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Results tagged with ac.commutative-algebra
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user 345
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
11
votes
Different definitions of the dimension of an algebra
In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian r …
- 3,545
8
votes
1
answer
1k
views
Primes in a (commutative) Jacobson ring
Recall that a commutative ring is Jacobson if every prime ideal is the intersection of the maximal ideals that contain it.
In the exercises of a commutative algebra course I gave I asked the student …
- 3,545
8
votes
3
answers
907
views
Generic Noether normalisation
Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional …
- 3,545
7
votes
Different definitions of the dimension of an algebra
Often the most useful dimension in non-commutative algebra is the length of the minimal injective resolution of the ring as a module over itself. In many important cases this is the same as the global …
- 3,545
6
votes
Maximal Ideals in Formal Laurent Series Rings?
I think that the question difficult as illustrated by Hailong's answer. I suspect that it will be hard to even find a nice parameterisation of the $H$-orbits of maximal ideals in your refined question …
- 3,545
3
votes
Accepted
Is the pair $(C([0 \;1]),\mathbb{C})$ a consecutive pair?
The answers to the question prime ideals in C([0,1]) explain why the answer is no, assuming that Yemon Choi was mistaken in his answer in believing that you are only interesting in continuous homomorp …
- 3,545
3
votes
Generic Noether normalisation
In case anyone else has the same question and discovers this page I have just found a more explicit reference for this result: Remark 3.4.4 of A Singular introduction to commutative algebra by Greuel …
- 3,545
3
votes
1
answer
922
views
How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $ …
- 3,545
2
votes
A ring such that all projectives are stably free but not all projectives are free?
Example 1.2.2 in Chapter 1 of Weibel's book in progress on K-theory http://www.math.rutgers.edu/~weibel/Kbook.html says that $R_2$ in the notation of your question has a stably free module that is not …
- 3,545