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Results tagged with ac.commutative-algebra
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user 1384
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
20
votes
3
answers
2k
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Which rings are subrings of matrix rings?
In this question, all rings are commutative with a $1$, unless we explicitly say
so, and all morphisms of rings send $1$ to $1$.
Let $A$ be a Noetherian local integral domain. Let $T$ be a non-zero $ …
- 40.3k
13
votes
Atiyah-MacDonald, exercise 2.11
Every few years this question re-appears in my life (as it just did this week) and I have to rediscover the Euler Characteristic proof which I half-remember. Let me just write my thoughts down here so …
- 40.3k
52
votes
Accepted
Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over int...
Yes: this is an old chestnut. Let me write $\oplus_n\mathbf{Z}$ for what you call $\mathbf{Z}[x]$ and $\prod_n\mathbf{Z}$ for what you call $\mathbf{Z}[[x]]$ (all products and sums being over the set …
2
votes
Accepted
Dense section of sheaves of modules
If U is an open set in X, but U isn't X, then there are non-zero sheaves on X whose support lies outside U. Now add O_X to one of these to get a counterexample.
- 40.3k
13
votes
Accepted
Why is the prime spectrum not useful in non-archimedean analytic geometry?
I am surprised that Brian got to this one first without making what I thought was another obvious comment: affinoids are Jacobson rings! A function which is zero at all points of an affinoid rigid spa …
- 40.3k
9
votes
Accepted
Z_p flatness and irreducible components.
Your proof seems wrong to me. I might be misunderstanding some things you wrote, but surely $\mathbf{Q}{}_p=\mathbf{Z}_p[X]/(pX-1)$ is finite type over $\mathbf{Z}_p$, and contains many elements which …
- 40.3k
13
votes
Class number measuring the failure of unique factorization
For Dedekind domains, like the integers of a number field, PID iff UFD. There's definitely a quantitative statement relating the class number to failure of PIDness: the higher the class number, the sm …
- 40.3k
4
votes
Elementary proof that projective space is a quotient
Look at the subspace of $\mathbf{A}^{n+1}$ cut out by your polynomials. This set is invariant under the diagonal action of $k^\times$. So the functions that vanish on it will be an ideal $I$ (the radi …
- 40.3k
7
votes
2
answers
1k
views
Picard group vs class group
The question.
Let $R$ be a commutative ring. Let $M$ be an $R$-module with the property that there exists an $R$-module $N$ such that $M\otimes_R N\cong R$. Does there always exist an ideal $I$ of $R$ …
- 40.3k
44
votes
A game on Noetherian rings
I computed the nimbers of a few rings, for what it's worth. I don't see any sensible pattern so perhaps the general answer is hopelessly hard. This wouldn't be surprising, because even for very simple …
- 40.3k
0
votes
Is tensoring with a module representable iff it is locally free of finite rank?
Contrary to what I guessed initially, I now think the question has a great answer: the functor is representable if and only if $M$ is locally free, and the proof is EGA I, 9.4.10.
Edit: this is an an …
- 40.3k
0
votes
Is tensoring with a module representable iff it is locally free of finite rank?
Here is an example where representability fails. If $R$ is an $A$-algebra representating $\otimes_AM$ on $A$-algebras, and if $B\to C$ is an injective map of $A$-algebras, then $R(B)\to R(C)$ will be …
- 40.3k
6
votes
Accepted
Can different modules have the same symmetric algebra? (answered: no)
I now believe a-fortiori's argument: translations are a problem, but, as a-fortiori observed, they are the only problem. Let me spell it out.
Say $f:Sym(M)\to Sym(N)$ is an isomorphism. For $m\in M$ …
- 40.3k
0
votes
Is (relatively) algebraically closed stable under finite field extensions?
Let me have a punt at this. $F$ alg closed inside $F'$ iff $\overline{F}\otimes_FF'$ is a field, right? So now it's easy because $\overline{L}$ is an algebraic closure of $F$, and I don't think I even …
- 40.3k
7
votes
Additive commutators and trace over a PID
Every matrix with trace zero over a PID is a commutator, according to the MR review of
Rosset, Myriam(IL-BILN); Rosset, Shmuel(IL-TLAV)
Elements of trace zero that are not commutators.
Comm. Algebra …
- 40.3k
21
votes
Accepted
Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p...
I don't think so. Consider the $\mathbb{Z}$-module $M$ be the additive subgroup of the rationals consisting of rationals with square-free denominator.
- 40.3k
5
votes
Infinite collection of elements of a number field with very similar annihilating polynomials
The answer "is" that the smallest $r$ is what it is, and what it is could well depend on $\alpha$. Let me also raise the possibility that there might be no simple "formula" relating $r$ to $\alpha$. T …
- 40.3k
9
votes
product of all F_p, p prime
How far does the logic route get you? Here's a plan of attack. Say we have such a max ideal $m$.
1) If $S$ is a subset of the primes then there's an idempotent $e_S$ in $R$, which is 1 at $p$ if $p\i …
- 40.3k
5
votes
Errata for Atiyah–Macdonald
The following slip on p82 was found by Kenny Lau when he was formalising Prop 7.8 in Lean: In the line "Substituting (1) and making repeated use of (2) shows that each element of C is..." there's an i …
2
votes
Accepted
A question about the invariants of a finite group
I think it's irreducible in $R$ as you suggest. Here's a sketch which I think works. If the polynomial factored in a non-trivial way, then because of the $t^n$ term the factors must have degree less t …
- 40.3k