All Questions
6,056 questions
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151
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Rectangular Newton polygon of a Jacobian pair
Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero.
By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an ...
- 2,837
1
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0
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62
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Invertibility under base change for the Weyl algebra instead of for the polynomial algebra
From Lemma 1.1.8, we obtain the following:
Assume that $R \subseteq S$ are commutative rings
and
$u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism
that has an invertible Jacobian, namely,
$Jac(u(x)...
- 2,837
1
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0
answers
46
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Integral closure of lexsegment ideal
Let $R=k[x_1,\ldots,x_d]$ where $k$ is a field and $I$ be a lexsegment ideal of $R$ and $l(I)=d$ (where $l(I)$ is analytic spread of $I$).
Is $I$ integrally closed?
If I is generated by elements ...
- 1,713
1
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0
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134
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$\Omega^1_{B/A}=0$ implies $A\subset B$ unramified
Let $(A,\mathfrak{p})\subset(B,\mathfrak{q})$ two local rings, such that $B$ is finitely generated as $A$-module. It's very well known, from Algebraic Geometry, that if $\Omega^1_{B/A}=0$ then the ...
- 1,252
1
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0
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184
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reduced-ness of the fibres of stein factorisation
Let $X\xrightarrow{f} Y\xrightarrow{g} Z$ be a stein factorisation.
It is known that the fibres of $f$ is connected.
Are the fibres of $f$ reduced?
user111251
1
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0
answers
240
views
On the coherence of $K[[X_1,X_2,...]]$
Recall that a commutative ring is coherent if every finitely generated ideal is finitely presented, or equivalently if every submodule of every finitely generated module is finitely presented.
Let $A ...
- 563
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40
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A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?
I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
- 111
1
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0
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103
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Separability in the absolute integral closure
Let $R$ be a Noetherian normal domain, and let $R^+$ denote the absolute integral closure of $R$, i.e. the directed union of all module-finite extension domains. I'm trying to understand what "...
1
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0
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82
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Maximum dimension of the socle of a colength n monomial ideal in k[x,y,z]
Let $k$ be field and fix a positive integer $n$. If $I\subseteq k[x,y,z]$ is a colength-n ideal, what is the maximum possible dimension of the socle of $k[x,y,z]/I$? Is there a formula for this in ...
- 11
1
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0
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133
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Finding Generators of an Ideal Over Number Field? [closed]
Is there any way or algorithm to find generators of an ideal over number field? (A algorithm that can be implemented and not expensive)
- 149
1
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0
answers
161
views
Projective dimension of a principal ideal
Let $R$ be a polynomial ring, $I$ a homogeneous ideal, and let $\operatorname{pd}(R/I)$ denote its projective dimension. Is there a characterization of homogeneous elements $a\in R\setminus I$ for ...
- 878
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0
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336
views
flatness and restrictions to DVR
Let $T$ be a scheme over $\mathbb{C}$ and $\mathcal{F}$ be any $\mathcal{O}_T$ module. Suppose for all morphisms $Spec~R\rightarrow T$, (where R is a DVR) the restriction $\mathcal{F}|_{Spec~R}$ is ...
user111251
1
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0
answers
62
views
Solutions to a certain Birkhoff-interpolation problem
$\newcommand{\CC}{\mathbb{C}}$
Let for $n > 1$ and $m = n-1$
$$
p = x^n + a_1 x^{n-1} + \cdots + a_m x
$$
be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$.
The ...
- 708
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0
answers
110
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When is the following sequence exact?
Let $R:=\mathbb{Z}_2[w_1,\ldots,w_n]$, $I$ some ideal in $R$ and $A=R/I$. I would very much want to show that a sequence of the form:
$0\to A/(\alpha)\overset{\cdot\beta}{\to}A/(\alpha\beta)\overset{...
- 11
1
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0
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29
views
Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (...
- 10.5k
1
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0
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60
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$Ext^i(D(R),R)$ for a certain commutative algebra
Let $k$ be a field which is not algebraic over a finite field and $a \in k$ an element of infinite multiplicative order.
Let $R=k[V,X,Y,Z]/I$ with $I=<V^2,Z^2,XY,VX+aXZ,VY+YZ,VX+Y^2,VY-X^2>.$
...
- 26.5k
1
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124
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Algebra Invariants of Schubert Calculus
For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...
- 449
1
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0
answers
57
views
Finite presentation for left-exact endofunctors
For a commutative unital ring R and a left-exact endofunctor L on the category of R-modules, what is, or should be, the definition of finite presentation for L ? Possibilities:
L( R ) is ...
- 141
1
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0
answers
128
views
Nielsen--Schreier for fields
Is it true that a subextension of a purely transcendental extension is itself purely transcendental?
In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
- 9,720
1
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0
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244
views
Quotient of polynomial ring over a Dedekind domain
Let $A$ be a Dedekind domain and let $B:=A[X]/(f(X))$, where $f(X)\in A[X]$ is some monic polynomial such that $B$ is a domain. If I take the canonical map $A\longrightarrow B$, then it nduces a ...
- 1,252
1
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0
answers
194
views
generators for a graded algebra
I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
- 673
1
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0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
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0
answers
551
views
An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$
We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$
${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
- 563
1
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0
answers
36
views
Quadratic suborders of an imprimitive quartic order
Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
- 26.9k
1
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127
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Is there a (nontrivial) known example of an algebra over a complete regular local ring with the following property?
I am working on some algebras over complete regular local algebras. But I am not sure whether such rings are worth to study. I am looking for some examples of these algebras. Let $(R,\mathfrak{m})$ be ...
- 191
1
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0
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95
views
Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
1
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0
answers
113
views
Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
- 1,615
1
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0
answers
151
views
Generators of the same degree in a graded ring and GIT quotient
Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
- 2,789
1
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1
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266
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Providing a grading for the polynomial ring over a commutative unital graded ring
Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ ...
user111524
1
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0
answers
77
views
Number of quartic rings with fixed data
In her paper "Quartic rings associated to binary quartic forms"
(https://doi.org/10.1093/imrn/rnr070), Wood showed that $\operatorname{GL}_2(\mathbb{Z})$-classes of integral binary quartic forms are ...
- 26.9k
1
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0
answers
47
views
estimate on degree of generators in cohomology of differential graded module
Let $R=\mathbb{C}[X_1,\ldots,X_r]$ be a polynomial ring and consider a finitely generated and free differential graded $R$-module $M$ with differential $d$. Lets say that the degree of the variables $...
- 11
1
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1
answer
177
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Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond
Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
- 10.5k
1
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0
answers
50
views
Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
- 10.5k
1
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0
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283
views
Analytic spread of an ideal
How to calculate analytic spread of the ideal $I=\left<xyw^2,xyz^2,xw^2+yz^2\right>$ in $\mathbb Q[x,y,z,w]?$
I think it is 3.
- 1,713
1
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0
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273
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Alexander Ostrowski's argument for filling the gap in Gauss's proof of the FTA
I've read in Smale "The fundamental theorem of algebra and complexity theory" and Cain "C. F. Gauss’s Proofs of the Fundamental Theorem of Algebra" regarding how there was a gap in Gauss's proof in ...
- 387
1
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0
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175
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Is every stably free module of commutative group ring free?
Is every stably free module of commutative group ring free? if not can you give me an example of commutative group ring with nonfree stably free module.
- 11
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0
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138
views
Monomorphism between two ideals
Let $I $ and $J $ be two ideals in a commutative ring $R $ with $1$. Is there any equivalent property for the fact that there are no $R $-module monomorphism from $I $ to $J $?
- 21
1
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0
answers
92
views
Torsion functors and weak assassins
Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module $M$, we set $\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^nx=0\}$, and we ...
- 6,700
1
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0
answers
318
views
Find the generators of a complete intersection maximal ideal
Let $k$ be a field (not necessarily algebraically closed), define the $k$-algebra
$$
B:=k[x_{0}, x_{1}, x_{2}, x_{3}]_{(F_{2}F_{3})}
$$
(degree 0 part of the localization), it's the coordinate ring of ...
- 1,906
1
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0
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190
views
A question about the coefficient of a specific monomial in the expansion of the following polynomial
By this question I asked before,
I know that for any nonnegative integers $a$, $b$, $c$, the coefficient of the monomial $x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1}$ in the expansion of
$$(x_1-x_2)^{2a+1}(x_2-...
- 1,997
1
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0
answers
112
views
Isotropic subgroups for both alternating forms [closed]
Isotropic subgroups for both alternating forms
- 11
1
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0
answers
112
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Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,713
1
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0
answers
101
views
A class number in a family of number fields
Given a number field $K$ and an order (not necessarily maximal) $\mathcal{O}$ on it, it's a difficult problem to count the number $h(\mathcal{O})$ of $\mathcal{O}$-fractional ideals in $K$ (I think).
...
- 389
1
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0
answers
58
views
A class of finitely generated semigroups
Let $G$ be a finitely generated group with a probability measure $\nu$. Suppose we have a finite first moment function $f:G\to \mathbb{R}$, i.e, such that $\Sigma_{g\in G}f(g)\nu(g)< \infty$. Then, ...
- 49
1
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0
answers
85
views
Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?
There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$,
$k$ is a field of characteristic zero,
the Polarization Theorem, Corollary 5.5 by A. Joseph.
...
- 2,837
1
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0
answers
167
views
Extensions of modules
Let $M$, $N$ be two modules over a ring $R$, suppose $M$ (resp. $N$) is a non-split extension of $M_2$ by $M_1$ (resp. of $N_2$ by $N_1$) as $R$-modules. We make the following assumption:
(1) $\...
- 41
1
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0
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81
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2 questions about "monogenic" coordinate rings of affine curves
Suppose $k$ is an algebraically closed field with characteristic 0 and let $X \subset \mathbb{A}^n$ be an irreducible curve.
If $f \colon X \to \mathbb{A}^1$ is finite of degree $d$, then the ...
- 243
1
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0
answers
215
views
Non-noetherian coherent local rings
Let $A$ be a non-neotherian UFD such that $A$ satisfies the following condition $(\sharp)$$\colon$
$(\sharp)$ For any prime ideal ${\frak P}$ of $A$ such that ${\mathrm{ht}}({\frak P}) < \infty$, ...
- 781
1
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0
answers
133
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Intersections of Noetherian regular local rings of finite Krull dimensions
Let us consider Noetherian regular local rings $R_i$ of finite Krull-dimensions for each $i \geq 1$ such that
\begin{equation*}
R_1 \supset R_2 \supset \cdots
\end{equation*}
Suppose each embedding $\...
- 781
1
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0
answers
49
views
Atomicity of the semidomain (under Dirichlet convolution) of arith. fncs to a subrig of the ring of integers of a totally real NF
Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the ...
- 10.5k