Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
39
votes
3
answers
8k
views
What is the "intuition" behind "brave new algebra"?
Y.I. Manin mentions in a recent interview
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
- 10.8k
6
votes
4
answers
2k
views
Is a torsion-free abelian group finitely generated, if all of its localizations at primes $p$ are finitely generated over $\mathbb{Z}_p$?
Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{...
- 1,527
5
votes
2
answers
770
views
Integrally closed factor rings and projective modules
I have a weird vision that comes from reading a paper by Raphael and Desrochers..
Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is ...
- 2,275
10
votes
2
answers
2k
views
Primary decomposition for modules
I am quite curious about the definition and applications of the primary decomposition for modules.
The definition of a primary submodule. (Let's assume we work over a commutative noetherian ring $R$ ...
user709
26
votes
2
answers
9k
views
Maximal ideals in the ring of continuous real-valued functions on ℝ
For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
- 6,734
1
vote
2
answers
1k
views
An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
- 1,801
35
votes
6
answers
9k
views
Do convolution and multiplication satisfy any nontrivial algebraic identities?
For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication -
$$(f\cdot g)(x) = f(x)g(x),$$
and convolution -
$$(f*...
- 5,992
22
votes
3
answers
3k
views
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 $...
- 41.4k
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]."
...
9
votes
3
answers
2k
views
Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 2,275
8
votes
2
answers
759
views
Can any countably generated k-algebra occur as the ring of global sections of some variety?
In the answer to this question we saw that there exists a nonsingular quasi-projective threefold over a field with non-finitely generated global sections.
I was talking about this previous ...
- 8,681
15
votes
7
answers
4k
views
Different definitions of the dimension of an algebra
I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...
- 118k
4
votes
1
answer
310
views
Solvable subgroups of groups of polynomial automorphisms
Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
- 41
16
votes
6
answers
1k
views
Solving polynomial equations when you know in which number field the solutions live
Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
- 28.1k
26
votes
3
answers
2k
views
When does the converse to Schur's Lemma hold?
Let $R$ be a commutative ring, let $A$ be an $R$-algebra, and let $M$ be an $A$-module. If $M$ is simple, then End$_{A-mod}(M)$ is a division ring.
A common use is when $R$ is the complex numbers $\...
- 3,103
8
votes
2
answers
625
views
Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
- 12.6k
17
votes
2
answers
2k
views
How much theory works out for "almost commutative" rings?
I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. ...
- 16k
17
votes
2
answers
5k
views
Exactness of filtered colimits
Are filtered colimits exact in all abelian categories?
In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase ...
- 1,500
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 1,056
2
votes
1
answer
226
views
Are non-maximal orders in number fields Grothendieck rings?
Recall that a ring homomorphism A->B is geometrically regular if for all primes p of A, the fiber of B over p is geometrically regular over k(p). A Grothendieck ring (or, G-ring) is one for which A_p->...
- 5,062
18
votes
9
answers
2k
views
What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 118k
4
votes
2
answers
758
views
What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
- 25.8k
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 11.2k
12
votes
2
answers
1k
views
Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 121
20
votes
3
answers
2k
views
Simple example of a ring which is normal but not CM
I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
- 156k
34
votes
2
answers
7k
views
What is the geometric meaning of integral closure?
More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its ...
- 118k
6
votes
3
answers
1k
views
Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 16k
17
votes
3
answers
1k
views
R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
- 273
41
votes
5
answers
3k
views
Are submersions of differentiable manifolds flat morphisms?
Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion?
Recall that ...
- 5,343
32
votes
2
answers
3k
views
Graded local rings versus local rings
A lot of times I see theorems stated for local rings, but usually they are also true for "graded local rings", i.e., graded rings with a unique homogeneous maximal ideal (like the polynomial ring). ...
- 10.7k
8
votes
3
answers
921
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
36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
- 10.5k
27
votes
13
answers
4k
views
Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...
- 27.5k
28
votes
5
answers
9k
views
Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
16
votes
5
answers
2k
views
Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?
Background: the Hochschild homology of an associative algebra is the homology of the complex
$$ \ldots \longrightarrow A \otimes A \otimes A \longrightarrow A \otimes A \longrightarrow A$$
where ...
- 7,800
20
votes
10
answers
7k
views
Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
6
votes
2
answers
673
views
are deformations of torsion modules always torsion?
Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$.
Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
- 44.7k
33
votes
5
answers
13k
views
Atiyah-MacDonald, exercise 2.11
Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to A^n$...
- 1,098
10
votes
5
answers
632
views
is there a good computer package for working with bicomplexes?
I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
- 44.7k
120
votes
5
answers
13k
views
What do epimorphisms of (commutative) rings look like?
(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "...
20
votes
4
answers
4k
views
What is interesting/useful about Castelnuovo-Mumford regularity?
What is interesting/useful about Castelnuovo-Mumford regularity?
- 10.5k
32
votes
6
answers
9k
views
What is the universal property of normalization?
What is the universal property of normalization? I'm looking for an answer something like
If X is a scheme and Y→X is its
normalization, then the morphism
Y→X has property P and any ...
37
votes
4
answers
12k
views
Finite extension of fields with no primitive element
What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...