All Questions
6,055 questions
4
votes
0
answers
238
views
When $\langle u,v,w \rangle$ is a maximal ideal in $\mathbb{C}[x,y]$?
Let $u,v,w \in \mathbb{C}[x,y]$ and let $\langle u,v,w \rangle$ be the ideal generated by $u,v,w$.
It is known that for two elements the following result holds:
$\langle u,v \rangle$ is a maximal ...
- 63.9k
2
votes
1
answer
185
views
Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 6,262
3
votes
3
answers
517
views
Cohen-Macaulay property for reducible schemes
I have the following question about certain schemes being Cohen-Macaulay.
Let $X$ be the union of all coordinate $k$-planes in
${\mathbb A}^N$. Is it CM?
Let $R$ be a collection of $k$-element ...
- 3,065
1
vote
0
answers
130
views
A basis of the weight space in the semi-invariant ring corresponding to the weight $\langle(2,3,2),\cdot\rangle$
I'm trying to understand Example 10.11.1 on page 225 of the book "An introduction to quiver representations" by Harm Derksen and Jerzy Weyman (see the attached screenshot below)
I want to ...
- 839
11
votes
1
answer
2k
views
geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings
Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...
- 103
3
votes
1
answer
206
views
Discrepancy between $\dim H^2(G, \mathrm{ad}(\bar \rho))$ and the number of relations in a minimal presentation of the universal deformation ring $R$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\ad{ad}\DeclareMathOperator\gen{gen}$Let $p$ be a prime and $G$ be a profinite group such that the pro-$p$ completion of every open subgroup is ...
- 148k
6
votes
0
answers
513
views
Finite extensions of $\mathbb Q_p$
Is there any finite extension of $\mathbb Q_p$ which is not the completion of a finite extension of $\mathbb Q$ at some place over $p$?
Analogously in equicharacteristic, if $k=\overline {\mathbb F_p}$...
- 113
1
vote
0
answers
125
views
Recovering a ring from its localization and completion with respect to a fixed element
Suppose I have a commutative ring $k$ and an element $x \in k$. Then I can form the localization $k[x^{-1}]$ of $k$ at the multiplicative subset $\{ 1, x, x^2, ... \}$, and I can form the completion $\...
- 61
12
votes
3
answers
4k
views
Existence of prime ideals and Axiom of Choice.
One of the must obvious equivalences of Axiom of Choice is the converse of Krull Theorem.
Bernhard Banaschewski in the Article titled by A New Proof that “Krull implies Zorn” showed a very simple ...
- 4,713
1
vote
0
answers
87
views
Factorial surfaces and smoothness
It is well-known that normal curves are smooth. Moreover, a UFD of Krull dimension one is regular. Is there any higher-dimensional analog?
For example, given a normal projective surface $S$ over $\...
- 389
2
votes
0
answers
67
views
When does an algebraically independent set "satisfy Noether normalization"?
Let $k$ be a field, $A$ a finitely generated $k$-algebra.
By Noether normalization, we know that there exists a finite morphism of $k$-algebras $\varphi : k[x_1, \ldots, x_d] \hookrightarrow A$, with $...
- 237
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
0
votes
0
answers
184
views
Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
3
votes
0
answers
73
views
Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$
I have a (fairly large) system of polynomial equations, of the form
$$
c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots
$$
(In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
- 18.7k
3
votes
2
answers
401
views
$R$-Module objects in cartesian closed categories
I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...
CommunityBot
- 1
5
votes
2
answers
1k
views
Surjectivity of the natural map of injective module to its localization
Lemma 3.3 page 214 in Hartshorne's Algebraic Geometry book states: "If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its localization $...
- 12.9k
2
votes
0
answers
157
views
Resolutions of semi free (or almost free) commutative dg algebras with finitely generated cohomology
Let $A^{\bullet}:=\{ \cdots \rightarrow A^i \overset{d^i}{\rightarrow} A^{i+1} \rightarrow \cdots \rightarrow A^{-1} \rightarrow A^0 \rightarrow 0 \rightarrow \cdots \}$ be a non-positively graded ...
- 889
5
votes
0
answers
160
views
$S$ and $T$ globally isomorphic semigroups, with $S$ (commutative and) cancellative, iff $S$ is isomorphic to $T$?
Denote by $\mathcal P(S)$ the semigroup obtained by equipping the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication ...
- 10.5k
4
votes
1
answer
273
views
Height of a conductor ideal
We say an extension of Noetherian rings $R\subset S$ is elementary subintegral if $S=R[b]$ for some $b\in S$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...
- 6,262
3
votes
0
answers
137
views
Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
- 33.8k
13
votes
4
answers
2k
views
Does smoothness descend along flat morphisms?
Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$?
If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after ...
- 761
2
votes
2
answers
1k
views
when tensor complex resolves S/I+J?
Assume that $I\subset k[x_1,\ldots,x_n]$ and $J\subset k[y_1,\ldots,y_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F_\cdot$ and $G_\cdot$, are ...
CommunityBot
- 1
0
votes
0
answers
70
views
Hensel lifting of roots of a biquadratic polynomial
Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
12
votes
2
answers
785
views
Description of p-adics tensor the reals
What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to?
where $\mathbb{Z}_p$ are the p-adic integers.
I am specially interested in the case $p=2$.
Do know that $\mathbb{Z}_p\otimes_{\...
- 148k
2
votes
1
answer
145
views
The presentations of finite complete local rings
Suppose that $R$ is a commutative ring such that there is a surjection $ \pi:\mathbf{Z}_p[[T_1,\cdots,T_n]]\to R$ of rings where $\mathbf{Z}_p[[T_1,\cdots,T_n]]$ is the ring of formal power series ...
- 6,262
21
votes
6
answers
3k
views
A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
- 187
6
votes
2
answers
345
views
Are finite projective modules over $R[t]$ free when $R$ is DVR?
Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$?
As far as I understand, this should be a ...
- 21
6
votes
0
answers
355
views
Integral group rings on which stably free modules are free
Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...
- 187
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
1
vote
0
answers
85
views
Regarding the common zeros of the system of equations
Consider the following two systems 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
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
13
votes
3
answers
978
views
Model Structure/Homotopy Pushouts in topological monoids?
Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.
Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
- 4,713
3
votes
1
answer
171
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
- 38.6k
1
vote
0
answers
374
views
Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
1
vote
0
answers
55
views
Can a surjective morphism between complete intersection rings be given by adding terms to a regular sequence?
Given a surjective morphism
$$\frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I}\twoheadrightarrow \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J}$$
where $I,J$ are genereated by regular sequences.
Question
Can ...
- 1,270
75
votes
9
answers
17k
views
Why is an elliptic curve a group?
Consider an elliptic curve $y^2=x^3+ax+b$. It is well known that we can (in the generic case) create an addition on this curve turning it into an abelian group: The group law is characterized by the ...
- 1,446
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
7
votes
3
answers
988
views
A polynomial identity for $\displaystyle \sum_{k=0} ^m (-1)^ka^{m-k}b^k$
I asked this question on MSE here.
I was solving this problem:
Show that if $\gcd(a, b) = 1$ and $p$ is an odd prime, then $\displaystyle \gcd\left(a+b, \frac{a^p+b^p}{a+b}\right)=1$ or $p$.
This ...
- 125
1
vote
0
answers
97
views
Algorithms to decompose a graded module over $R[x]$, where $R$ is a PID
There is a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors. A few equivalent definitions are given below. The question is what computer algorithms ...
- 3,065
4
votes
0
answers
302
views
What is known about the number of elements needed to generate a given ideal in $k[X_1,\dots,X_n]$?
In Algebraic Geometry by J.S. Milne, after he proves Hilbert's Basis Theorem, he makes the following aside:
One may ask how many elements are needed to generate a given a given ideal $\mathfrak a$ in ...
- 961
10
votes
1
answer
533
views
How is Taylor-Wiles patching "horizontal Iwasawa theory"?
I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the ...
- 37k
1
vote
1
answer
463
views
Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
- 21
13
votes
4
answers
4k
views
Reference for tensor products of fields
Does anybody know a reference for basic properties of tensor products of (finite) algebraic extensions of fields?
Ideally, I would like a description of $L \otimes_k K$ for arbitrary finite ...
- 233
0
votes
1
answer
118
views
$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
- 6,262
4
votes
0
answers
128
views
Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
- 223
3
votes
1
answer
185
views
Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 21.5k
3
votes
1
answer
200
views
Size of the Groebner basis and change of coordinates
Given an ideal $I$ of $k[x,y]$, the size of the Groebner basis depends on the monomial ordering. For example, if
$$
I = \langle x^3y^4 , x^2 + y^2 \rangle,
$$
then the Groebner basis with the ...
- 63.9k
22
votes
8
answers
5k
views
Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
- 309
0
votes
0
answers
60
views
Symbolic polyhedron of a monomial ideal
$\DeclareMathOperator\maxAss{maxAss}\DeclareMathOperator\conv{conv}$Let $I$ be a non-zero monomial ideal and $P$ $\subseteq$ $\mathbb R_+ ^ {n+1}$ be its symbolic polyhedron: then
$$
\alpha(P)= \min \{...
- 12.9k
3
votes
0
answers
156
views
Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
- 10.4k