All Questions
6,055 questions
5
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1
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208
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Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
- 889
1
vote
1
answer
87
views
Topological modules over a locally compact ring
Let $R$ be a locally compact, separably metrizable ring (commutative with an identity) and let $M$ be a closed submodule of $R \oplus R$. Is the projection of $M$ onto the first coordinate closed?
- 63.9k
2
votes
0
answers
125
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Smoothness of locus of triples $(B_1,B_2,i)$ in Nakajima's notes
In section 1.4 of Nakajima's notes on Lectures on Hilbert Schemes, it is mentioned that $(\mathbb A^2)^{[n]}$ is identified with the space of triples $\{(B_1,B_2,i)\}/GL_n$. Here $B_1,B_2$ are $n\...
11
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2
answers
769
views
Are topological PID's Noetherian?
Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...
- 5,409
2
votes
0
answers
123
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An alternative proof that Buchsbaum rings are generalized Cohen-Macaulay
Let $(R,\mathfrak{m})$ be a Noetherian local ring. $R$ is said to be Buchsbaum if, for each ideal $\mathfrak{q}$ generated by a full system of parameters, the number $\lambda_R(R/\mathfrak{q})-e_{\...
- 5,429
5
votes
2
answers
2k
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Is a valuation domain PID when its maximal ideal is principal?
It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?
1
vote
1
answer
131
views
Completion of $\mathbb F_q(T)$
It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
- 63.9k
3
votes
3
answers
598
views
uniform bound of the number of generators of prime ideals
These questions are inspired from the well known fact (by Sally et. al.) as follows:
Theorem 1. Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension one. Then the minimal number of ...
- 4,713
2
votes
0
answers
105
views
Geometrizing saturation construction
Edit: My original question quickly got one close request, so I edited it to add some context and motivation.
Consider homogeneous polynomials $J_1,\dots,J_r\in k[\bar x]$. I want to construct a ...
- 650
5
votes
2
answers
478
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 41.1k
3
votes
1
answer
372
views
Subalgebras of quadratic algebras that are not quadratic
Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
- 16.9k
2
votes
3
answers
688
views
Metrizability of $\mathfrak{a}$-adic topology
Let $A$ be a ring, $\mathfrak{a}\subset A$ an ideal. Then is the $\mathfrak{a}$-adic topology on $A$ necessarily a metric space? I can see that it is true when $A$ is a DVR, but is it true in general?
2
votes
0
answers
68
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Number of solutions of overdetermined quadratic polynomial equations
Given $m$ linearly independent quadratic polynomials over the complex field in $n$ variables with $m>n$ and such that the number of zeros, say $N$, is finite, is there a known or conjectured strict ...
- 1,207
20
votes
1
answer
519
views
Concept associated to the Eudoxus reals
I am aware of three different constructions of the field of real numbers :
The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the ...
- 4,985
5
votes
2
answers
3k
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What is a fat point?
In our scriptum we're talking about singularities. And there is the term "fat point" (for example of "tangent of fat point") . I cannot find any definition :-/ Has somebody an idea?
- 5,407
1
vote
1
answer
120
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Embedding noetherian domains in a PID with finite index
The starting point of this post is the following question:
Embedding number fields in fields with class number 1
It is shown that in the answers that , given an number field $K$, we cannot necessarily ...
- 2,391
1
vote
1
answer
134
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If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$.
Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions:
(1) $(f,g)$ is a maximal ideal of ...
- 7,722
4
votes
1
answer
666
views
Example of existence of small CM algebra in mixed characteristic
A Noetherian local ring $R$ admits a small Cohen-Macaulay (CM) algebra if there is an injective map of rings $R\hookrightarrow S$ such that every system of parameters of $R$ becomes a regular sequence ...
CommunityBot
- 1
1
vote
1
answer
117
views
When is the Tor-dimension of $R/(r)$ strictly smaller than that of $R$?
Let $R$ be a ring (commutative with unit) which I assume Noetherian and regular. In particular, the homological dimension of $R$ is the same as its Krull dimension.
I am looking for results in ...
- 1,810
1
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1
answer
130
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Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?
Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ?
This would be a ...
- 63.9k
4
votes
1
answer
208
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
- 23.7k
2
votes
0
answers
118
views
polynomials with no repeated factors
Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in ...
- 3,062
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
6
votes
1
answer
1k
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If some powers of polynomials are linearly independent, does it imply higher powers are also independent?
Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly ...
- 148k
36
votes
3
answers
2k
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Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 3,932
2
votes
1
answer
165
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$K_0((k[x]/(x^2))[y])$
Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
- 56.9k
1
vote
0
answers
87
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Projective dimension and certain subideals
My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...
12
votes
2
answers
572
views
Regularity of local ring
Let $(R, \mathfrak{m})$ be a Noetherian local ring. It is well known that $R$ is regular iff $\operatorname{pd}(R/\mathfrak{m}) < \infty$ (i.e. $R/\mathfrak{m}$ has finite projective dimension).
...
- 12.9k
3
votes
0
answers
107
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Non-tree models of Lagrange inversion polynomials
The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
- 10.5k
1
vote
0
answers
150
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Invariant polynomials under a non-standard group action
There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
- 63.9k
4
votes
1
answer
194
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When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
- 23k
1
vote
1
answer
450
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Is a proper map of varieties which is a bijection on points an isomorphism?
Suppose that I have a proper morphism $f: X \to Y$ of varieties (i.e. reduced separated schemes of finite type). I am given that (a) on a dense open $U \subseteq Y$, $f$ is an isomorphism (i.e. $X\...
- 28.7k
0
votes
1
answer
137
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$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$.
Claim:
$\mathbb{C}(...
- 542
11
votes
1
answer
4k
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Does completion commute with localization?
Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
5
votes
0
answers
138
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Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)
Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
- 32.5k
7
votes
2
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269
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Double dual of free $\mathbb{Z}_{(p)}$-modules
For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D_{(p)}A=\text{Hom}(A,\mathbb{Z}_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $...
- 32.5k
1
vote
1
answer
153
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Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$?
The following question appears in MSE without answers.
Let $f_1,f_2,g_1,g_2 \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle \subsetneq \mathbb{C}[x,y]$,
...
- 6,237
28
votes
2
answers
863
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$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 35.2k
4
votes
0
answers
242
views
A question about Euclidean domains
An integral domain $R$ is a Euclidean domain if there is a degree function $$\deg : R-\{0\} \to \mathbb{Z}_{\ge 0} $$ such that
For every $a,b\in R$ with $b\ne 0$ there are $q,r\in R$ such that $$ a=...
- 755
2
votes
0
answers
436
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Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?
Let $M$ be a smooth manifold and let $A=C^\infty(M).$
We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
- 22.3k
1
vote
0
answers
159
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Why N-1 and N-2 rings are called like that?
In the Stacks Project, Tag 032F, we find:
Definition. Let $R$ be a domain with field of fractions $K$.
We say $R$ is N-1 if the integral closure of $R$ in $K$
is a finite $R$-module.
We say $R$ is N-...
1
vote
1
answer
1k
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Finiteness of the integral closure of an integral domain in its field of fractions
I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as
follows: If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain finite ...
3
votes
0
answers
176
views
The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
- 187
0
votes
1
answer
162
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$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?
The following question is a direct continuation of this question:
Let $u,v \in \mathbb{C}[x,y]$.
Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
- 2,959
12
votes
4
answers
7k
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Definition of étale for rings
Let $A \to B$ be a ring extension.
What is the definition of $B/A$ étale ?
When $A$ is a field, do we get a nice characterization ?
- 35.5k
10
votes
2
answers
2k
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why are subextensions of Galois extensions also Galois?
Generally a Galois extension is defined to be an algebraic extension that is also normal & separable. It is then shown that in the sequence of field extensions $L|M|K$ if $L|K$ is Galois then $L|M$...
- 148k
2
votes
1
answer
116
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On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
CommunityBot
- 1
1
vote
0
answers
78
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For a finite locally free $A\to B$ when does the different equal the Noether different?
(Cross-posted from MSE.)
All rings commutative with $1$. Let $A\to B$ be an $A$-algebra which is finite projective, meaning $B$ is finitely generated projective as an $A$-module, so there is the trace ...
- 215
6
votes
1
answer
193
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Ideal-like filter on a ring not generated by ring ideals
Suppose one has a filter (a collection of subsets closed under increasing the size of the set and under finite intersection) $F$ on a ring $R$. Say that $F$ is (ring) ideal-like if for every set $U \...
- 4,316
4
votes
1
answer
278
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If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?
The following question is a direct continuation of this elaborate question; it is mentioned there at the end:
Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
- 2,959