Modules.

Using the general results on modules over $\mathbb {C}{}[X]$, outline a scheme for proving the theorem on JNF (see [ВА II]).

Using the general results on modules over $\mathbb {Z}$, outline a scheme for proving the theorem on finitely generated abelian groups.

Let $A=P\left[X_{1}, \ldots , X_{n}\right]$ be the ring of polynomials in $n$ variables over a field $P$. A sequence $(f_{1}, \ldots , f_{r})$ of $r$ polynomials $f_{i} \in A$ is called unimodular if $A f_{1}+A f_{2}+\ldots +A f_{r}=A$, i.e.,

for some $u_{i} \in A, 1 \leqslant i \leqslant r$.

Let, further, $V$ be a module of finite type over $A$. In connection with certain subtle questions from algebraic geometry, the French mathematician J.-P. Serre (1955) put forward the conjecture

which was given the following elegant form: every relation (*) can be written as the equality

for suitable $u_{i j} \in A$. This statement, despite its apparent simplicity, was proved only in 1976, independently by A.A. Suslin (Russia) and D. Quillen (USA). Try to carry this out for small $n$ and $r$.

The main idea is to study the action of the group $\mathrm{GL}\left(r, A\left[X_{1}, \ldots , X_{n-1}\right]\right)$ on the set of unimodular sequences and to use induction on $n$. One can become acquainted with the proof from the original article: Suslin A.A.// DAN SSSR. - 1976. - Vol. 229, No. 5. - pp. 1063-1066; or from the talk at N. Bourbaki's seminar: Ferrand D.// Sém. N. Bourbaki, 28 ème annee, 1975/76, Juin 1976. The exposition is quite elementary. At what price this was achieved can be judged from an earlier talk at N. Bourbaki's seminar: Bass H.// Sém. N. Bourbaki, 26 ème annee, 1973/74, Juin 1974. The cited literature contains statements of unsolved problems. This whole range of questions is very good for discussion in a special seminar.