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

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41
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5answers
4k views

Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot. One thing that strikes me ...
73
votes
5answers
6k 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, "...
35
votes
6answers
1k views

On the universal property of the completion of an ordered field

I have been trying to write up some notes on completion of ordered fields, ideally in the general case (i.e., not just completing $\mathbb{Q}$ to get $\mathbb{R}$ but considering the completion via ...
18
votes
12answers
2k 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 ...
0
votes
2answers
278 views

Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
57
votes
2answers
5k views

How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? ...
40
votes
9answers
11k views

Irreducibility of polynomials in two variables

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper. Does anyone know of similar results in the same vein? How about ...
74
votes
2answers
7k views

When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...
30
votes
12answers
9k 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 ...
67
votes
1answer
5k views

A short proof for $\dim(R[T])=\dim(R)+1$

For a commutative ring $R$ we clearly have $\dim(R[T]) \geq \dim(R)+1$. If $R$ is noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
16
votes
5answers
3k 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 ...
20
votes
2answers
2k views

Criteria for irreducibility of polynomial

If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible? Thank you very much, best
16
votes
5answers
3k views

Noether's normalization lemma over a ring A

Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, ...
31
votes
2answers
2k views

Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation. The question is in two parts. The first, as stated ...
15
votes
2answers
2k views

Ideals of the ring of smooth functions

The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
13
votes
4answers
3k views

A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/ ...
25
votes
2answers
3k 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? ...
14
votes
1answer
5k views

Rank of a module

What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...
9
votes
4answers
2k views

When are dual modules free?

Let $A$ be a commutative integral domain, with fraction field $K$. Let $T$ be a torsion-free finitely generated $A$ module, so $T \otimes_A K$ is a finite dimensional vector space $V$. Let $T^*$ be ...
12
votes
1answer
1k views

Questions about spectra of rings of continuous functions

I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
9
votes
0answers
417 views

What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory? In my view, that means it should start off with unpunctured surfaces (and in fact,...
6
votes
1answer
378 views

dual of Z^I for uncountable I

Let $I$ be an infinite set. There is a homomorphism of abelian groups $\mathbb{Z}^{(I)} \to \hom(\mathbb{Z}^I,\mathbb{Z})$ which sends the basis element $e_i$ to the projection $p_i$. If $I$ is ...
16
votes
3answers
856 views

Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
11
votes
0answers
395 views

What is the state of art in Groebner bases

How big polynomial systems can we deal with? How do you know when you don't even have to try? Motivation: Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
8
votes
3answers
2k views

Zero divisor conjecture and idempotent conjecture

Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$. The wiki ...
6
votes
3answers
763 views

Are epimorphisms from a division ring isomorphisms ?

According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67: If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in ...
6
votes
2answers
792 views

von neumann algebras and measurable spaces

I've read some pages on links between von neumann (VN) algebras and measurable spaces (Spectra of $C^*$ algebras and Non-commutative geometry from von Neumann algebras?), but I can't get the following:...
4
votes
2answers
928 views

Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
16
votes
1answer
580 views

Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular? I mean by regular that the localization at every prime ideal is a regular local ...
9
votes
1answer
533 views

Homomorphisms from powers of Z to Z

I believe it is known that if I is a set of non-measurable cardinality, then any homomorphism $Z^I\to Z$ factors through a finite power. Here $Z$ is the group of integers. Can anyone give a ...
4
votes
1answer
296 views

an algebraic variety for a boolean circuit

There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$. I mean there is polynomial reduction $F$ such that for every boolean ...
4
votes
1answer
281 views

Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...
0
votes
1answer
286 views

Iwasawa invariants

Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are $F_p[[...
5
votes
1answer
196 views

Assuming $depth M\ge depth N$, what can one say about $depth M_p$ and $depth N_p$?

Let $(R,m)$ be a Noetherian local ring􀀀, $M$ and $N$ finite $R$-modules, $p$ a prime ideal,􀀀 and $I$ an ideal such that $IM\neq M$. Definition: The common length of the maximal $M$-sequences in $I$...
3
votes
2answers
217 views

Divisibility of the degree of an extension by the degree of its residual field

Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and $...
45
votes
2answers
2k views

a categorical Nakayama lemma?

There are the following Nakayama style lemmata: (the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...
50
votes
2answers
4k views

What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?

One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...
33
votes
7answers
4k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{k!l!}...
22
votes
6answers
4k views

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 ...
32
votes
3answers
5k 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 ...
70
votes
11answers
4k views

Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
35
votes
3answers
4k 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 ...
26
votes
4answers
4k views

Flatness and local freeness

The following statement is well-known: $A$ a commutative Noetherian ring, $M$ a finitely generated $A$-module. Than $M$ is flat if and only if $M_{\mathfrak{p}}$ is free for all $\mathfrak{p}$. My ...
28
votes
3answers
7k views

When is the product of two ideals equal to their intersection?

Consider a ring $A$ and an affine scheme $X=SpecA$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union $V(...
14
votes
9answers
1k 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 ...
28
votes
10answers
7k views

Rings in which every non-unit is a zero divisor

Is there a special name for the class of (commutative) rings in which every non-unit is a zero divisor? The main example is $\mathbf{Z}/(n)$. Are there other natural or interesting examples?
19
votes
5answers
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To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field?

In his answer to a question about simple proofs of the Nullstellensatz (Elementary / Interesting proofs of the Nullstellensatz), Qiaochu Yuan referred to a really simple proof for the case of an ...
18
votes
2answers
4k 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 ...
10
votes
1answer
821 views

geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring. What is the geometric interpretation of: 1- Gorenstein rings 2- Complete intersections 3- Regular rings? and how can I realize differences by geometric ...
30
votes
2answers
2k views

How can I define the product of two ideals categorically?

Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms ...