Questions tagged [ac.commutative-algebra]

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

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Algebraic stacks from scratch [closed]

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
30 votes
2 answers
3k views

Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. (http://en.wikipedia.org/wiki/Peano_axioms#First-...
Russell Easterly's user avatar
30 votes
0 answers
893 views

On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies (Reg) Every finitely generated ideal of $R$ has finite projective dimension. Clearly this gives the usual ...
Laurent Moret-Bailly's user avatar
29 votes
2 answers
5k views

Examples of algebraic closures of finite index

So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices? ...
Andrew Homan's user avatar
29 votes
2 answers
5k views

Regular, Gorenstein and Cohen-Macaulay

All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on; It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
Ehsan M. Kermani's user avatar
29 votes
5 answers
9k views

Local complete intersections which are not complete intersections

The following definitions are standard: An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X_1,\ldots, X_n]$. ...
Adam K's user avatar
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3 answers
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What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear ...
Harry Gindi's user avatar
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29 votes
4 answers
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Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
Tim Campion's user avatar
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2 answers
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Elementary proof of Nakayama's lemma?

Nakayama's lemma is as follows: Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M$...
Phillip Williams's user avatar
28 votes
5 answers
3k views

Does Smith normal form imply PID?

Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain? If this fails, suppose we ...
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28 votes
6 answers
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Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$. I ...
Laurent Lessard's user avatar
28 votes
1 answer
2k views

SOS polynomials with integer coefficients

A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
Gautam's user avatar
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28 votes
3 answers
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Why is "h" the notation for class numbers?

A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\...
KConrad's user avatar
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28 votes
3 answers
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Non finitely-generated subalgebra of a finitely-generated algebra

Ok, I feel a little bit ashamed by my question. This afternoon in the train, I looked for a counter-example: — $k$ a field — $A$ a finitely generated $k$-algebra — $B$ a $k$-subalgebra of $A$ that ...
28 votes
2 answers
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What are applications of commutativity theorems for rings?

Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...
KConrad's user avatar
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28 votes
4 answers
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What are traces?

Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
Anonymous Coward's user avatar
28 votes
3 answers
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Equivalent definitions of invertible modules

Let $R$ be commutative unital ring, and $M$ an $R$-module. $M$ is called invertible (a.k.a. projective module of rank one), if it is finitely generated, and $M_{\mathfrak{p}} \cong R_{\mathfrak{p}}$ ...
eb80's user avatar
  • 523
28 votes
1 answer
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Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
Gorav Jindal's user avatar
28 votes
2 answers
3k views

Maximal Ideals in Formal Laurent Series Rings?

Setup: Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x_1,\ldots,x_n]]$ denote the commutative ring of formal power series over $k$ in $x_1,\ldots,x_n$. We know that there is ...
Ed Letzter's user avatar
28 votes
1 answer
1k views

What are retracts of polynomial rings?

Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring? ...
Todd Trimble's user avatar
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28 votes
2 answers
808 views

$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$

Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)? Edited to add: As no answers are forthcoming, does anyone ...
Pace Nielsen's user avatar
28 votes
0 answers
836 views

The field of fractions of the rational group algebra of a torsion free abelian group

Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions. ...
Jeremy Rickard's user avatar
27 votes
5 answers
3k views

Class number measuring the failure of unique factorization

The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question: Is there a ...
Dror Speiser's user avatar
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27 votes
13 answers
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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 ...
Chris Schommer-Pries's user avatar
27 votes
5 answers
5k views

Why does the (S2) property of a ring correspond to the Hartogs phenomenon?

Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this ...
Yuhao Huang's user avatar
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27 votes
5 answers
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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=...
Anton Geraschenko's user avatar
27 votes
1 answer
2k views

What was commutative algebra before (modern) algebraic geometry?

Reading "H. Matsumura - Commutative Ring theory" I had the impression that the definitions were all made to mean something in algebraic geometry afterwards. I wonder what was commutative algebra ...
João Dos Reis's user avatar
27 votes
2 answers
2k views

A sum involving roots of unity

Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that \begin{align*} \sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}. \end{align*} Since $\...
Chitsai Liu's user avatar
  • 2,153
27 votes
2 answers
1k views

Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
Graham Leuschke's user avatar
26 votes
5 answers
2k views

Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?

The question Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...
Tom Leinster's user avatar
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26 votes
5 answers
14k views

Flat module and torsion-free module

All rings in this question are integral. It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. ...
Liu Hang's user avatar
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26 votes
2 answers
2k views

Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do ...
Pierre-Yves Gaillard's user avatar
26 votes
1 answer
4k views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
urelement's user avatar
  • 363
26 votes
3 answers
2k views

Invariance of $\mathbb{Z}[x]$ under a self-equivalence of the category of commutative rings with 1

Let $\mbox{Rings}$ be the category of commutative rings with $1$. Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ such that $$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
Nico Bellic's user avatar
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 $\...
cdouglas's user avatar
  • 3,083
26 votes
4 answers
3k views

Nilradicals without Zorn's lemma

It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$. Every proof I found (e.g. in the classical "Commutative Algebra" by ...
Daniele Turchetti's user avatar
26 votes
2 answers
2k views

Uniqueness of the "algebraic closure" of a commutative ring

There are several ways to generalize the notion of "algebraic closure" from fields to arbitrary commutative rings. A good overview is On algebraic closures by R. Raphael. I am more ...
Martin Brandenburg's user avatar
26 votes
1 answer
890 views

Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?

I am working on some combinatorics problems. One of my problems leads to the following question: Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \...
The Nguyen's user avatar
25 votes
5 answers
2k views

Exotic principal ideal domains

Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that ...
25 votes
4 answers
3k views

Is a domain all of whose localizations are noetherian itself noetherian ?

Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ? The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory: Let ...
KBuck's user avatar
  • 538
25 votes
2 answers
8k 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 ...
Alon Amit's user avatar
  • 6,414
25 votes
5 answers
3k views

Algebraic description of compact smooth manifolds?

Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
Jason DeVito - on hiatus's user avatar
25 votes
1 answer
4k views

The Rabinowitz Trick

The recent question about problems which are solved by generalizations got me thinking about the Rabinowitz trick, which is used to prove a statement of Hilbert's Nullstellensatz, specifically, the ...
Grant Rotskoff's user avatar
25 votes
5 answers
3k views

is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
John Salvatierrez's user avatar
25 votes
3 answers
2k views

product of all F_p, p prime

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements. Is it true that $R$ has a quotient by a maximal ideal which is a field of ...
Wanderer's user avatar
  • 5,113
25 votes
1 answer
2k views

History of Koszul complex

This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie ...
Sasha Pavlov's user avatar
  • 1,535
25 votes
1 answer
3k views

Underlying structure behind the infamous IMO 1988 Problem 6

This is the infamous Problem 6 from the 1988 IMO which has recently been popularised by the YouTube channel Numberphile: Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{...
user avatar
25 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 ...
24 votes
4 answers
4k views

Serre's theorem about regularity and homological dimension

One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional. I'd like to ask a somewhat ...
Mariano Suárez-Álvarez's user avatar
24 votes
6 answers
5k views

Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")? There are simple formulas describing Pythagorean n-tuples for n=3,4,6: n=3. The formula ...
mikhail skopenkov's user avatar