All Questions
6,053 questions
0
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468
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Finite extensions of residue fields of Henselian DVRs
Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
CommunityBot
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2
votes
2
answers
2k
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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
158
views
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
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1
vote
0
answers
207
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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 ...
- 2,356
1
vote
0
answers
136
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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 ...
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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 $p>0$, $$A:=k[x_1,...
- 7,377
2
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1
answer
159
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Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
CommunityBot
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6
votes
0
answers
152
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Can Harrison cohomology be written using Ext?
Just like Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras, it'll be nice (or disappointing?) if Harrison cohomology can be expressed in terms of Ext'...
- 16.2k
3
votes
0
answers
89
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Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
2
votes
1
answer
149
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Finitely generated modules over completion
Let $k$ be a field, $A$ a finitely generated $k$-algebra and $I \subset A$ an ideal with $I$-adic completion $\hat{A} = \varprojlim A/I^n$. Is every finitely generated $\hat{A}$-module the completion ...
- 16.2k
3
votes
1
answer
180
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How to find equations of $\mathbb{C}^*$-curves
Fix positive integers $t_1,t_2,t_3$.
Suppose we have a $\mathbb{C}^*$ action on $\mathbb{C}^3\setminus\{0\}$ defined by
$$\mathbb{C}^* \times \mathbb{C}^3 \setminus \{0\} \to \mathbb{C}^3 \setminus \{...
4
votes
1
answer
330
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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
4
votes
2
answers
491
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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
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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
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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
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0
answers
104
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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
4
votes
0
answers
147
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Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
CommunityBot
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1
vote
0
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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 ...
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1
answer
243
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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
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votes
0
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114
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
- 63.9k
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 $1 \leq j \leq m$
$$ \operatorname{trdeg}_{\mathbb{F}(x_1,\ldots,x_j)}\{f_1,\ldots,...
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2
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0
answers
171
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Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
CommunityBot
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40
votes
1
answer
2k
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Rigid non-archimedean real closed fields
Update. The question has been recently answered in the positive by David Marker and Charles Steinhorn (as in indicated in Marker's answer). Note that Remark 3 below is now expanded by reference to a ...
- 3,530
3
votes
0
answers
161
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On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have
$$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$
So the ...
- 15.6k
3
votes
0
answers
92
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Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
3
votes
4
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944
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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
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0
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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(\...
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13
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2
answers
1k
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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
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1
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326
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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 ...
- 2,356
1
vote
1
answer
67
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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
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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 ...
- 2,356
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
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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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0
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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
3
votes
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
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0
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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
5
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0
answers
191
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Do most semigroups have a zero?
It is widely believed in finite semigroup theory that asymptotically almost all finite semigroups $S$, up to isomorphism, are 3-nilpotent, i.e., they satisfy $\#\{abc\,:\,a,b,c\in S\} = 1$. My ...
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9
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0
answers
180
views
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
602
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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
5
votes
3
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851
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What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
- 17k
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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-...
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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 ...
- 35.5k
1
vote
1
answer
52
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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