**116**

votes

**3**answers

6k views

### A Game on Noetherian Rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring $\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ...

**90**

votes

**7**answers

8k views

### How to memorise (understand) Nakayama's lemma and its corollaries?

Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...

**64**

votes

**11**answers

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 ...

**57**

votes

**2**answers

5k 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 ...

**55**

votes

**5**answers

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, ...

**44**

votes

**2**answers

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 ...

**43**

votes

**2**answers

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 $M$ ...

**43**

votes

**5**answers

2k views

### Bizarre operation on polynomials

There I was, innocently doing some category theory, when up popped a totally outlandish operation on polynomials. It seems outlandish to me, anyway. I'd like to know if anyone has seen this ...

**40**

votes

**11**answers

3k views

### How to introduce notions of flat, projective and free modules?

In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...

**39**

votes

**20**answers

9k views

### Errata for Atiyah-Macdonald

Is there a good errata for Atiyah-Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies ...

**39**

votes

**2**answers

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 ...

**39**

votes

**2**answers

2k views

### Is primary decomposition still important?

On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says
In the modern treatment, with its
emphasis on ...

**39**

votes

**7**answers

3k 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 ...

**38**

votes

**7**answers

4k views

### “Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...

**37**

votes

**7**answers

5k views

### Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders ...

**35**

votes

**9**answers

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 ...

**33**

votes

**1**answer

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 ...

**32**

votes

**8**answers

8k views

### Modern algebraic geometry vs. classical algebraic geometry

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...

**32**

votes

**3**answers

2k views

### What does it mean geometrically that an element in a domain is irreducible?

Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...

**31**

votes

**5**answers

889 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 ...

**31**

votes

**1**answer

1k views

### Is the radical of an irreducible ideal irreducible?

I originally posted this to math.stackexchange.com
here. I got a partial answer, but I now suspect that the complete answer is much harder than I thought, so I'm posting it here.
Fix a commutative ...

**30**

votes

**9**answers

4k views

### Why is an elliptic curve a group?

Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the ...

**30**

votes

**3**answers

2k views

### Why do we care whether a PID admits some crazy Euclidean norm?

An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > ...

**30**

votes

**1**answer

1k views

### Is there a “purely algebraic” proof of the finiteness of the class number?

The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching ...

**29**

votes

**5**answers

2k views

### How to think about CM rings?

There are a few questions about CM rings and depth.
Why would one consider the concept of depth? Is there any geometric meaning associated to that? The consideration of regular sequence is okay to ...

**29**

votes

**7**answers

3k views

### What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...

**29**

votes

**5**answers

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 ...

**29**

votes

**2**answers

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 ...

**29**

votes

**4**answers

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 ...

**28**

votes

**3**answers

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 ...

**28**

votes

**3**answers

2k views

### Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And ...

**27**

votes

**1**answer

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 ...

**26**

votes

**2**answers

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 ...

**26**

votes

**3**answers

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 ...

**26**

votes

**2**answers

924 views

### If a field extension gives affine space, was it already affine space?

Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a
field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the
algebraic closure $\overline F$.
Does it ...

**26**

votes

**0**answers

591 views

### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...

**26**

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 ...

**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 ...

**25**

votes

**1**answer

2k views

### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...

**24**

votes

**9**answers

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?

**24**

votes

**5**answers

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 ...

**24**

votes

**4**answers

2k views

### Algebraic machinery for algebraic geometry

Hello everybody,
I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative ...

**24**

votes

**3**answers

2k views

### Are there more Nullstellensätze?

Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. ...

**23**

votes

**4**answers

1k 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 ...

**23**

votes

**1**answer

1k 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 ...

**22**

votes

**11**answers

6k 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 ...

**22**

votes

**3**answers

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 ...

**22**

votes

**3**answers

2k views

### 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 ...

**22**

votes

**3**answers

3k views

### Matrix factorizations and physics

I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...

**22**

votes

**5**answers

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 ...