All Questions
5,985 questions
2
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Inversion of Laurent series
For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...
- 10.5k
2
votes
1
answer
157
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Computing the minimal polynomial of roots of polynomials with algebraic coefficients
Let $p(x) = \sum_{i=0}^{n} c_i x^i$ with $c_i \in \mathbb{A}$ with $q_i(c_i) = 0$ and all $q_i \in \mathbb{Q}[x]$ being minimal polynomials of the coefficients.
Let $r$ be a zero of $p(x)$. Is there ...
CommunityBot
- 1
1
vote
0
answers
206
views
When is the derived category of a ring generated by injective modules
Are there any equivalent conditions on a ring to the condition that the localizing subcategory of $D(R)$ generated by injective modules is the entire category? Are there any non-examples in Boolean ...
- 2,346
1
vote
0
answers
135
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Local cohomology and image of $1$ under the canonical map from Ext to local cohomology
Let $R$ be a commutative Noetherian local ring, and $S$ be an $R$-algebra. Let $x_1,\dots,x_t$ be elements, in the maximal ideal of $R$, which is a regular sequence on both $R$ and $S$, and let $I$ be ...
- 412
5
votes
1
answer
264
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Non-existence of power divided structure on a maximal ideal of truncated polynomial rings (example from Koblitz)
In 3.3 of Berthelot-Ogus's book Notes on Crystalline Cohomology, an example from Koblitz is exhibited without proof.
Let $k$ be a field (or a commutative ring) with characteristic $p>0$, $$A:=k[x_1,...
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2
votes
1
answer
159
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Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
CommunityBot
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6
votes
0
answers
151
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Can Harrison cohomology be written using Ext?
Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext'...
- 16.2k
2
votes
1
answer
149
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Finitely generated modules over completion
Let $k$ be a field, $A$ a finitely generated $k$-algebra and $I \subset A$ an ideal with $I$-adic completion $\hat{A} = \varprojlim A/I^n$. Is every finitely generated $\hat{A}$-module the completion ...
- 16.2k
3
votes
1
answer
180
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How to find equations of $\mathbb{C}^*$-curves
Fix positive integers $t_1,t_2,t_3$.
Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by
$$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{...
4
votes
1
answer
330
views
An assertion of Mahler
Let $\rho$ be an integer greater than $1$. In the article "Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen" https://link.springer.com/article/10.1007/...
- 105k
3
votes
1
answer
368
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Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
CommunityBot
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4
votes
2
answers
490
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Krull dimension of completions in non-noetherian setting (especially completed perfections)
What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology?
An example of the sort of "nice" topological ring I'm looking for is a ...
- 362
4
votes
0
answers
180
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Deminormal and Gorenstein
Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein?
for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf
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36
votes
4
answers
5k
views
What is interesting/useful about big Witt Vectors?
$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...
3
votes
3
answers
387
views
Basic question about completion of local ring
Let $(A,m)$ be a Noetherian local ring and $\tilde{A}$ its completion by the maximal ideal $m$.
Are the following three statements true?
(i)
If $\tilde{A}$ is free over $A$, then $A\cong \tilde{A}$
(...
- 1,144
5
votes
1
answer
151
views
Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 16.2k
1
vote
0
answers
104
views
Is a normal domain a filtered colimit of Noetherian normal domains?
As described in the title, is any normal domain a filtered colimit of Noetherian normal domains? It will be great if one can show this, even with additional conditions, or if one can provide a ...
- 1,024
0
votes
1
answer
114
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Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
CommunityBot
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0
votes
1
answer
396
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What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
CommunityBot
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1
vote
0
answers
216
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Dimension under change of ground field
I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?
Let $k\subset K$ an extension of algebraically closed fields of ...
- 19
10
votes
1
answer
243
views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 460
2
votes
1
answer
262
views
Randomly fixing elements and transcendence degree
Given $f_1,\ldots,f_n \in \mathbb{F}_q[x_1,\ldots,x_m]$ such that $\deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
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2
votes
0
answers
169
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Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
CommunityBot
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40
votes
1
answer
1k
views
Rigid non-archimedean real closed fields
Update. The question has been recently answered in the positive by David Marker and Charles Steinhorn (as in indicated in Marker's answer). Note that Remark 3 below is now expanded by reference to a ...
- 3,530
3
votes
4
answers
944
views
What conditions are needed for $-\otimes_A B$ to be faithful?
For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-...
5
votes
0
answers
211
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On the natural map $\mathrm{Br}(R) \rightarrow \mathrm{Br}(S)$ of Brauer groups
$\DeclareMathOperator\Br{Br}$Let $R$ be a commutative ring, and let $\Br(R)$ be the Brauer group of $R$ as defined by Auslander and Goldman. Let $S$ be a commutative $R$-algebra, and consider the ...
- 63.9k
3
votes
0
answers
181
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Conditions for an open mapping between spectra
Let $(A,\mathfrak{m})$ be a Locally Noether Ring, and $\hat{A} = \varprojlim A/ \mathfrak{m}^{n}$ .Furthermore, let $f : A \to \hat{A}$ be a canonical morphism, and consider the mapping $f^{*} : Spec(\...
- 912
13
votes
2
answers
1k
views
When does a quasicoherent sheaf vanish?
Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the ...
- 2,806
9
votes
1
answer
324
views
Nonzero module with vanishing derived fibers
What's an example of a nonzero $R$-module with vanishing derived fibers at all points of $\mathrm{Spec}(R)$?
This was asked in When does a quasicoherent sheaf vanish?
but the answer there only says ...
- 2,346
1
vote
1
answer
67
views
An example of a commutative ring $R$ which has a proper right ideal which is not a right SIP $R$-module
Recall that a module $M_R$ ($R$ is a ring with unity) is called SIP if the intersection of any two summands of $M$ is also a summand. I asked before if there exists a commutative ring which is not an ...
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2
votes
1
answer
181
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Idempotent algebras over absolutely flat ring
Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...
- 2,346
2
votes
1
answer
232
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Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 63.9k
3
votes
0
answers
33
views
Algorithm to determine closedness of orbits?
Consider a reductive group $G$ acting on an affine variety $X$. It is known that for every $x\in X$, we have $G.x\subseteq\overline{G.x}$ is open dense. Then $\partial({G.x})\subseteq X$ is a closed ...
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5
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0
answers
185
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Stone–Weierstrass theorem for topological fields
It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull ...
- 12.9k
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
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3
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1
answer
168
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Does there exists a "local slice" for an action $ \widehat{\mathbb{G}}_a $ on $ \operatorname{Spf}(\widehat{A}) $ (char zero)?
Every action $ \beta $ of $ \mathbb{G}_a $ on a variety $ \operatorname{Spec}(A) $ over a field of characteristic zero is obtained from a locally nilpotent derivation $ \delta $ via $ f(t_0 \ast x) = \...
- 912
1
vote
1
answer
104
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Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...
- 16.2k
3
votes
1
answer
242
views
Points of multiplicative groups
Let $R$ be a discrete valuation ring with residue field $k$. Denote by $\mathbb G_m:= R[x,1/x]$ the multiplicative group over $R$ and $\mathbb G_{m,k}:= k[x,1/x]$. If $B$ is a flat local $R$-algebra, ...
- 542
6
votes
0
answers
632
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Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
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0
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176
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$\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, with each pair of $\{f,g,x\}$ not generating $\mathbb{C}(x,y)$
Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$.
Write,
$f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$
and
$g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$,
for some $n,m ...
- 2,837
9
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0
answers
180
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How should we picture the set of monomial orders (= positive monoid orders on $\mathbb{N}^k$)?
Motivation: So apparently there's some sort of sport competition currently going on where I live, which leads to countries being given an element of $\mathbb{N}^3$ called a “medal count”, and not ...
- 32.5k
14
votes
0
answers
601
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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). ...
CommunityBot
- 1
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votes
0
answers
53
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A question on bounding the size of the polynomial
Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-...
- 341
29
votes
6
answers
8k
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How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 221
25
votes
7
answers
3k
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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 ...
- 35.4k
1
vote
1
answer
51
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Exceptional Lenz-Barlotti classes IVa.3 and IVb.3
On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...
- 1,300
4
votes
1
answer
185
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Order of pole of Poincaré series
Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
- 12.9k
3
votes
0
answers
189
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
1
answer
92
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On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 6,262
0
votes
0
answers
48
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A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214