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

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59
votes
5answers
5k 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, ...
44
votes
2answers
4k views

How would you solve this tantalizing Halmos problem?

1-ab invertible => 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? Geometric series. In ...
31
votes
5answers
939 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 ...
0
votes
2answers
249 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 ...
25
votes
9answers
5k 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?
31
votes
5answers
2k 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 ...
16
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 ...
13
votes
6answers
2k views

Duals and Tensor products

Let $A$ be a commutative ring with a unit element. Let $M$ and $N$ be $A$-modules. Let $M^v$ and $N^v$ be the dual modules. In general, do we have $M^v \otimes N^v \cong (M\otimes N)^v$? It is ...
15
votes
5answers
2k 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, ...
9
votes
4answers
3k views

One point in the post of Terence Tao on Ax-Grothendieck theorem

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/ ...
29
votes
2answers
1k 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 ...
4
votes
2answers
659 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 $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
10
votes
0answers
309 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 ...
14
votes
1answer
386 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 ...
11
votes
2answers
876 views

Is every projective $\mathbf{Z}[x]$-module free?

Is every finitely generated projective $\mathbf{Z}[x]$-module free?
6
votes
2answers
1k 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 ...
0
votes
1answer
270 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 ...
4
votes
1answer
194 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 ...
1
vote
0answers
136 views

Descending chain condition for radical ideals

For which integral domains $R$ (not filed) the ring $R[x_1, \ldots, x_n]$ satisfies descending chain condition for radical ideals? I am not expert in Ring Theory and I need an answer to construct some ...
36
votes
9answers
8k 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 ...
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 ...
65
votes
11answers
3k 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 ...
26
votes
3answers
4k 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 ...
39
votes
2answers
3k 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 ...
23
votes
3answers
4k 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 ...
21
votes
6answers
3k 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 ...
28
votes
3answers
3k 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 ...
13
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 ...
19
votes
5answers
2k views

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 ...
26
votes
2answers
1k 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 ...
33
votes
1answer
1k views

Categorical definition of the ideal product within the category of rings

This is an extension of this question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I*J)$ out of $R \to R/I$ and $R \to R/J$ in ...
7
votes
1answer
450 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 ...
17
votes
2answers
3k views

Noetherian rings of infinite Krull dimension?

Since Noetherian rings satisfy the ascending chain condition, every such ring must contain infinitely many chains of prime ideals s.t. the heights of these chains are unbounded. The only example I ...
8
votes
5answers
2k views

When a formal power series is a rational function in disguise

Given a formal power series $f \in k[[X]]$, where $k$ is a commutative field, is there any good way to tell whether or not $f\in k(X)$? Edit: To clarify, "good way to tell" means "computable ...
16
votes
2answers
1k views

Extra principal Cartier divisors on non-Noetherian rings? (answered: no!)

On the way to defining Cartier divisors on a scheme $X$, one sheafifies a presheaf base-presheaf of rings $\mathcal{K}'(U)=Frac(\mathcal{O}(U))$ on open affines $U$ to get a sheaf $\mathcal{K}$ of ...
22
votes
5answers
1k views

Explicit elements of $K((x))((y)) \setminus K((x,y))$

In an answer to the popular question on common false beliefs in mathematics Examples of common false beliefs in mathematics. I mentioned that many people conflate the two different kinds of formal ...
9
votes
2answers
737 views

Complete intersections and flat families

If I have a flat family $f \colon X \to T$ such that some fiber is (locally) a complete intersection, does that imply that there is an open set $U$ in $T$ such that the fibers above $U$ are (locally) ...
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 ...
16
votes
3answers
1k 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 A-algebra ...
15
votes
3answers
1k views

Total ring of fractions vs. Localization

Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a ...
8
votes
5answers
2k views

Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ...
24
votes
5answers
1k views

(Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting. The problem is to produce an example of the following ...
10
votes
4answers
932 views

Flatness of normalization

Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5). What happens if we ...
27
votes
4answers
1k views

Why Cohen-Macaulay rings have become important in commutative algebra?

I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects. I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm ...
16
votes
6answers
1k views

Nonfree projective module over a regular UFD?

What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free? In fact I'll be at least ...
14
votes
2answers
563 views

injectivity of torsion submodules of injectives

Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on ...
10
votes
1answer
618 views

Modules and Square Zero Extensions

Let $R$ be a commutative ring, $RMod$ its category of modules and $CRing$ the category of commutative rings. There's an embedding $RMod \rightarrow CRing/R$ that sends an $R$-module $M$ to the ring ...
8
votes
1answer
656 views

A local ring not a quotient of a regular local ring

In his book Commutative Ring Theory, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional. What is an example of a ...
7
votes
2answers
770 views

Generalization of finitely generated, finitely presented modules?

Let $R$ be a commutative ring and $M$ an $R$-module. The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$. Similarly, $M$ is finitely presented iff there is an ...
1
vote
6answers
9k views

Are submodules of free modules free? [closed]

Are all submodules of free modules free? I would like a reference to a proof or counterexample please.