# 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**

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

**Definition 3.2**

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

**Remark 3.3**

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 3.4**

**Proposition 3.5**

## 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

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 *Ã©tale*.

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**

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**

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

**Corollary 3.9**

**(3.4) Existence of smooth points**

Let $k$ be a field.

**Lemma 3.10**

^{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 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**

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**

*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**

*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**

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 $A/{\mathfrak m}$.

**(3.6) Smoothness and regularity**

Let $k$ be a field.

**Lemma 3.16**

**Lemma 3.17**

**Theorem 3.18**

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 $d$.

May 13,

2019

**Corollary 3.19**

**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**

$X$ 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.

**(3.7) Modules of differentials**

Let $A$ be a ring.

**Definition 3.22**

*$A$-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 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, 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 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**

**Corollary 3.26**

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

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 3.27**

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 Ã©tale*, 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**

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

which is exact.

If moreover $g$ is formally smooth, then the sequence

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

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.

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

**Remark 3.31**

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

**Definition 3.32**

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 Ã©tale*, 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**

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 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

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**

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 3.36**

**Proposition 3.37**

**Proposition 3.38**

**Proposition 3.39**

**(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

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**

May 27,

2019

**Theorem 3.42**

**Proposition 3.43**

**Theorem 3.44**

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**

**Proposition 3.46**

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**

**Proposition 3.48**

**Corollary 3.49**

**Proposition 3.50**

**Proposition 3.51**

**Proposition 3.52**

**(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**

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**

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 $d$*.Let $R$ be a ring. A

*graded $R$-algebra*is a graded ring $A$ together with a ring homomorphism $R\to A$.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 $d$.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 $d$*.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 $d$.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**

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

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

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

**Example 3.56**

**Definition 3.57**

*relevant*if $A_+ \not\subseteq {\mathfrak p}$.

**Definition 3.58**

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

**Lemma 3.59**

**Proposition 3.60**

June 17,

2019

**Definition 3.61**

*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**

**(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

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

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

**Lemma 3.64**

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

**Proposition 3.65**