## 109.22 Finite locally free modules

Definition 109.22.1. 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.

Exercise 109.22.2. 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 109.22.3. 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 109.22.2).

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

Exercise 109.22.4. Show that the class groups of the following rings are trivial

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

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

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

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

Exercise 109.22.5. 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 109.22.6. Let $A$ be a ring.

1. 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 $(A, M, \varphi )$”.

2. 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 $\det (\varphi \circ \psi ) = \det (\psi \circ \varphi )$.

3. 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 109.22.7. 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.

1. Define $\text{Trace}_{B/A}$ and $\text{Norm}_{B/A}$ using $\text{Trace}$ and $\det$ from Exercise 109.22.6.

2. 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 $V(f) = \{ {\mathfrak p} \mid f \in {\mathfrak p}\}$.)

3. (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 $\text{Norm}_{B'/A'}(b \otimes 1)$.

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

5. 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” $A = {\mathbf Q}[x] \subset A' = {\mathbf Q}(\zeta _ n)(x^{1/n})$.)

Comment #4612 by Cecilia on

In Definition 107.22.1 for a finite locally free A-module, it should be "for every p $\in$ Spec(A), $\exists$ an $f \in$ Spec(A)" instead of "f $\in$ A" because otherwise it does not make sense to localize A at f.

Comment #4614 by Cecilia on

Or is it that in this definition, localization at a point $f \in$ A means that f is a non-nilpotent element and the set S = {$f^n$ : n = 0,1,...}? Thanks in advance!

Comment #4616 by on

OK, I understand your confusion. When we have a ring $A$ and an element $f \in A$, then the notation $A_f$ and $M_f$ always mean: $S^{-1}A$ and $S^{-1}M$ where $S$ is the multiplicative subset $S = \{1, f, f^2, f^3, \ldots\}$ of $A$ even when $f$ is nilpotent (in which case $A_f$ is the zero ring and $M_f$ is the zero module). Definition 109.22.1 really means for every $\mathfrak p$ there is an $f \in A$, $f \not \in \mathfrak p$ such that $M_f$ is a finite free $A_f$-module. But note that any module over the zero ring is both zero and free and finite and invertible (either by convention or because this follows from the definition of a free or finite or invertible module).

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