All Questions
612 questions
4
votes
5
answers
2k
views
What properties define open loci in families?
This question is somehow related to the question What properties define open loci in excellent schemes?.
Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over ...
- 16.1k
3
votes
1
answer
280
views
Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?
Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
- 2,851
3
votes
0
answers
407
views
Extending reals with logarithm of zero: properties and reference request
If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...
- 10.1k
3
votes
2
answers
164
views
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
- 10.5k
2
votes
1
answer
348
views
Rings such that all quotients by prime ideals are PIDs?
Let $R$ be a commutative ring such that for every prime ideal $P$ of $R$, the ring $R/P$ is a PID. Do you know how these rings are called or another characterization of them?
I know there are a lot ...
- 321
2
votes
1
answer
205
views
Deformation of isolated singularities and non zero divisors
Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
2
votes
0
answers
238
views
A special type of ideals
I am looking for some references that contained a study of ideals with the following *-property:
Let $I $ be an ideal of a commutative ring with ideantity. The ideal $I $ has the *-property if $I\...
- 61
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
1
vote
1
answer
388
views
Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding
The two sets are, of course, supposed infinite.
This question is related to that one
Commutation of tensor products with inverse limits in a specific case
where it received a (partial) answer ($A$ ...
- 3,738
1
vote
0
answers
274
views
Does analytic isomorphism imply local isomorphism?
If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
- 912
0
votes
1
answer
347
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[[...
- 1,209
0
votes
1
answer
473
views
A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
- 2,837
226
votes
4
answers
16k
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 ...
- 148k
222
votes
8
answers
35k
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 ...
- 14.3k
75
votes
9
answers
17k
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 ...
- 9,366
74
votes
3
answers
7k
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 ...
- 65.4k
72
votes
14
answers
22k
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 ...
70
votes
2
answers
9k
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 ...
- 44.7k
62
votes
5
answers
10k
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 ...
- 11.2k
59
votes
4
answers
12k
views
Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...
- 63.1k
50
votes
0
answers
2k
views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 39.7k
48
votes
4
answers
4k
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]$. ...
- 65.4k
47
votes
2
answers
5k
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) > 0$,...
- 65.4k
46
votes
4
answers
8k
views
What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
- 11.2k
45
votes
2
answers
3k
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 ...
- 63.1k
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 63.9k
44
votes
4
answers
3k
views
Do rings of smooth functions differ from rings of continuous functions?
Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
- 6,962
42
votes
4
answers
8k
views
Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 30.5k
42
votes
4
answers
3k
views
What is the Krull dimension of the ring of holomorphic functions on a complex manifold?
Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$
My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?
...
- 47.5k
41
votes
5
answers
3k
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 ...
- 5,343
40
votes
1
answer
2k
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 ...
- 401
40
votes
1
answer
3k
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 ...
- 47.5k
39
votes
2
answers
6k
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 ...
- 30.5k
39
votes
3
answers
8k
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 ...
- 10.8k
38
votes
2
answers
6k
views
Over which fields are symmetric matrices diagonalizable ?
The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof:
Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...
- 567
37
votes
3
answers
3k
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 ...
- 47.5k
36
votes
3
answers
2k
views
The roots of unity in a tensor product of commutative rings
For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...
- 4,535
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...
35
votes
6
answers
9k
views
Do convolution and multiplication satisfy any nontrivial algebraic identities?
For (suitable) real- or complex-valued functions $f$ and $g$ on a (suitable) abelian group $G$, we have two bilinear operations: multiplication -
$$(f\cdot g)(x) = f(x)g(x),$$
and convolution -
$$(f*...
- 5,992
34
votes
4
answers
5k
views
What is the right definition of the Picard group of a commutative ring?
This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
- 65.4k
33
votes
2
answers
7k
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 ...
- 331
33
votes
5
answers
13k
views
Atiyah-MacDonald, exercise 2.11
Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to A^n$...
- 1,098
32
votes
4
answers
2k
views
Do there exist non-PIDs in which every countably generated ideal is principal?
The title pretty much says it all: suppose $R$ is a commutative integral domain such that every countably generated ideal is principal. Must $R$ be a principal ideal domain?
More generally: for ...
- 65.4k
31
votes
2
answers
2k
views
Should Krull dimension be a cardinal?
A totally ordered finite set $\quad \mathcal P_0 \varsubsetneq \mathcal P_1\varsubsetneq \dots \mathcal \varsubsetneq \mathcal P_n \quad$ of prime ideals of a ring $A$ is said to be a chain of ...
- 47.5k
31
votes
1
answer
2k
views
Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?
This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
- 4,897
31
votes
8
answers
21k
views
Reference book for commutative algebra
I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:
More comprehensive than Atiyah–Macdonald
More readable than Matsumura (maybe better ...
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
29
votes
2
answers
2k
views
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 ...
- 50.6k
28
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 $\...
- 2,183
28
votes
5
answers
9k
views
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=...
