The Stacks project

111.40 Invertible sheaves

Definition 111.40.1. Let $X$ be a locally ringed space. An invertible ${\mathcal O}_ X$-module on $X$ is a sheaf of ${\mathcal O}_ X$-modules ${\mathcal L}$ such that every point has an open neighbourhood $U \subset X$ such that ${\mathcal L}|_ U$ is isomorphic to ${\mathcal O}_ U$ as ${\mathcal O}_ U$-module. We say that ${\mathcal L}$ is trivial if it is isomorphic to ${\mathcal O}_ X$ as a ${\mathcal O}_ X$-module.

Exercise 111.40.2. General facts.

  1. Show that an invertible ${\mathcal O}_ X$-module on a scheme $X$ is quasi-coherent.

  2. Suppose $X\to Y$ is a morphism of locally ringed spaces, and ${\mathcal L}$ an invertible ${\mathcal O}_ Y$-module. Show that $f^\ast {\mathcal L}$ is an invertible ${\mathcal O}_ X$ module.

Exercise 111.40.3. Algebra.

  1. Show that an invertible ${\mathcal O}_ X$-module on an affine scheme $\mathop{\mathrm{Spec}}(A)$ corresponds to an $A$-module $M$ which is (i) finite, (ii) projective, (iii) locally free of rank 1, and hence (iv) flat, and (v) finitely presented. (Feel free to quote things from last semesters course; or from algebra books.)

  2. Suppose that $A$ is a domain and that $M$ is a module as in (a). Show that $M$ is isomorphic as an $A$-module to an ideal $I \subset A$ such that $IA_{\mathfrak p}$ is principal for every prime ${\mathfrak p}$.

Definition 111.40.4. Let $R$ be a ring. An invertible module $M$ is an $R$-module $M$ such that $\widetilde M$ is an invertible sheaf on the spectrum of $R$. We say $M$ is trivial if $M \cong R$ as an $R$-module.

In other words, $M$ is invertible if and only if it satisfies all of the following conditions: it is flat, of finite presentation, projective, and locally free of rank 1. (Of course it suffices for it to be locally free of rank 1).

Exercise 111.40.5. Simple examples.

  1. Let $k$ be a field. Let $A = k[x]$. Show that $X = \mathop{\mathrm{Spec}}(A)$ has only trivial invertible ${\mathcal O}_ X$-modules. In other words, show that every invertible $A$-module is free of rank 1.

  2. Let $A$ be the ring

    \[ A = \{ f\in k[x] \mid f(0) = f(1) \} . \]

    Show there exists a nontrivial invertible $A$-module, unless $k = {\mathbf F}_2$. (Hint: Think about $\mathop{\mathrm{Spec}}(A)$ as identifying $0$ and $1$ in ${\mathbf A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$.)

  3. Same question as in (2) for the ring $A = k[x^2, x^3] \subset k[x]$ (except now $k = {\mathbf F}_2$ works as well).

Exercise 111.40.6. Higher dimensions.

  1. Prove that every invertible sheaf on two dimensional affine space is trivial. More precisely, let ${\mathbf A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$ where $k$ is a field. Show that every invertible sheaf on ${\mathbf A}^2_ k$ is trivial. (Hint: One way to do this is to consider the corresponding module $M$, to look at $M \otimes _{k[x, y]} k(x)[y]$, and then use Exercise 111.40.5 (1) to find a generator for this; then you still have to think. Another way to is to use Exercise 111.40.3 and use what we know about ideals of the polynomial ring: primes of height one are generated by an irreducible polynomial; then you still have to think.)

  2. Prove that every invertible sheaf on any open subscheme of two dimensional affine space is trivial. More precisely, let $U \subset {\mathbf A}^2_ k$ be an open subscheme where $k$ is a field. Show that every invertible sheaf on $U$ is trivial. Hint: Show that every invertible sheaf on $U$ extends to one on ${\mathbf A}^2_ k$. Not easy; but you can find it in [H].

  3. Find an example of a nontrivial invertible sheaf on a punctured cone over a field. More precisely, let $k$ be a field and let $C = \mathop{\mathrm{Spec}}(k[x, y, z]/(xy-z^2))$. Let $U = C \setminus \{ (x, y, z) \} $. Find a nontrivial invertible sheaf on $U$. Hint: It may be easier to compute the group of isomorphism classes of invertible sheaves on $U$ than to just find one. Note that $U$ is covered by the opens $\mathop{\mathrm{Spec}}(k[x, y, z, 1/x]/(xy-z^2))$ and $\mathop{\mathrm{Spec}}(k[x, y, z, 1/y]/(xy-z^2))$ which are “easy” to deal with.

Definition 111.40.7. Let $X$ be a locally ringed space. The Picard group of $X$ is the set $\mathop{\mathrm{Pic}}\nolimits (X)$ of isomorphism classes of invertible $\mathcal{O}_ X$-modules with addition given by tensor product. See Modules, Definition 17.25.9. For a ring $R$ we set $\mathop{\mathrm{Pic}}\nolimits (R) = \mathop{\mathrm{Pic}}\nolimits (\mathop{\mathrm{Spec}}(R))$.

Exercise 111.40.8. Let $R$ be a ring.

  1. Show that if $R$ is a Noetherian normal domain, then $\mathop{\mathrm{Pic}}\nolimits (R) = \mathop{\mathrm{Pic}}\nolimits (R[t])$. [Hint: There is a map $R[t] \to R$, $t \mapsto 0$ which is a left inverse to the map $R \to R[t]$. Hence it suffices to show that any invertible $R[t]$-module $M$ such that $M/tM \cong R$ is free of rank $1$. Let $K$ be the fraction field of $R$. Pick a trivialization $K[t] \to M \otimes _{R[t]} K[t]$ which is possible by Exercise 111.40.5 (1). Adjust it so it agrees with the trivialization of $M/tM$ above. Show that it is in fact a trivialization of $M$ over $R[t]$ (this is where normality comes in).]

  2. Let $k$ be a field. Show that $\mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3, t]) \not= \mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3])$.

Comments (2)

Comment #7166 by Harun KIR on

In Definition 110.40.7, there is an assertion that for a ring we set Yes, it is really an assertion. So it should be out of the definition. You may claim that you are just defining as you did. But it highly causes to some issues in the common literature. By the way, in the left side you have a group structure for a ring. Is the same thing true for the left side for a ring if you think what really it is. At any rate, we first should define as usual, then we should prove that they are really the same. As your previous works, it follows from Bourbaki, Commutative Algebra.

Comment #7306 by on

This is the chapter on exercises, so I am going to be a bit sloppy here. But yes in principle I agree with you. The Picard group of a ring is "officially" defined in Section 15.117 (it doesn't have a formal definition environment, so you could complain on that page if you like).

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