0

B. Hartley and T.O. Hawkes define the invariant factors of a matrix over a PID, and they also give an algorithm for computing the aforementioned factors.

So far so good, but they also define the invariant factors of a finitely generated module over a PID in terms of its invariant factor decomposition, and I haven't been able to puzzle out the connection between these definitions. If I want to find the invariant factor decomposition of a finitely generated module over PID, how do I turn this problem into a matrix problem so that I can apply the algorithm?

goblin GONE
  • 67,744

1 Answers1

1

Short answer: “A finitely presented module is the cokernel of a matrix.”

A finitely generated module $M$ over a PID $R$ has generators, say $n$ of them. That means that there is some homomorphism $h:R^n \to M$ sending $e_i$ to the $i$th generator. The first isomorphism theorem says that $M \cong R^n / \ker(h)$.

Now $\ker(h)$ is also finitely generated (a fact about PIDs, even noetherian rings) by some formal linear combinations of generators of $M$ that turn out to be 0 in $M$ even though they are not 0 in $R^n$. In other words, there is a homomorphism $f:R^m \to R^n$ so that $\ker(h) = \operatorname{im}(f)$ so that $M \cong R^n / \operatorname{im}(f)$, the so-called cokernel of $f$.

You should find that the invariant factors of the matrix of $f$ are the invariant factors of the cokernel of $f$, $M$.

Jack Schmidt
  • 55,589