All Questions
Tagged with big-list ac.commutative-algebra
18 questions
4
votes
0
answers
95
views
List of equivalent conditions for the invariant subalgebra to be polynomial
Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
- 3,318
2
votes
1
answer
318
views
Property of a commutative ring that is determined by the prime ideals of the ring
Robert Gilmer, in his paper "Commutative rings in which each prime ideal is principal", says:
Some well known theorems indicate that certain ideal-theoretic structure properties of a ...
- 1,355
34
votes
8
answers
4k
views
Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
5
votes
0
answers
788
views
Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
5
votes
3
answers
810
views
Update on "Hopf algebras: their status and pervasiveness" by Hazewinkel
Hazewinkel wrote this article in 2005. Perhaps it's time for an update.
For example, updating item
34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...
27
votes
3
answers
1k
views
Graded analogues of theorems in commutative algebra
Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...
18
votes
3
answers
2k
views
Alternate proofs of Hilberts Basis Theorem
I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.
If $R$ is a noetherian ring, then so is $R[X]$.
or its sister version
If $R$ is a noetherian ...
12
votes
7
answers
1k
views
Properties of rings that have an elegant description in terms of the associated category of modules
Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...
1
vote
0
answers
282
views
Computation of Quillen K-groups for mod R
The recent paper K-Groups for Rings of Finite Cohen-Macaulay Type by H. Holm allows us to compute the Quillen $K$-group $K_1(\text{mod}\hspace{.1 cm}R)$ as a quotient of the abelianization of the ...
19
votes
3
answers
2k
views
Using schemes to prove things about rings
I apologize for asking a big list question, I've tried to avoid doing so for a while. I'll give my justification in a moment.
The question is as follows:
What are examples of strict applications ...
23
votes
6
answers
2k
views
Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
15
votes
4
answers
1k
views
What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 5,846
101
votes
31
answers
29k
views
Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
6
votes
2
answers
1k
views
Gaining intuition for how submodules behave
I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
- 2,830
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
72
votes
14
answers
22k
views
Elementary / Interesting proofs of the Nullstellensatz
Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...
84
votes
31
answers
70k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...