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

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0
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2answers
239 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 ...
52
votes
5answers
4k 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, ...
22
votes
9answers
4k 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?
14
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
2k 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/ ...
0
votes
1answer
252 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 ...
11
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2answers
788 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 ...
1
vote
0answers
127 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 ...
42
votes
2answers
3k 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
9answers
7k 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 ...
24
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4answers
3k 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 ...
63
votes
10answers
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 ...
37
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 ...
19
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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 ...
20
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 ...
13
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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 ...
18
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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 ...
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 ...
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 ...
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 ...
28
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 ...
7
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 ...
32
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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 ...
16
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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 ...
15
votes
2answers
2k 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 ...
9
votes
2answers
699 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
1k 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 ...
14
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 ...
14
votes
2answers
544 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 ...
12
votes
2answers
352 views

Do all subtraction-free identities tropicalize?

If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like ...
12
votes
5answers
1k views

An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following: Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace) $$\int_0^{\infty}: ...
9
votes
1answer
367 views

When is the reduced subscheme of a Cohen-Macaulay scheme also Cohen-Macaulay?

Let $X$ be a Cohen-Macaulay scheme (I will be interested in the case when this is ${\rm Spec}(A/I)$ where $A$ is a polynomial ring over a field and $I$ is a homogeneous ideal). I would like to know ...
6
votes
2answers
569 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 ...
27
votes
1answer
2k views

Is every connected scheme path connected?

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following. Let's ...
12
votes
2answers
1k views

Why are injective modules more complicated than projective modules?

For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...
10
votes
5answers
553 views

Is complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$. $$ h_a=\text{ sum of all monomials of degree ...
10
votes
2answers
702 views

About integer polynomials which are sums of squares of rational polynomials…

I have the following question for which I haven't been able to find any reference or proof. Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two ...
8
votes
1answer
3k 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 ...
7
votes
1answer
504 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 ...
7
votes
6answers
1k views

What's an example of a transcendental power series?

Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? I am looking for elementary example (so there should be a proof of transcendence that does ...
6
votes
0answers
346 views

“Consecutive” irreducible polynomials

If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then it is easy to see that for any integer $m$, at least one of the polynomials $P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb ...
6
votes
6answers
2k views

When is a blow-up non-singular?

Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the blow-up $\operatorname{Bl}_{Z}(X)$ non-singular? The blow-up of a non-singular variety along a ...
5
votes
5answers
1k views

Alternative proof of unique factorization for ideals in a Dedekind ring

I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
3
votes
2answers
527 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 $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
1
vote
4answers
1k views

A proof for a statement about polynomial automorphism

I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...
11
votes
6answers
870 views

When can we prove constructively that a ring with unity has a maximal ideal?

Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...