Definition 111.22.1. Let $A$ be a ring. Recall that a *finite locally free* $A$-module $M$ is a module such that for every ${\mathfrak p} \in \mathop{\mathrm{Spec}}(A)$ there exists an $f\in A$, $f \not\in {\mathfrak p}$ such that $M_ f$ is a finite free $A_ f$-module. We say $M$ is an *invertible module* if $M$ is finite locally free of rank $1$, i.e., for every ${\mathfrak p} \in \mathop{\mathrm{Spec}}(A)$ there exists an $f\in A$, $f \not\in \mathfrak p$ such that $M_ f \cong A_ f$ as an $A_ f$-module.

## 111.22 Finite locally free modules

Exercise 111.22.2. Prove that the tensor product of finite locally free modules is finite locally free. Prove that the tensor product of two invertible modules is invertible.

Definition 111.22.3. Let $A$ be a ring. The *class group of $A$*, sometimes called the *Picard group of $A$* is the set $\mathop{\mathrm{Pic}}\nolimits (A)$ of isomorphism classes of invertible $A$-modules endowed with a group operation defined by tensor product (see Exercise 111.22.2).

Note that the class group of $A$ is trivial exactly when every invertible module is isomorphic to a free module of rank 1.

Exercise 111.22.4. Show that the class groups of the following rings are trivial

a polynomial ring $A = k[x]$ where $k$ is a field,

the integers $A = \mathbf{Z}$,

a polynomial ring $A = k[x, y]$ where $k$ is a field, and

the quotient $k[x, y]/(xy)$ where $k$ is a field.

Exercise 111.22.5. Show that the class group of the ring $A = k[x, y]/(y^2 - f(x))$ where $k$ is a field of characteristic not $2$ and where $f(x) = (x - t_1) \ldots (x - t_ n)$ with $t_1, \ldots , t_ n \in k$ distinct and $n \geq 3$ an odd integer is not trivial. (Hint: Show that the ideal $(y, x - t_1)$ defines a nontrivial element of $\mathop{\mathrm{Pic}}\nolimits (A)$.)

Exercise 111.22.6. Let $A$ be a ring.

Suppose that $M$ is a finite locally free $A$-module, and suppose that $\varphi : M \to M$ is an endomorphism. Define/construct the

*trace*and*determinant*of $\varphi $ and prove that your construction is “functorial in the triple $(A, M, \varphi )$”.Show that if $M, N$ are finite locally free $A$-modules, and if $\varphi : M \to N$ and $\psi : N \to M$ then $\text{Trace}(\varphi \circ \psi ) = \text{Trace}(\psi \circ \varphi )$ and $\det (\varphi \circ \psi ) = \det (\psi \circ \varphi )$.

In case $M$ is finite locally free show that $\text{Trace}$ defines an $A$-linear map $\text{End}_ A(M) \to A$ and $\det $ defines a multiplicative map $\text{End}_ A(M) \to A$.

Exercise 111.22.7. Now suppose that $B$ is an $A$-algebra which is finite locally free as an $A$-module, in other words $B$ is a finite locally free $A$-algebra.

Define $\text{Trace}_{B/A}$ and $\text{Norm}_{B/A}$ using $\text{Trace}$ and $\det $ from Exercise 111.22.6.

Let $b\in B$ and let $\pi : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ be the induced morphism. Show that $\pi (V(b)) = V(\text{Norm}_{B/A}(b))$. (Recall that $V(f) = \{ {\mathfrak p} \mid f \in {\mathfrak p}\} $.)

(Base change.) Suppose that $i : A \to A'$ is a ring map. Set $B' = B \otimes _ A A'$. Indicate why $i(\text{Norm}_{B/A}(b))$ equals $\text{Norm}_{B'/A'}(b \otimes 1)$.

Compute $\text{Norm}_{B/A}(b)$ when $B = A \times A \times A \times \ldots \times A$ and $b = (a_1, \ldots , a_ n)$.

Compute the norm of $y-y^3$ under the finite flat map ${\mathbf Q}[x] \to {\mathbf Q}[y]$, $x \to y^ n$. (Hint: use the “base change” $A = {\mathbf Q}[x] \subset A' = {\mathbf Q}(\zeta _ n)(x^{1/n})$.)

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