All Questions
6,057 questions
6
votes
0
answers
345
views
Uncountable Mittag-Leffler condition?
Let $(X_\alpha)_{\alpha <\kappa}$ be an inverse system of abelian groups.
If $\kappa = \omega$ (or by extension if $\kappa$ is of countable cofinality), then the Mittag-Leffler condition is a ...
- 64k
11
votes
1
answer
498
views
Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?
$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$Question 1: What is an example of a sequence $(X_\alpha)_{\alpha<\kappa}$ of ...
- 3,411
2
votes
2
answers
475
views
On the notion of torsion-freeness in semigroup theory
The following seems to be the "official" notion of torsion-freeness in the context of semigroups:
TF1. A (multiplicatively written) semigroup $\mathfrak A$ is torsion-free if there do not ...
- 18.7k
4
votes
2
answers
328
views
Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way?
I'm looking for an inverse system $(X_\alpha)_{\alpha < \omega_1}$ of vector spaces (EDIT: over a finite field) such that, for some $\lambda \geq 2$ with $\lambda < \lambda^{\omega_1}$ (I ...
- 64k
2
votes
0
answers
148
views
etale locally infinitesimal lifting property
For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...
- 3,472
8
votes
2
answers
3k
views
Equivalent definitions of arithmetically Cohen-Macaulay varieties
Let $X\subset \mathbb{P}^n$ be a projective algebraic variety with coordinate ring $R$.
$X$ is said to be arithmetically Cohen-Macaulay if $R$ is a Cohen-Macauly ring. A equivalent definition is that ...
CommunityBot
- 1
5
votes
0
answers
190
views
Can an infinite abelian $p$-group be tall and thin?
Does there exist an abelian $p$-group $A$ with countable Ulm invariants and uncountable height?
Here by height, I mean the minimal ordinal $\rho$ such that $p^\rho A$ is divisible [1]. For an ordinal ...
- 64k
14
votes
2
answers
1k
views
About integer polynomials which are sums of squares of rational polynomials...
I have the following question for which I haven't been able to find any reference or proof.
Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two ...
- 109k
3
votes
0
answers
79
views
The type number of an algebra
I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...
- 323
2
votes
0
answers
492
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
- 739
9
votes
2
answers
815
views
Ideal norm in orders
Let $\overline{T}$ be a Dedekind ring such that $\overline{T}/\overline{I}$ is finite for every nonzero ideal $\overline{I}$ of $\overline{T}$. Let $T$ be a subring of $\overline{T}$ with the same ...
- 7,893
6
votes
1
answer
499
views
Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
- 26.5k
3
votes
2
answers
249
views
"Completion property" in $(\beta\omega,+)$
Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, ...
- 29k
6
votes
4
answers
2k
views
Applications of the prime avoidance lemma
I was wondering if the prime avoidance lemma is very useful or just a nice result. So far I know just only one application: let $R$ be a commutative noetherian ring and $I$ be a proper ideal of $R$. ...
- 14.6k
4
votes
1
answer
222
views
Addition and Rudin-Keisler ordering in $\beta \omega$
$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...
- 73.2k
11
votes
4
answers
1k
views
Explicit large finite fields in characteristic $2$
Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is ...
- 3,446
1
vote
1
answer
96
views
$m$ contains a non-zero-divisor on $R/(I:m^{\infty}) $
Let $(R,m)$ be a noetherian local ring with the maximal ideal $m$ and $I$ an $R$-ideal such that $\sqrt{I}\neq m$. Then, is it true that $m$ contains a non-zero-divisor on $R/(I:m^{\infty})$ . The ...
- 2,089
3
votes
0
answers
102
views
Binary quartic forms with vanishing invariants: ring theoretic interpretation
Let $F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 xy^3 + a_0 y^4 \in \mathbb{R}[x,y]$ be a binary quartic form, and let $V(\mathbb{R})$ be the 5-dimensional $\mathbb{R}$-vector space of such forms....
- 26.9k
1
vote
0
answers
201
views
Is there a natural topology on $\mathbb{C}(t)[x_1,\ldots, x_n]$ with this property?
Is there a good topology on $A=\mathbb{C}(t)[x_1,\ldots, x_n]$ so that $A$ is a topological algebra with the following property:
For any $N>0$ and a polynomial $F\in\mathbb{C}[x_1,\ldots, x_n]$ ...
- 329
1
vote
0
answers
355
views
On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
- 494
1
vote
0
answers
189
views
Existence of integral extension of DVR satisfying some conditions
Let $X^{\prime}$ and $X$ be integral noether schemes over $\mathbb{C}$, and $p:X^{\prime}\rightarrow X$ be a surjective morphism.
Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its ...
- 297
9
votes
2
answers
650
views
Definition of subcoalgebra over a commutative ring
Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...
- 7,746
5
votes
1
answer
213
views
Can the strongest Hensel lemma over integer rings imply smoothness over $\mathbb Z_p$?
Let $X \rightarrow \mathbb Z_p$ be a flat finite type morphism, with reduced special fiber and smooth generic fiber.
Assume $X(O_K) \rightarrow X(O_K/m_K)$ is surjective for all fintie extension $K$ ...
- 461
10
votes
1
answer
637
views
Discrete logarithm for polynomials
Let $p$ be a fixed small prime (I'm particularly interested in $p = 2$), and let $Q, R \in \mathbb{F}_p[X]$ be polynomials.
Consider the problem of determining the set of $n \in \mathbb{N}$ such that $...
- 1,362
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 16.2k
7
votes
2
answers
2k
views
Picard group vs class group
The question.
Let $R$ be a commutative ring. Let $M$ be an $R$-module with the property that there exists an $R$-module $N$ such that $M\otimes_R N\cong R$. Does there always exist an ideal $I$ of $R$ ...
- 1,052
6
votes
1
answer
208
views
Cayley-Hamilton over super rings
If $R$ is a commutative ring, then the Cayley-Hamilton theorem states that any endomorphism $\phi: R^{n} \rightarrow R^{n}$ of a rank $n$ free module satisfies its own characteristic polynomial, in ...
- 63.9k
3
votes
0
answers
247
views
Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
- 739
5
votes
1
answer
226
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
- 26.5k
6
votes
1
answer
448
views
What is good $t$-adic like topology on $\mathbb{C}(t)$?
Each function $f\in\mathbb{C}(t)$ can be rewritten in the form $f = a_{k}t^{k}+\ldots+a_0+a_1t+\ldots$, $k\in\mathbb{Z}$ and it is possible to define the topology with the open prebase at zero
$V_{n,v,...
- 291
0
votes
0
answers
257
views
How to prove the map of rings $\mathcal{R} \to \mathcal{R'}$ is flat?
We fix a finite extension $K$ of $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ and residue field $\kappa$. Consider the ring of witt vectors $W(\kappa)$ over the residue field $\...
- 930
1
vote
0
answers
81
views
Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\
- 473
26
votes
1
answer
2k
views
Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?
I am working on some combinatorics problems. One of my problems leads to the following question:
Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \...
- 82.7k
2
votes
0
answers
81
views
Coarsening is still henselian
Let $(K,v)$ be a valued field where $\Gamma$ is its valued group. Let $\Delta$ be convex subgroup of $\Gamma$ and consider the coarse valuation $\hat{v}:= K \rightarrow \Gamma/\Delta$ which sends each ...
- 195
22
votes
3
answers
2k
views
Discriminant of characteristic polynomial as sum of squares
The characteristic polynomial of a real symmetric $n\times n$ matrix $H$ has $n$ real roots, counted with multiplicity.
Therefore the discriminant $D(H)$ of this polynomial is zero or positive.
It is ...
- 52.3k
18
votes
4
answers
1k
views
Bass' stable range condition for principal ideal domains
In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v_1, \ldots, v_n$ that generate the unit ideal there are numbers $r_1, \ldots ...
- 1,388
8
votes
2
answers
592
views
Solving the field membership problem using Grobner bases
Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield?
Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set ...
- 63.9k
7
votes
1
answer
236
views
Functions on Stone spaces as "enveloping algebra" of Boolean algebra
I'm looking for references for the following closely related facts:
Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap ...
- 38.6k
1
vote
0
answers
106
views
Homomorphisms and indecomposable decompositions of finite modules over polynomial rings [closed]
I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\...
- 63.9k
9
votes
1
answer
607
views
Bézout ring with non-trivial Picard group?
[I asked this on stackexchange here a few weeks ago to no response]
A ring is called Bézout when its finitely generated ideals are principal.
Q: Is there a nice example of a Bézout ring $R$ with ...
- 4,985
1
vote
0
answers
62
views
Local cohomology for infinitely generated modules
Let $R$ be a local (with maximal ideal $m$) commutative Gorenstein ring of dimension $d$.
Then for any $0 \leq i \leq d$ there are isomorphisms for the local cohomology:
$H_m^i(M) \cong D Ext_R^{d-i}(...
- 26.5k
32
votes
7
answers
5k
views
Invariant polynomials under a group action (hidden GIT)
Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...
- 6,237
3
votes
0
answers
83
views
associated primes and distinguished triangles
Let $S$ be a commutative unital Noetherian ring. Let $D^+(S)$ be the derived category of bounded-below cochain complexes. Given $M^\bullet\in D^+(S)$, define
$$\operatorname{Ass}(M^\bullet):= \...
- 3,079
11
votes
0
answers
286
views
Does every finite poset have a rigid endomorphism?
Crossposted on Mathematics.
In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism ...
- 4,416
2
votes
0
answers
227
views
Tate module of elliptic curves; Commuting Hom functor and tensor product in the second coordinate
Let $\Lambda$ be the Iwasawa Algebra of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_{cyc}$ of $\mathbb{Q}$. Let $\widehat{\Lambda}$ be its Pontryagin dual (i.e the ...
- 313
6
votes
3
answers
594
views
Spectrum of a ring (studied by Krull?) of rational functions
Let $k$ be an algebraically closed field and $\mathbb A^2_k=\operatorname {Spec}k[x,y]$ the affine plane over $k$.
Consider the ring $R \subset k(x,y)$ of the rational functions on the plane defined ...
- 16.2k
54
votes
8
answers
58k
views
Modern algebraic geometry vs. classical algebraic geometry
Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...
CommunityBot
- 1
6
votes
0
answers
461
views
Strict Henselization vs base-change to algebraic closure
Let $x$ be a smooth $k$-point on a variety $X$ over a field $k$ of characteristic $0$.
Is the strict Henselized local ring $\mathcal{O}_{X,x}^{\mathrm{sh}}$ the same as $\mathcal{O}_{X,x}^{\mathrm{h}} ...
- 15.4k
1
vote
0
answers
270
views
Almost ring theory and derivations
I don't understand the definition of $\boldsymbol{\Omega}_A$ in the context of almost rings. In Gabber and Ramero https://arxiv.org/pdf/math/0409584.pdf it is covered in 9.6.12. How is $\boldsymbol{\...
- 16.2k
2
votes
0
answers
55
views
A small lemma in Schlessinger's criterion paper
In the construction of a hull in Schlessinger's paper, one small lemma used is not clear in my opinion. That should be stated as follows:
Let $(R,m)$ be a Noetherian complete local ring, $I_1\supset ...
- 41