Find all projective modules.

Question

I asked a similar question earlier but I will be more specific this time.

Let $A= \mathbb{C}[x]/(x^n -1)$. How can I describe all the principal projective modules of $A$?

If $M= (x)$ is a principal/cyclic $A$-module it can be proven that $M \cong A/\text{Ann}_{A}(x)$, but from here I am not making any progress, I am trying to use different equivalent conditions for a projective module but no further success, any hints?

Answer 1

$\operatorname{Ann}_A(a)$ (don't use the letter $x$ to denote two different things) has to be generated by an idempotent of $A$ since the exact sequence: $$0\longrightarrow\operatorname{Ann}_A(a) \longrightarrow A\longrightarrow Aa\longrightarrow 0$$ splits. So it comes down to finding all idempotents of $\mathbf C[x]/(x^n-1)\simeq\mathbf C^n$.

Some details:

$\operatorname{Ann}_A(a)$ is generated by an idempotent because it is finitely generated, and for any prime ideal $\mathfrak p\in\operatorname{Spec} A$, $A_{\mathfrak p}\dfrac a1$ is a projective, hence free, $A_{\mathfrak p}$-module, so $$\bigl(\operatorname{Ann}_A(a)\bigr)_{\mathfrak p}=\operatorname{Ann}_{A_{\mathfrak p}}\Bigl(\frac a1\Bigr)= (0)\enspace\text{or}\enspace A_{\mathfrak p}.$$ Now an ideal which is locally $0$ or $1$ is generated by an idempotent.

Answer 2

This is a solution based on the representation theory of finite groups indicated in the comment of Kyle Miller.

Let $G$ be the cyclic group of order $n$, generated by an element $g \in G$. Then we can prove that $$f \colon \mathbb CG \overset{\sim}{\to} A,\ g \mapsto x$$ is an isomorphism of $\mathbb C$-algebras. Hence $A$ is semisimple by the Maschke theorem and projective indecomposable $A$-modules are precisely simple $A$-modules. Now let $\zeta_n$ be a $n$-th root of unity and set $$e_i := \frac{1}{n}\sum_{0 \le j < n} \zeta_n^{-ij}g^j$$ for $0 \le i < n$. These are primitive idempotents of $\mathbb CG$ and $e_i\mathbb CG = \mathbb Ce_i$ are simple $\mathbb CG$-modules by the representation theory of finite groups. Conclusion: every projective $A$-module is semisimple and it is direct sum of some of $f(e_i)A = \mathbb Cf(e_i)$.