All Questions
1,966 questions with no upvoted or accepted answers
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Rlim versus tensor product
Let $R$ be a coherent ring, and let $(M_n)_{n\geq 1}$ and $(N_n)_{n\geq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 \lim M_n=R^1 \lim N_n = 0$, is it true also that $R^1 ...
- 13.1k
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313
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Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
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265
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Chevalley restriction theorem: group vs lie algebra version
Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and ...
- 689
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220
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Finitely generated commutative rings with the same profinite completion
Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...
- 1,418
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419
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Are most semigroups nilpotent of degree 3?
A semigroup is nilpotent of degree 3 if every product of 3 elements gives the same result. In 2012, Andreas Distler and James D. Mitchell wrote that:
It is part of the folklore of semigroup theory ...
- 22.3k
8
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119
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Catenarity and epimorphisms of rings
Let $R$ be a commutative ring. The following are well-known:
If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary.
If $R$ is catenary and $S\subseteq R$ is ...
- 6,700
8
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548
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Foundational Questions on Adic Spaces
There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
- 2,923
8
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500
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Curvilinear locus in the Hilbert scheme of points
Let $X$ be a smooth complex projective variety of dimension $d$. Then the Hilbert scheme of $n$ points $X^{[n]}$ is not irreducible in general, but it has always the main component $X^{[n]}_{sm}$ of ...
- 577
8
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438
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If $A$ is normal and $\Omega^1_{B/A}=0$ then $B$ is normal
Let $A\subseteq B$ be two noetherian domains with fraction fields $k$ and $L$, respectively. Assume that $A$ is normal and $B$ is finite as $A$-module. I'm asking myself if $B$ is also normal if $\...
- 1,252
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127
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universally open and connected fibers
Let $A$ be a coherent ring, and consider the map:
$Spec(A[[t]])\rightarrow Spec(A)$,
in particular, we know that it's flat. Is it universally open? Does it have connected fibers?
N.B: Easy if A is ...
- 3,472
8
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213
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"Rings" with partially defined addition in Algebra or Algebraic Geometry?
Were there ever considered sets $P$ with associative multiplication and a partially defined commutative, associative addition, $+: U\to P,$ $U\subset P\times P$, such that $x(y+z)=xy+xz$ when the left ...
- 2,390
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575
views
Polynomial maps over $\mathbb{Z}$
It is know that an injective polynomial map $f:\overline{\mathbb{Q}}^{n} \longrightarrow \overline{\mathbb{Q}}^{n}$ is an bijection with inverse regular (Cynk-Rusek theorem). My question is following:
...
- 348
8
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480
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Connections and curvature in commutative algebra
Since on any commutative algebra $R$ over ring $S$ we have module of Kahler differentials $(\Omega_{R/S},d)$ which extends to the algebraic de-Rham complex $(\Omega^\bullet,d),$ it is natural to ...
- 1,615
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350
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What does the characteristic polynomial of an element in a finite flat $R$-algebra tell you?
Let $R$ be a noetherian ring, and $B$ a finite locally free $R$-algebra. Since $B$ is locally free, for every $b\in B$, multiplication by $b$ gives an $R$-linear endomorphism of $B$ as a locally free $...
- 7,527
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439
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Involutions on power series $\mathbb{C}[[X_1,\ldots,X_n]]$
For the ring of formal power series $\mathbb{C}[[X_1,\ldots,X_n]]$ over complex numbers,
is it true that any automorphism of order $2$ is after change of co-ordinates
given by $X_i\mapsto \pm X_i$?
- 161
8
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446
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Capitulation of ideal classes in general Dedekind Domains
I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up:
Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...
- 1,588
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623
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Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?
All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb Z\...
- 403
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337
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flatness and derived completion
Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...
- 3,472
8
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297
views
scheme of commuting matrices
Let $k$ be any field. Let $r$ and $n$ be two positive integers.
Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices $...
- 183
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363
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When is $I \otimes_A \hat{A} \cong I\hat{A}$?
Let $A$ be a commutative ring and $I$ a (finitely generated) ideal in $A$. We denote by $\hat{A}$ the $I$-adic completion of $A$, i.e. $\hat{A} = \varprojlim(A/I \leftarrow A/I^2 \leftarrow \ldots)$.
...
- 189
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4k
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Kunneth spectral sequence
In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also hold ...
- 1,589
8
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430
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name for a degree-like invariant of a power series over a commutative ring
Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f =...
- 1,802
8
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429
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History of the characterization of commutative Artin rings
When it comes to the world of "classical" (pre-homological) Noetherian commutative algebra, I tend to think of most of the results (Krull's intersection theorem, the principal ideal theorem, etc.) as ...
- 639
8
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answers
210
views
Smallest class of rings closed under familiar operations
Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some ...
- 5,759
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1k
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Two definitions of smoothness?
This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.
...
- 3,555
8
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307
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Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them
I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
- 2,527
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3k
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elementary exact sequence of normal sheaves
Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$.
I'm trying to show that there is an exact sequence of normal bundles.
$0 \rightarrow N_{Z/Y} \rightarrow N_{Z} \...
- 337
8
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answers
494
views
"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,595
7
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202
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Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.1k
7
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131
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When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
Is there any simple algorithm or criterion to check it?
I have chosen the complex ...
- 171
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225
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Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,287
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274
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Is there a relevant universal property for Fitting ideals?
Let $S$ be a scheme and $\mathscr F$ a quasi-coherent sheaf on $S$, locally finitely presented. Set $S_{-1}=S$ and for all $n\geqslant 0$, let $S_n$ be the closed subscheme of $S$ defined by the $n$-...
- 1,584
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167
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Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
- 71
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295
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A minimal semigroup generating subset of the additive reals
I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
- 2,023
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138
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
- 41.9k
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365
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Residue field of a ring does not depend upon the maximal ideal
Let $\mathbb{K}$ be a field and let $A$ be a $\mathbb{K}$-algebra. We will say that $A$ is residually $\mathbb{K}$ if for every maximal ideal $\mathfrak{m}$ we have that the structural morphism $\...
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275
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Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
- 696
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194
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Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 695
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284
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Generic behavior of "polynomialish" models of $\mathsf{Q}$
(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.)
Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
- 20.5k
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235
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Brauer group of the Henselization
Let $R$ be a Noetherian local ring and let $R^h$ be its Henselization. What can we say about the kernel and range of the map
$$
\operatorname{Br}(R) \rightarrow \operatorname{Br}(R^h)?
$$
Are there ...
- 81
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169
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Do ideals in regular local rings contain prime ideals of smaller heights?
Suppose $R$ is a regular local ring. If $I$ is ideal of height strictly larger than $h$, does $I$ contain a height $h$ prime ideal?
I’m particularly interested in the case of mixed characteristic ...
- 476
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344
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Irreducibility of a palindromic polynomial
I have strong reasons to believe that the palindromic polynomial $p_n(x)$ defined by
$$p_n(x) = x^{2n}+2x^{2n-1}+3x^{2n-2}+ \cdots+ nx^{n+1}+(n+3)x^{n}+nx^{n-1}+\cdots+2x+1$$
is irreducible in $\...
7
votes
0
answers
273
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Is there a Swan-style description of topological K-homology?
A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\...
- 1,020
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0
answers
823
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On Grothendieck's abstract definition of differential operators
I have heard that there is the following abstract definition due to Grothendieck of differential operators on a module $M$ over a commutative associative unital algebra $A$ over a field of ...
- 21.8k
7
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answers
181
views
Classification of Frobenius algebras of small dimensions
Despite (commutative) Frobenius algebras over a field $K$ being a very popular class of algebraic objects, it seems no attempt of classification (up to $K$-algebra isomorphism) for them has been ...
- 26.5k
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260
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Generating the monoid of injective endomorphisms of the free group
Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injective group endomorphisms $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set ...
- 121
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659
views
Row rank and column rank of matrix with entries in a commutative ring
Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is ...
- 356
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0
answers
127
views
Contractible affine surfaces of log Kodaira dimension 2
The first examples of contractible smooth affine algebraic surfaces (over the complex numbers) of log Kodaira dimension 2 were constructed in a famous paper of Ramanujam
https://www.jstor.org/stable/...
- 3,758
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205
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Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers
Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
7
votes
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answers
181
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Self-flat modules
(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a ...
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