All Questions
1,947 questions with no upvoted or accepted answers
111
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0
answers
17k
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A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...
- 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
46
votes
0
answers
1k
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 ...
- 3,249
31
votes
0
answers
1k
views
On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
- 19.6k
29
votes
0
answers
875
views
The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...
- 35.2k
23
votes
0
answers
682
views
CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
- 17.7k
18
votes
0
answers
382
views
Deforming a basis of a polynomial ring
The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...
- 27.8k
17
votes
0
answers
1k
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monomorphisms and epimorphisms of local rings
I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.
...
- 63.1k
16
votes
0
answers
448
views
Are dualizable modules finitely generated?
Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...
- 161
15
votes
0
answers
376
views
Reducible polynomials of the shape $f(t^2)$, where $f$ is irreducible
Let $f(x) \in \mathbb{Z}[x]$ be a monic, irreducible polynomial. What are necessary and sufficient conditions for $g(t) = f(t^2)$ to be reducible over $\mathbb{Q}$?
For instance, if $f(x) = x-1$ then $...
- 26.9k
15
votes
0
answers
720
views
If a polynomial ring is a finite flat module over some subring, is that subring itself a polynomial ring?
A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring ...
- 6,612
15
votes
0
answers
718
views
Bloch-Kato conjecture and Wiles' numerical criterion
I already asked this question some days ago on https://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.
In the ...
- 16.2k
14
votes
0
answers
601
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
0
answers
432
views
Surprisingly only real points on intersection of certains quadrics
Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by
\begin{align}
X_e &= 0\\
X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\
X_g &...
- 22.5k
14
votes
0
answers
821
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
14
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0
answers
1k
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Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
14
votes
0
answers
1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
14
votes
0
answers
567
views
(When) is isomorphism on differentials enough to guarantee that a map is étale?
I'm sorry if this is too easy for MO.
Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...
- 16.1k
14
votes
0
answers
956
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 ...
- 8,597
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
13
votes
0
answers
260
views
Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
13
votes
0
answers
748
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
13
votes
0
answers
542
views
When does the product equal the sum?
Let $R$ be a commutative ring with identity and $R^n$ be the direct sum of $R$. Find all $a_1, a_2, \cdots, a_n \in R$ such that $$a_1 + a_2 + \cdots + a_n = a_1a_2\cdots a_n,$$
or, in other words, if ...
- 1,549
13
votes
0
answers
501
views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
- 6,612
13
votes
0
answers
474
views
Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
- 8,717
13
votes
0
answers
495
views
Are the supports of $Ext^i(M,N)$ eventually periodic?
Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules.
Question: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic ...
- 30.5k
12
votes
0
answers
254
views
When do (or don't) residue fields generate the derived category of a ring?
Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in ...
- 3,785
12
votes
0
answers
295
views
Connections in terms of tangent ($\infty$-)categories?
Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, ...
user20948
12
votes
0
answers
433
views
Abelian categories have become the language of homological algebra. Why haven't Zariski categories become the language of commutative algebra?
I'm not seeing much mention of Zariski categories in the literature. There is no article on Zariski categories in nLab, which would seem to be an obvious place to have such an article. What has ...
- 629
12
votes
0
answers
3k
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Algebraic Closure of the field of rational functions
Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is $\...
- 1,891
12
votes
0
answers
288
views
How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?
One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
- 1,554
12
votes
0
answers
2k
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Finding ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer
As the title says, I'm trying to find ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer. I suspect $(2^k, 2^{k-1} x, 2^{k-2} x^2, ..., x^k)$ is generated by no fewer than $n=k+1$ ...
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
11
votes
0
answers
615
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
11
votes
0
answers
333
views
"Small" zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$
If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
- 23k
11
votes
0
answers
629
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Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
11
votes
0
answers
324
views
Subrings of invariants in divided power algebras
I am wondering to what extent the functors "ring of invariants under a group action $G$"
and "divided power envelope with respect to a $G$-stable ideal" commute.
To be precise, let $R$ be a ...
- 1,609
11
votes
0
answers
1k
views
Reverse mathematics strength of identically zero polynomials are the zero polynomial
According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
user5810
10
votes
0
answers
189
views
Is every UFD a filtered colimit of Noetherian UFDs?
I'm wondering how one could prove or disprove that any non-Noetherian UFD is a filtered colimit of Noetherian UFDs. This would allow for some absolute Noetherian approximation to be applied for ...
- 804
10
votes
0
answers
443
views
History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring
This is a repost. So far, I've received no answers on HSM Stack Exchange; maybe I do in MO.
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (for a ...
10
votes
0
answers
194
views
Singularity category of a hypersurface associated to $M_{11}$
For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
- 16.2k
10
votes
0
answers
195
views
Local cohomology and residues of rational functions at 0 and $\infty$
Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where
$s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector
space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...
- 50.8k
10
votes
0
answers
452
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
10
votes
0
answers
474
views
Is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p$ coherent?
The question is as in the title:
Is the ring $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p = \mathbb{Z}_p \otimes_{\mathbb{Z}_{(p)}} \mathbb{Z}_p$ coherent?
As shown in the related question, the ...
- 3,785
10
votes
0
answers
561
views
Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...
10
votes
0
answers
574
views
How general are Gröbner degenerations?
While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
- 3,513
10
votes
0
answers
312
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
10
votes
0
answers
201
views
Valuation with values in a semiring?
The notion of "valuation" on a ring $R$ is peculiar in that as typically presented, it is really two notions, neither of which subsumes the other.
A valuation can be a homomorphism $v: (R,\times) \to ...
- 63.9k
10
votes
0
answers
241
views
Has anyone seen this construction of dg algebras?
Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication
$$ ...
- 40.2k