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Questions tagged [ac.commutative-algebra]

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

2
votes
2answers
964 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 ...
31
votes
6answers
6k 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*...
19
votes
3answers
2k 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 $...
36
votes
17answers
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]." ...
8
votes
2answers
1k 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 \...
7
votes
2answers
577 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 ...
12
votes
6answers
2k 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 ...
4
votes
1answer
302 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?
14
votes
6answers
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 ...
25
votes
3answers
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 $\...
8
votes
2answers
520 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 ...
15
votes
2answers
1k 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. ...
13
votes
2answers
2k 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 ...
47
votes
7answers
4k 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 ...
2
votes
1answer
194 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->...
14
votes
9answers
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 ...
3
votes
2answers
488 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 ...
47
votes
4answers
5k 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 ...
10
votes
2answers
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 ...
17
votes
3answers
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'...
30
votes
2answers
5k 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 ...
5
votes
3answers
767 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 ...
12
votes
3answers
931 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 ...
32
votes
4answers
2k 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 ...
26
votes
2answers
2k 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). ...
6
votes
3answers
551 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 ...
28
votes
3answers
2k 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 $\...
20
votes
12answers
3k 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 ...
22
votes
5answers
5k views

Can a quotient ring R/J ever be flat over R?

If R is a ring and J⊂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=0. For a less ...
12
votes
5answers
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 ...
11
votes
8answers
3k views

Resources on Invariant Theory

Hi, So my question is pretty much summed up by the summary - 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 ...
5
votes
2answers
645 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 (...
21
votes
5answers
8k 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 $A^m \to A^n$...
9
votes
5answers
607 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 ...
92
votes
5answers
8k 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, "...
10
votes
2answers
2k views

What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?
24
votes
6answers
5k 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 ...
28
votes
4answers
8k 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 ...