be a homogeneous element. Then the map
\[ D_+(f) \to \operatorname{Spec}A_{(f)}, \quad {\mathfrak p}\mapsto ({\mathfrak p}A_f)\cap A_{(f)} \]
is a homeomorphism.

Proposition 3.60
There is a unique sheaf ${\mathscr O}$ of rings on $\operatorname{Proj}(A)$ such that
\[ \Gamma (D_+(f), {\mathscr O}) = A_{(f)} \]
for every homogeneous element $f\in A$ and with restriction maps given by the canonical maps between the localizations. The ringed space $(\operatorname{Proj}(A), {\mathscr O})$ is a separated scheme which we again denote by $\operatorname{Proj}(A)$.

June 17,
2019

Definition 3.61
Let $R$ be a ring, and let $X$ be an $R$-scheme. We say that $X$ is projective over $R$ (or that the morphism $X\to \operatorname{Spec}R$ is projective), if there exist $n\ge 0$ and a closed immersion $X\to \mathbb {P}^n_R$ of $R$-schemes.

Theorem 3.62
Let $R$ be a ring, and let $X$ be a projective $R$-scheme. Then $X$ is proper over $R$.

(3.13) Quasi-coherent modules on $\operatorname{Proj}(A)$

Let $A$ be a graded ring, $X=\operatorname{Proj}A$. If $M$ is a graded $A$-module, there is a unique sheaf $\widetilde{M}$ of ${\mathscr O}_X$-modules such that

\[ \Gamma (D_+(f), \widetilde{M}) = M_{(f)} \]

for every homogeneous element $f\in A$, and such that the restriction maps for inclusions of the form $D_+(g)\subseteq D_+(f)$ are given by the natural maps between the localizations. This sheaf is a quasi-coherent ${\mathscr O}_X$-module.

Example 3.63

Let $A(n)$ be the graded $A$-module defined by $A(n) = \bigoplus _{d\in \mathbb {Z}} A_{n+d}$. We set ${\mathscr O}_X(n) = \widetilde{A(n)}$. If $A=R[T_0, \dots , T_n]$ for a ring $R$, so that $X=\mathbb {P}^n_R$, then this notation is consistent with our previous definition.

For $f\in A_d$, multiplication by $f^k$ defines an isomorphism

\[ {\mathscr O}_{X|D_+(f)} \overset {\sim }{\to }{\mathscr O}_X(n)_{|D_+(f)}. \]

In particular, if $A$ is generated as an $A_0$-algebra by $A_1$, then ${\mathscr O}_X(n)$ is a line bundle.

Assume, for the remainder of this section, that $A$ is generated as an $A_0$-algebra by $A_1$. So $X = \bigcup _{f\in A_1} D_+(f)$, and ${\mathscr O}_X(n)$ is a line bundle.

For an ${\mathscr O}_X$-module ${\mathscr F}$, write ${\mathscr F}(n) :={\mathscr F}\otimes _{{\mathscr O}_X}{\mathscr O}_X(n)$, and define a graded $A$-module $\Gamma _*({\mathscr F})$ by

\[ \Gamma _*({\mathscr F}) = \bigoplus _{n\in \mathbb {Z}} \Gamma (X, {\mathscr F}(n)). \]

Lemma 3.64
For a graded $A$-module $M$, there is a natural map $M\to \Gamma _*(\widetilde{M})$. For an ${\mathscr O}_X$-module ${\mathscr F}$, there is a natural map $\widetilde{\Gamma _*({\mathscr F})}\to {\mathscr F}$. If ${\mathscr F}$ is quasi-coherent, then the latter map is an isomorphism.

Call a graded $A$-module $M$ saturated, if the map $M\to \Gamma _*(\widetilde{M})$ is an isomorphism.

Proposition 3.65
The functors $M\to \widetilde{M}$ and ${\mathscr F}\to \Gamma _*({\mathscr F})$ define an equivalence of categories between the category of saturated graded $A$-modules and the category of quasi-coherent ${\mathscr O}_X$-modules.

  1. The statement is true in general, but in the lecture we proved it only with the additional assumption that $X$ and $Y$ are integral.