All Questions
6,055 questions
0
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130
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Levi-Civita field in unusual basis
Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}...
- 10.1k
0
votes
1
answer
114
views
On a sum of squares representation
We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$
$$r^2=|4pq|$$
holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
- 1,826
0
votes
1
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581
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A lemma on a sequence of three morphisms
Few months ago, I proved that when there is three morphisms of modules over a commutative ring with zero composition, i.e., a sequence
$$A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{\...
- 109
0
votes
1
answer
266
views
Is a coslice (slice under) category a full subcategory of it ambient category?
Let $\operatorname{C}$ be a category and $c\in \operatorname{C}$ an object. Consider the coslice (sometimes called slice under) category ${\operatorname c}/{\operatorname C}$. My question is whether ${...
- 455
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1
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435
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Regular local artinian k-algebra with residue field k is k
I am reading an article. There is a step in which I suspect that they use a "result" that "Let $A$ be a local artinian $k$-algebra with residue field $k$. If $A$ is regular then $A$ is nothing but $k$....
- 119
0
votes
1
answer
305
views
Integral morphism between universally closed and separated schemes
Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}_p)$.
Assume:
$f$ is integral
both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}_p$
$f$ mod $p^n$ is an ...
user113393
0
votes
1
answer
353
views
Strange subscheme in ${\mathrm{Spec}} R \times {\Bbb A}^1_{\Bbb C}$
Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as
$R \...
- 563
0
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1
answer
114
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Adic filtration and integral closure
Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...
- 3,079
0
votes
1
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79
views
Degree of a field extension with a rational solution
Let $S$ be a system of polynomial equations over $\mathbb{F}_q$.
Assume that $S$ has a solution in $\overline{\mathbb{F}_q}$.
Denote by $k$ the minimal number such that $S$ has $\mathbb{F}_{q^k}$-...
- 2,576
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votes
2
answers
524
views
Almost complete intersection ideal and $d$-sequence
In a Noetherian local ring $R$, an ideal $I$ is called an almost complete intersection ideal if $\mu(I)=\text{ht}(I)+1$.
Q) Is it true that $I$ is generated by a $d$-sequence?
- 1,713
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1
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154
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$R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1,$ show that $R_f \cong S[X,X^{-1}]$ [closed]
Suppose $R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1.$ Then I want to show that $R_f \cong S[X,X^{-1}],$ where $S=(R_f)_0$ and $X$ transcendental over $S.$
I wanted to use the isomorphism $...
- 111
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1
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63
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The semiring generated by $\mathbb{R}_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$
Let $S$ be the semiring in $\mathbb{R}[x,y]$ generated by the nonnegative real numbers $\mathbb{R}_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$, i.e.
$$S = \left\{ \sum_{i,j,k,\ell \ge 0 } a_{i,...
user94803
0
votes
1
answer
305
views
How does normalization behave on closed subschemes?
I'm trying to understand Harbater's "Mock Covers and Galois Extensions".
There, a "mock cover" of a domain $S$ is a finite map $p : $ Spec $T\rightarrow $ Spec $S$, where $T$ is a torsion-free $S$-...
- 7,527
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1
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354
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Does the integral closure of a normal local ring in a finite extension of its fraction field have finite projective dimension?
Let $A$ be a normal local domain with field of fractions $K$. Let $L$ be a finite separable extension of $K$ (if relevant, I'm happy to assume all possible ramification is tame), and let $B$ be the ...
- 7,527
0
votes
1
answer
381
views
Milnor numbers and mixed multiplicities
section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration $\{m^rJ^s\...
- 1,713
0
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1
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245
views
Properties of Betti number of ideal
Notations:
$R$- Noetherian graded ring and $I,J$ homogeneous ideals in $R$
Definition:
The projective dimension of $R/I$, denoted $pd(R/I)$, is the length of a minimal free graded resolution of $R/...
- 381
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1
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195
views
Can you detect homological dimensions from homology?
Suppose you are given a bounded chain complex $M$ over a commutative ring $R$.
Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies?
For ...
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1
answer
357
views
Is there a complete local analogue of the Artin-Tate lemma?
The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a $...
- 1,802
0
votes
1
answer
217
views
quasi-projective and separated as topological properties
Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?
(By the way, projective implies a ...
- 2,384
0
votes
1
answer
279
views
Noetherianess of a finite module over a noethrian ring without Axiom of Choice
All rings are assumed to be commutative with 1.
We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...
- 1,169
0
votes
1
answer
173
views
if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?
definition from Bruns-Herzog:
It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true:
if $ \lambda (I)= \dim R$, can one claim ...
- 1,355
0
votes
1
answer
253
views
Connected curve
Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...
- 315
0
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1
answer
455
views
Iwasawa theory for Mazur's deformation ring R
The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...
- 87
0
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1
answer
805
views
Does Noether normalization hold more general? [duplicate]
Noether normalization tells us that a finitely generated $k$-algebra is an integral extension of a polynomial algebra over the field $k$.
My question is whether this still holds if we replace the ...
0
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1
answer
159
views
Chain of Ideals of same height
I have been wondering about the following (and allready posted a similar question, see Dimension of ring completion wrt to a decreasing chain of ideals):
Let $R$ be the ring of formal power series ...
- 71
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1
answer
1k
views
Relations between automorphisms of field of rational functions and Mobius Transfomation
Proposition: If $F$ is a field, let $F[x]$ be the ring of all polynomials whose coefficients are in $F$. The fraction field of $F[x]$, denoted $F(x)$, is defined to be the ratios $r(x) = f(x)/g(x)$ ...
- 8,071
0
votes
2
answers
335
views
What does a singular simplex with real coefficient mean [closed]
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental ...
- 689
0
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1
answer
111
views
Explicit representation of $R[\frac{x}{y}]$ where $x, y\in R$ for non-Euclidean PIDs $R$?
It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the quotient field) is a ...
- 147
0
votes
2
answers
155
views
example re torsionless quotients of abelian groups
Recall that an abelian group $G$ is $Z$-torsionless if for all $a\in G$ ($\neq 0$) there is a homomorphism of $\phi\in Hom(G,Z) = G^*$ so that $\phi(a)\neq 0$
Suppose $S$ is a subgroup of torsionless ...
- 345
0
votes
1
answer
340
views
Length of a module
Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?
- 11
0
votes
1
answer
241
views
Sums of Squares and Totally Positive Numbers
In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of ...
- 3
0
votes
1
answer
232
views
Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
- 8,512
0
votes
1
answer
425
views
Are maximal Cohen-Macaulay modules supported everywhere?
Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:
If $\omega$ is a canonical ...
- 2,857
0
votes
1
answer
475
views
How to use Nakayama [closed]
Hi there,
Let R be a local commutative ring. If M and N are two R-modules with the condition that their direct sum is equal to R^n. How do I use Nakayama to show that M and N are in fact free R-...
- 1
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 1
0
votes
1
answer
2k
views
Dimension of tensor product of modules
$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\...
- 2,857
0
votes
1
answer
219
views
Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 2,837
0
votes
1
answer
177
views
On the solutions of system of homogeneous polynomials of degree $d$ in $n$ variables
Consider the following two system of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
- 1,269
0
votes
1
answer
172
views
Is the integral closure of a henselian local domain of dimension $1$ again local?
Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$.
If $R$ is henselian, then is $\overline R$ also a local ...
- 412
0
votes
1
answer
153
views
Unitary representation of a group of automorphism on an abelian algebra
Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
- 33
0
votes
1
answer
80
views
Ideal membership and change of fields
Let $R=k[x_1,...,x_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f_1,...,f_m)$ a homogeneous ideal.
With Macaulay2, one can compute the Groebner basis of $I$ when $...
- 103
0
votes
1
answer
221
views
Are zero dimensional ideals radical?
I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano.
Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
- 121
0
votes
1
answer
295
views
Depth of almost complete intersection rings
Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...
0
votes
1
answer
265
views
Quiver representations over any commutative ring
I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this ...
- 839
0
votes
1
answer
127
views
Software to compute generators of a module over polynomial ring
Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.
Consider the subset
$$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}...
- 21.8k
0
votes
1
answer
154
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Nullstellensatz and nilpotence of a module
Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated.
The rational abelianization of $N$, $H_1(N)$ is a $\mathbb{C}[H]$-module, ...
- 75
0
votes
1
answer
49
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More vocabulary for periodic elements in monoids
Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.
An easy division algorithm argument shows that if $m$ is the ...
- 18.7k
0
votes
1
answer
248
views
Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called
We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?
0
votes
1
answer
208
views
Separable non-flat simple ring extension
Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \...
- 2,837