All Questions
665 questions
3
votes
0
answers
67
views
Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...
- 10.5k
3
votes
1
answer
257
views
Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $L(R)$ ...
- 2,605
3
votes
1
answer
3k
views
About the Definition of Flat Morphism (Flat Sheaf)
I have a confusion about the definition of flat sheaf of module.
Let $f: X \rightarrow Y$ be a morphism of schemes and $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$ module. Then $\mathcal{F}$ is flat ...
- 165
3
votes
1
answer
413
views
Do there exist these real polynomials?
Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients, and $n>20$ a natural number such that
$$\left(\sum\limits_{k=0}^{n} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$
I have ...
- 4,074
3
votes
1
answer
196
views
Number of free summands of finite local extensions
Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
etale on the punctured spectrum
not flat / etale at the origin
and such that the residue fields $R/m = S/n$...
- 20.5k
3
votes
1
answer
618
views
When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
- 3,079
3
votes
1
answer
240
views
Split monomorphisms 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. Further $X\subseteq M$ and for ...
- 696
3
votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 2,857
3
votes
0
answers
334
views
Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
3
votes
1
answer
404
views
How many monoids with $n$ arrows exist?
How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
- 41
3
votes
0
answers
161
views
Making the powerset into a topological monoid
Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
- 11.8k
3
votes
0
answers
151
views
Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
- 10.5k
3
votes
2
answers
810
views
What is the divisibility theory for Bezout Domains?
There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is ...
- 30.1k
3
votes
1
answer
736
views
Can height one maximal ideals in the normalization contract to non-height one primes in the base?
Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
- 1,802
3
votes
1
answer
881
views
Are essentially smooth schemes noetherian?
Let $k$ be a field. I am unable to find a precise definition of essentially smooth $k$ schemes, but I will stick to this definition below, since this is exactly what I need:
Definition: A $k$-scheme $...
- 525
3
votes
1
answer
378
views
Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
$\...
- 11.8k
3
votes
1
answer
162
views
Classification of finitely generated chain groups
An ordered pair $\ \mathbf X := (X\ d)\ $ is called a chain group $\ \Leftarrow:\Rightarrow\ X\ $ is an abelian group, $\ d:X\rightarrow X\ $ is an abelian group endomorphism, and $\ d\circ d= 0$.
A ...
- 4,786
3
votes
0
answers
354
views
Weak assassins and essential morphisms
Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.
Over non-noetherian rings the notion of associated ...
- 6,700
3
votes
1
answer
2k
views
Are quotients of affine schemes by finite groups faithfully flat?
Let $R$ be a (Noetherian) ring, and $G$ a finite group acting on $R$. Consider the subring $R^G$. Is the map $R^G\rightarrow R$ faithfully flat?
If not, does this become true if we restrict to ...
- 2,824
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
1
answer
224
views
Rings or algebras with many nilpotent elements and efficient computation
Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...
- 25.4k
3
votes
1
answer
420
views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
- 87
3
votes
3
answers
1k
views
Topology on $p$-adic period ring in an article by Fontaine
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathcal{C}$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathcal{C}/p,$$ where the transition maps in ...
3
votes
0
answers
235
views
Is there a converse of Abhyankar-Moh-Suzuki theorem?
The following question is the same as this question; I ask it here, since I have not got any comments there (I really apologize if I should have waited some more time before asking it here; it is just ...
- 2,837
3
votes
1
answer
186
views
If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?
EDIT: The question was originally about general Noetherian rings instead of PID's. Thanks to YCor for pointing out how wrong this was in the comments below (1 2 3).
Question 1: Let $R$ be a PID. ...
- 64k
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
3
votes
1
answer
785
views
Presentation of the tautological bundle of the Grassmannian
Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the ...
- 235
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
3
votes
1
answer
285
views
Cancellative semigroup on a distributive lattice
Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?
- 1,145
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
3
votes
0
answers
303
views
Pseudomodules, "general coherence theorem"
A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
- 2,312
2
votes
1
answer
514
views
When does $R [x]/I $ has infinitely many idempotents?
Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...
- 51
2
votes
0
answers
102
views
Characterizing all simple algebraic ring extensions of $\mathbb{C}[x]$ having no prime elements
Let $w$ be an algebraic element over $\mathbb{C}[x]$, with minimal polynomial $f(t)=c_mt^m+\cdots+c_1t+c_0$, $c_i \in \mathbb{C}[x]$.
Is it possible to characterize (in terms of the $c_j$'s) all ...
- 2,837
2
votes
1
answer
1k
views
common roots of bivariate polynomial equations
Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...
- 59
2
votes
1
answer
310
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
- 2,837
2
votes
0
answers
139
views
Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$
The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...
- 2,406
2
votes
1
answer
222
views
$C=A \cdot B$ matrices and exact sequence of $DVR$-modules
I'm looking for a proof or a reference for the following statement:
Let $R$ be a DVR (Discrete Valuation Ring) and $p$ a prime element, and let $\mathfrak a$, $\mathfrak b$ and $\mathfrak c$ be ...
- 291
2
votes
2
answers
974
views
Smoothness of hypersurfaces in Grassmannians
I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.
Grassmanians of planes
The $(2,n)$-Grassmannian, denoted $Gr(2,n)$...
- 13k
2
votes
0
answers
268
views
K-theory and completion
I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...
- 165
2
votes
0
answers
524
views
Irreducibility over the field of fractions of a quotient of a polynomial ring
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61
2
votes
0
answers
436
views
Why is $\Omega_k(C^\infty(M))\to\Omega^1(M)$ surjective?
Let $M$ be a smooth manifold and let $A=C^\infty(M).$
We consider module of Kahler differentials $\Omega_k(A)$ and module of 1-forms $\Omega^1(M).$ Denote Kahler differential by $d_k$ and classical ...
- 1,615
2
votes
0
answers
122
views
First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
2
votes
2
answers
241
views
If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 10.5k
2
votes
1
answer
337
views
Application of the adjoint functor theorem to get the right adjoint of the forgetful functor from $\delta$-rings to rings (the Witt vectors)
I am studying $\delta$-rings and Witt vectors from [K] (the definition of $\delta$-ring is [K, 2.1.1]), and I am having trouble verifying that everything in Kedlaya's definition for the Witt vectors ...
2
votes
1
answer
182
views
Terminology for the equation $a=a+b$ in commutative semigroups
Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...
2
votes
1
answer
279
views
Fixed points for action of finite group acting on Noetherian ring is a local Noetherian ring
Let $R$ be a local Noetherian ring which contains the field $\mathbb{Q}$ of rational numbers, let $G$ be a finite group acting on $R$, and let $R^G \subseteq R$ be the fixed points for the action of $...
user64623
2
votes
2
answers
589
views
Graded-irreducible ideals are irreducible?
One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?
Let $I$ be a graded ideal in a polynomial ring over a field....
- 1,313
2
votes
0
answers
106
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
2
votes
1
answer
122
views
If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
- 10.5k
2
votes
1
answer
640
views
How to show that the intersection of two certain affine varieties is reduced?
$\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is ...
- 165