All Questions
6,056 questions
1
vote
1
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118
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Symbolic power of an ideal associated to non-singular algebraic set
Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...
- 1,713
1
vote
2
answers
368
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How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]
Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$.
Let $G$ be the group ...
- 4,746
3
votes
1
answer
474
views
Structures of subgroups of a finite abelian p-group
$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $G=\mathbb{Z}/p^{i_1}\times\cdots\times\mathbb{Z}/p^{i_r}$ with $i_1\leq\ldots\leq i_r$ be a finite abelian $p$-group. Then there can be many ...
- 37.4k
7
votes
2
answers
457
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The soccer splitting problem in arbitrary commutative ring
There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...
- 7,831
53
votes
3
answers
6k
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Is it true that, as $\Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other?
Is it true that, in the category of $\mathbb{Z}$-modules, $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[x],\mathbb{Z})\cong\mathbb{Z}[[x]]$ and $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}[[x]],\mathbb{Z}...
- 5,359
1
vote
0
answers
171
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Separable field extensions and base change
Suppose that there are field extensions
\begin{array}{ccc}
k & \longrightarrow & K \\
\downarrow & & \downarrow \\
L & \longrightarrow & M
\end{array}
where $M$ is generated by ...
- 635
4
votes
2
answers
458
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Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$
Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...
- 642
1
vote
1
answer
169
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Open affine subscheme of a direct limit of smooth algebras
Let $R$ be (assumed to be commutative, Noetherian) a regular local ring. Let $A$ be a direct limit of $R$-smooth algebras, such that the transition maps are $R$-étale.
Let $U= Spec(B)$ be an affine ...
- 12.9k
1
vote
1
answer
210
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Primary decomposition of huge ideals using M2/Singular
I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
- 71
5
votes
1
answer
248
views
Artin-Rees lemma for multiplicative subsets?
The classical Artin-Rees lemma tells the following. Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Let $M$ be a finitely generated $R$-module and $N\subset M$ be a submodule. ...
- 6,262
5
votes
1
answer
408
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On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
CommunityBot
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4
votes
1
answer
327
views
Detecting closed immersions on fibers
Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...
CommunityBot
- 1
5
votes
1
answer
411
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Origin of Laguerre geometry?
Laguerre geometry is described as either the geometry of oriented lines and circles in the Euclidean plane, equipped with a certain unusual symmetry group (see https://en.wikipedia.org/wiki/...
- 108k
4
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0
answers
248
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Artin's "Versal Deformations and Algebraic stacks": Question concerning proof of Theorem 3.3
I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....
- 749
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 $...
- 166
2
votes
1
answer
347
views
Good prime ideals in tensor products of local rings
Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...
- 1,144
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1
answer
142
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Is there a Banach algebra which $A^2$ is not dense in $A$ but $(A^{**})^2$ is dense in $A^{**}$?
Is there a Banach algebra which $A^2=\langle a_1a_2 ; a_1,a_2\in A\rangle$ is not dense in $A$ but $(A^{**})^2$ is dense in $A^{**}$?
- 25.8k
4
votes
0
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115
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Integral face ring of the triangulation of a sphere
The integral face ring of a (finite) simplicial complex $K$ on $m$ vertices is the quotient ring
$$\mathbb{Z}[K]=\mathbb{Z}[v_1,...,v_m]/\mathcal{I}_K$$
where $\mathcal{I}_K$ is the ideal generated by ...
- 208
1
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0
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170
views
What is this algebraic object (special case of a semigroup)?
Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
- 1,832
1
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0
answers
57
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Families of polynomials given by tuples of binary forms with finitely many reducible members
Let $G_1, \cdots, G_n \in \mathbb{Z}[x,y]$ be binary forms, and put $\mathbf{G} = (G_1, \cdots, G_n)$. Consider the family of monic polynomials
$$\displaystyle \mathcal{F}_\mathbf{G} = \{x^n + G_1(p,q)...
- 26.9k
0
votes
0
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160
views
Are irreducible components of regularly embedded varieties regularly embedded?
Suppose I have a (reduced) subvariety $V \hookrightarrow X$ of a smooth variety $X$ such that $V$ is regularly embedded in $X$. (i.e. is locally cut out by a regular sequence of $\operatorname{codim}(...
- 111
3
votes
1
answer
311
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Functors that preserve monoids
In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
- 63.1k
8
votes
0
answers
398
views
A criterion for rational singularities in mixed characteristic
Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...
- 10.5k
1
vote
1
answer
285
views
Product absolute value in rings of integers
Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...
- 47.4k
3
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1
answer
510
views
How can every divisor be reached by a sequence of blow-ups?
The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties].
$X$ : an algebraic variety over a field $k$.
$(R,m)$ : a DVR of the quotient field $K(X)$ ...
CommunityBot
- 1
10
votes
3
answers
1k
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What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
- 63.9k
3
votes
3
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1k
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Topology on $p$-adic period ring in an article by Fontaine
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathcal{C}$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathcal{C}/p,$$ where the transition maps in ...
- 63.9k
6
votes
0
answers
399
views
Unbounded derived Nakayama lemma
Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...
- 15.9k
6
votes
2
answers
496
views
Tensor of finite-dimensional algebra over perfect field is semisimple
Let $K$ be a field and let $\Lambda_{1}$ and $\Lambda_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J_{1}$ and $J_{2}$ respectively. How to show or where can I find the proof of ...
- 38.6k
5
votes
0
answers
255
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Is the category of topologically free $k[[h]]$-modules locally presentable?
$\newcommand{\colim}{\operatorname{colim}}$
Let $k$ be a field (of characteristic 0, say) and $M$ be a module over $R=k[[h]]$. Recall that the $h$-adic completion of $M$ is
$$
\hat M:=\lim M/h^nM,
$$
...
- 35.5k
3
votes
1
answer
200
views
Interpolation of scheme-theoretic endomorphisms of closed fibers
Let $S$ be a scheme and $f : X\to S$ be an $S$-scheme. This question asks for examples of maps of sets $X(S) \to X(S)$ that do not come from an $S$-scheme endomorphism of $X$, but that, roughly, ...
- 9,061
2
votes
1
answer
437
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Extending functors between K-algebras to schemes
Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...
- 40.2k
3
votes
1
answer
235
views
Category of modules over internal monoid is abelian
I have asked the following question on MSE a few days ago, but without any success.
I am interested in proving the following statement:
Let $\mathcal{A}$ be a tensor category. Then the category of ...
- 275
2
votes
1
answer
98
views
Can the differentials in a minimal free resolution ever have a "long" row of $0$'s?
Assume just for sake of simplicity that $R = k[x_1 , \dots , x_n]$ is a standard graded polynomial ring over a field. If one considers the ideal
$$I = \left({x}_{1}{x}_{3},{x}_{2}^{2},{x}_{2}{x}_{3},{...
- 173
3
votes
1
answer
267
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Do there exist irreducible elements in this domain?
I asked this question on MSE. Here also I have the same motive in the question.
Let $D= \{\,a_1x^{r_1} + \cdots + a_n x^{r_n} \, \vert \, a_i \in \mathbb{C} \text{ for } i= 1,2,\dots,n \text{ and ...
- 63.9k
9
votes
3
answers
3k
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Free resolution dimension?
Is there a notion of a dimension associated to free resolutions like projective and injective dimensions associated to projective and injective resolutions? My guess is that it coincides with ...
CommunityBot
- 1
4
votes
1
answer
474
views
Original sources for two theorems by Bass, Matlis and Papp
It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...
11
votes
1
answer
339
views
Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions
Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled ...
- 10.5k
2
votes
1
answer
141
views
Reference to basic facts on non-Archimedean local fields
I need a reference to the following claims which, I believe, are correct and well known to experts (I am not one of them).
Let $K$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of ...
- 8,866
1
vote
1
answer
403
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Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
- 7,127
2
votes
1
answer
160
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The minimal complexes and the direct limit
In my research I confronted with the following question. I do not have a counterexample, so I ask it here if anyone knows. I would like a positive answer, however a negative answer seems more likely ...
- 173
5
votes
4
answers
660
views
Relative version of Hilbert syzygy theorem
I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a ...
- 15.6k
2
votes
0
answers
253
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Jacobson density theorem and Riesz representation theorem
My question is propably too vague to answer, the reader is advised.
There are these two mentioned theorems which you may recognise in other forms, but these are the ones that I have been taught.
...
- 1,848
3
votes
1
answer
390
views
The kernel from $A_\mathrm{inf}$ to $\mathcal{O}_{\mathbb{C}_K}$
I tried to understand this paper on page 31.
Let $K$ be an finite extension of $\mathbb Q_p$ and $\overline{K}$ be its algebraic closure; $\mathcal{O}_{\overline{K}}$ is the ring of integers of $\...
- 7,377
2
votes
0
answers
358
views
Dimension of dominant morphism of schemes
In differential geometry, there exists an immersion of a curve into a sphere whose image is dense. I wanted to know if this paradox still exists in algebraic geometry. I think the answer should be no.
...
- 635
0
votes
0
answers
72
views
Explicit representative for an extension class
Let $A$ be a regular local ring and $I\subset A$ a complete intersection ideal.
We have the natural map $\delta:Hom_A(I,A/I)\rightarrow Ext_A^1(A/I,A/I)$.
For a given $\alpha\in Hom_A(I,A/I)$ is there ...
- 1,463
1
vote
0
answers
152
views
Image of pullback for Brauer groups
If a have a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ where $X$ is a projective, geometrically integral $k$-scheme. Then this gives rise to a pullback map
\begin{align*}
\pi^{*}:\text{Br}(k(...
- 481
4
votes
1
answer
205
views
Let A be an Artin algebra. What happens if the limit and inverse limit are the same in mod A?
Let $A$ be an Artin algebra and $\text{mod}\,A$ the category of finite length modules. Further, let $X_0 \longrightarrow X_1 \longrightarrow X_2 \longrightarrow ...$ and $Y_0 \longleftarrow Y_1 \...
- 19.6k
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
2
votes
1
answer
148
views
Terminology for a ring where every right cancellable element is cancellable
Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
- 218