All Questions
665 questions
19
votes
3
answers
3k
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 ...
- 63.1k
19
votes
2
answers
742
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 min(min($x+...
- 19.7k
19
votes
5
answers
7k
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 ...
19
votes
3
answers
1k
views
How to construct a constructive proof from a non-constructive proof using prime ideals?
The sum of two nilpotent elements of a commutative ring is nilpotent. This can be checked by a direct calculation using the binomial theorem. In fact, this calculation shows the stronger statement $x^...
- 5,482
19
votes
2
answers
5k
views
Characterizations of UFD and Euclidean domain by ideal-theoretic conditions
This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ...
- 7,217
19
votes
2
answers
566
views
Ostrowski's Theorem for topological rings?
Ostrowski's theorem classifies all absolute values on a number field $K$.
Questions:
More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?
In ...
- 64k
19
votes
2
answers
5k
views
Are morphisms from affine schemes to arbitrary schemes affine morphisms?
To put this question in precise language, let $X$ be an affine scheme, and $Y$ be an arbitrary scheme, and $f : X \rightarrow Y$ a morphism from $X$ to $Y$. Does it follow that $f$ is an affine ...
- 387
19
votes
3
answers
4k
views
Generalized Euler phi function
Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\...
- 337
18
votes
4
answers
4k
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 ...
- 3,704
18
votes
5
answers
8k
views
Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
- 17.9k
18
votes
3
answers
1k
views
The isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$
In a recent conversation with a colleague, the following question arose:
What is the isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$? That is to say, what is $...
- 323
18
votes
4
answers
1k
views
Bass' stable range condition for principal ideal domains
In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v_1, \ldots, v_n$ that generate the unit ideal there are numbers $r_1, \ldots ...
- 245
18
votes
2
answers
1k
views
Equivalence of "Weyl Algebra" and "Crystalline" definitions of rings of differential operators between modules?
Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...
user61522
18
votes
3
answers
2k
views
If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?
Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that ...
- 3,245
18
votes
5
answers
2k
views
Is a complete homogeneous symmetric polynomial irreducible?
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 }...
- 446
18
votes
3
answers
2k
views
Alternate proofs of Hilberts Basis Theorem
I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.
If $R$ is a noetherian ring, then so is $R[X]$.
or its sister version
If $R$ is a noetherian ...
18
votes
9
answers
2k
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 ...
- 118k
17
votes
1
answer
2k
views
Geometric interpretation of filtered rings and modules
Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...
- 13.2k
17
votes
1
answer
2k
views
Non-isomorphic rings that are localizations of each other
Do there exist commutative rings $A$ and $B$ and multiplicative subsets $S\subseteq A$, $T\subseteq B$ such that $A\not\simeq B$ but $S^{-1}A \simeq B$ and $T^{-1} B\simeq A$?
This question comes ...
- 622
17
votes
1
answer
2k
views
Composing left and right derived functors
I would appreciate either an explanation or a reference for what is going on here.
Motivation:
Let $f : X \rightarrow Y$ be a morphism of algebraic varieties. The derived projection formula implies ...
- 1,209
17
votes
1
answer
8k
views
Different definitions of the rank of a module
I have seen different definitions of a rank of a module $M$ over a commutative ring $R$.
Here in nLab, for quite general modules, the rank is defined locally at $p\in \mathrm{Spec}(R)$ as the ...
- 1,128
16
votes
3
answers
797
views
For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $...
- 563
16
votes
4
answers
3k
views
Is there a simple method to test a local ring to be Cohen Macaulay?
Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. ...
- 1,207
16
votes
1
answer
2k
views
Does ZF prove that all PIDs are UFDs?
Main Question:
Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain?
The proofs I've seen all use dependent choice.
Minor Questions:
Does ZF + ...
user5810
16
votes
2
answers
2k
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 "...
- 11.3k
16
votes
4
answers
3k
views
Cardinal of maximal linearly independent subsets of a free module
Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero ...
- 2,857
16
votes
1
answer
733
views
Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 1,615
16
votes
2
answers
1k
views
The symmetric monoidal category of finite sets
It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite ...
- 63.1k
16
votes
2
answers
2k
views
Witt-vector vectors
I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...
- 12.9k
15
votes
5
answers
4k
views
Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension
Let $(A,m_A)$ and $(B,m_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}...
- 1,609
15
votes
6
answers
1k
views
An example of a series that is not differentially algebraic?
Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
- 16.9k
15
votes
2
answers
881
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 ...
- 6,700
15
votes
2
answers
2k
views
Is every poset the poset of prime ideals of a ring?
The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.
My question was inspired from ...
- 899
15
votes
6
answers
2k
views
Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
- 65.4k
15
votes
1
answer
2k
views
Automorphisms of $P(\Bbb N)$
I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...
- 1,445
15
votes
2
answers
1k
views
Exact sequence of monoids
What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
http://www.math.ucla.edu/~balmer/Pubfile/...
- 3,009
14
votes
2
answers
921
views
Why is the symmetric monoidal structure on invertible modules strict?
Let $N$ be an object in a symmetric monoidal category. Then the braid map $N\otimes N\to N\otimes N$ is almost never the identity, and this is the obstruction to making a symmetric monoidal category ...
- 31.2k
14
votes
2
answers
3k
views
Maximal ideal and Zorn's lemma
It is known that any nonzero ring A (say commutative with 1) has a maximal ideal. The proof uses Zorn's lemma.
Now I heard some people saying that if we assume A to be noetherian, then we don't need ...
- 1,271
14
votes
0
answers
603
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
14
votes
2
answers
1k
views
Economical hard word problem
Can anyone give me an example of a very simple word problem, where by "simple" I mean that it has very few generators and relations, that is nevertheless insoluble. To make the question easier, I am ...
- 29k
14
votes
4
answers
2k
views
Applications of Govorov-Lazard Theorem?
I asked this question on SE a long time ago, but never received an answer:
The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely ...
- 16.2k
14
votes
1
answer
641
views
First order decidability of rings vs Diophantine decidability
Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that:
$\bullet$ the first order theory of $R$ is undecidable, but
$\bullet$ the positive existential (= ...
- 19.6k
14
votes
5
answers
995
views
How can I write down polynomial relations that define when a polynomial is a square?
It's easy to tell when a polynomial is squarefree (or not): that's just the question of the vanishing of the discriminant, which can be dealt with as the resultant of $f$ and $f'$. However, given a ...
- 16k
14
votes
2
answers
984
views
Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
- 37.8k
14
votes
3
answers
2k
views
Intuition for Model Theoretic Proof of the Nullstellensatz
I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
- 15.4k
13
votes
1
answer
7k
views
Chinese Remainder Theorem for rings: why not for modules?
This is a followup to Analog to the Chinese Remainder Theorem in groups other than Z_n. . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese ...
- 33.8k
13
votes
1
answer
1k
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 ...
- 10.7k
13
votes
1
answer
228
views
Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 85.6k
13
votes
1
answer
3k
views
When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
13
votes
1
answer
797
views
Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent?
Let $\mathbb{Q}_p$ denote the field of fractions of $\mathbb{Z}_p$. By the answers to this quesition the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ cannot be a Noetherian ring (...
- 3,784