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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

21 votes
4 answers
4k views

Why are finitely generated modules over principal artin local rings direct sums of cyclic mo...

I am looking for a proof of the following fact: If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules. (Apparently such rings $ …
Keenan Kidwell's user avatar
16 votes
Accepted

Chinese Remainder Theorem for rings: why not for modules?

The second result you're talking about is also sometimes called the Chinese remainder theorem, and can be derived from the Chinese remainder theorem for rings by "tensoring the CRT isomorphism" with $ …
Keenan Kidwell's user avatar
6 votes
Accepted

discrete valuation ring and ring of witt vectors

I think this is possible, if I understand your question correctly. If $R$ is a discrete valuation ring of mixed characteristic $(0,p)$ with residue field $k$ and maximal ideal $pR$, then $R$ is a Cohe …
Keenan Kidwell's user avatar
3 votes
Accepted

Pontryagin dual

Let $\Lambda=\mathbf{Z}_p[[T]]$. The group $M[p]$ is dual to $M^\vee/pM^\vee$. So if $M[p]$ is finite, then $M^\vee/pM^\vee$ is finite, and the form of Nakayama's lemma applicable to profinite modules …
Keenan Kidwell's user avatar
3 votes
Accepted

Finitely Generated Commutative Z-algebra

This follows from the Nullstellensatz (or the version of it), which says that a finitely generated algebra over a Jacobson ring is Jacobson. This can be found in, for example, Eisenbud's book on commu …
Keenan Kidwell's user avatar