All Questions
6,057 questions
4
votes
1
answer
138
views
Decide whether there are "linear" relations between quadrics
Let $k$ be an algebraically closed field of characteristic $0$. For a homogeneous ideal $I=(q_1,\dots, q_k)\subset k[x_0,\dots,x_n]$ generated by quadrics, is there a method to decide whether the ...
- 30.6k
3
votes
0
answers
123
views
Dimension of the socle of the first local cohomology module
Let $M$ be a graded $\mathbb{C}[z_0,\dots,z_n]$-module. Using local duality one can show that
$$
\dim_\mathbb{C} (\text{soc} H_\mathfrak{m}^1(M))_k = \beta_{n,k+n+1}(M).
$$
Here $H_\mathfrak{m}^1(M)$ ...
- 294
5
votes
1
answer
334
views
Short proof a monoid is a group iff every splitting is right homogeneous
In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...
- 63.9k
1
vote
1
answer
229
views
Generic Galois alteration of an arithmetic model with semistable special fiber
Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations ...
- 1,144
1
vote
1
answer
126
views
Do these sorts of submonoids go by a particular name?
Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows:
$$r(x)=\{y\in M:xy=x\}$$
$$l(x)=\{y\in M:yx=x\}$$
Do these sorts of sub-monoids go by a particular name?...
- 63.9k
6
votes
2
answers
1k
views
Kernel of evaluation map into field of quotients
Let $R$ be an integral domain and for $a \in R$ denote by $\text{eval}_a: R[X] \to R$ evaluation at $a$. It's well-known (and easy to see) that
$$\ker(\text{eval}_a)=(X-a).$$
The next more ...
- 7,893
5
votes
1
answer
273
views
Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$
Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...
- 55.9k
6
votes
1
answer
262
views
Is there a finite extension with a non-trivial class group of any PID?
Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?
- 28.7k
0
votes
0
answers
75
views
Homomorphic image of $B_{\lambda}^o(S)$ is the Brandt $\lambda^o$-extension of some monoid with zero
Let $S$ be a monoid with zero and $I_{\lambda}$ be an indexed set, then $B_{\lambda}(S) = \{ (\alpha, s , \beta ) : \alpha , \beta \in I_{\lambda}, s\in S \} \cup \{0\}$ is a semigroup and $J = \{ (\...
- 63.9k
1
vote
0
answers
973
views
Intersection of principal ideals
Let $x,y$ be nonzero elements in a commutative ring $R$. Is $(x)\cap (y)$ always finitely generated?
What if we further assume that $R$ is an integral domain? Can we construct non-Noetherian non-local ...
8
votes
1
answer
372
views
Commutative ring $R$ with no nontrivial idempotents, with a localization $R_r$ with infinitely many idempotents
I am looking for a commutative ring $R$ with $1$ such that $R$ has no idempotents and there exists $r\in R$ such that the localization ring $R_r$ has infinitely many idempotents.
- 4,745
6
votes
1
answer
245
views
Transcendent basis for the field of multisymmetric functions
It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is,
rational function in variables $x_{11}, \ldots, x_{1m}, \ldots, x_{n1}, \ldots, x_{nm}$ ...
- 16.9k
1
vote
1
answer
154
views
Locally isomorphic algebras over a Dedekind domain
Let $R$ be a Dedekind domain. Let $A$ and $B$ be two finitely generated domains over $R$. Assume that for every maximal ideal $\mathfrak{p}\subset R$ the $R_{\mathfrak{p}}$-algebras $A_{\mathfrak{p}}$ ...
- 28.7k
0
votes
0
answers
93
views
A semifield of characteristic zero may have a finite number of elements
A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.
I ...
- 63.9k
2
votes
1
answer
381
views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
- 1,052
1
vote
0
answers
96
views
Degree reduction in decompositions of multivariate polynomials
Is the following statement true?
Let $m,n,d$ be natural numbers. Then there exists a natural number $D=D(d,m,n)$ with the following property: If a polynomial $P(x_1,\dots,x_n)$ of total degree $d$ ...
- 3,599
0
votes
1
answer
74
views
Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples
$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
Is it true that there are ...
- 1,826
6
votes
1
answer
269
views
Algebraization of Bayesian networks?
The algebraization of classical propositional logic is Boolean algebra.
Bayesian networks are a generalization of classical propositional logic with probability truth-values.
What is the ...
- 63.9k
5
votes
0
answers
162
views
Classifying toposes of theories of rings that aren't local rings
The standard uses of toposes in algebraic geometry come from sites that look roughly like the syntactic sites of theories of local rings that they classify. This isn't particularly surprising, since ...
- 3,816
8
votes
0
answers
419
views
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
1
vote
1
answer
111
views
non-archimedean valuations on graded rings
Let $R$ be a commutative (non-trivially) graded ring. By a non-archimedean valuation I mean a map $v: R \to \Gamma \cup {0}$ such that for all $x,y \in R$, we have $v(x+y) \leq \max\{v(x),v(y)\}$, $v(...
- 9,061
7
votes
1
answer
292
views
Is $Tor_A(k,k)$ a bicommutative Hopf algebra?
Let $A$ be a commutative (or graded commutative) algebra over a field $k.$ In some sources, such as Mcleary's book on spectral sequences, Corollary 7.12, pg. 248, it is claimed that $\text{Tor}_A(k,k)$...
- 16.9k
5
votes
1
answer
403
views
Classification of finitely generated modules over non-commutative rings
Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...
- 35.2k
6
votes
2
answers
685
views
Integral domain over which any non-constant, one variable, irreducible polynomial has degree 1
Let $R$ be an integral domain such that every non-constant, irreducible polynomial $f(X) \in R[X]$ has degree $1$.
Q. is it true that $R$ is a field?
If $0 \ne a \in R$ , then $X^2-a$ is ...
- 103
3
votes
0
answers
186
views
Is cup product of cycle classes on Noetherian regular excellent scheme compatible with intersection
Let $\mathcal{X}$ be a Noetherian regular integral excellent scheme. Let $Y$ and $Z$ be algebraic cycles of codimension $c$ and $d$ on $\mathcal{X}$.
Let $n$ be a positive integer invertible on $\...
- 43
7
votes
0
answers
824
views
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
9
votes
1
answer
313
views
Concerning $k \subset L \subset k(x,y)$
The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset ...
- 12.9k
2
votes
0
answers
105
views
Reference request: Differential graded structures in mixed characteristic
I am looking for references/papers on differential graded structures and their applications in mixed characteristic. The following I have discuss differential graded algebras in the general, not in ...
- 327
8
votes
1
answer
642
views
Property of the trace on finitely generated projective modules
Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
- 28.7k
5
votes
1
answer
400
views
Dual of $End_A(M)$
Let $A$ be a finitely generated $\mathbb C$-algebra and an integral domain. Assume also $A$ is Gorenstein. Let $M$ be a finitely generated torsion-free $A$ module.
Is it true that $Hom_A(End_A(M), A)\...
CommunityBot
- 1
1
vote
0
answers
153
views
A structure on the groupoid of algebraic closures
Given a field $k$ let $\Omega(k)$ be the set of algebraic closures of $k$.
$\Omega(k)$ is obviously a groupoid. At each element $\bar{k}$ of $\Omega(k)$ we have its automorphism group over $k$, ...
- 63.9k
3
votes
2
answers
363
views
When is a monomial rational map on the projective space birational?
Let $k$ be an algebraically closed field of characteristic $0$.
For $\alpha :=(a_1,\dots,a_{n+1})\in \mathbb N^{n+1}_{\ge 0}$ , let $\bar x^{\alpha}:= x_1^{a_1} \dots x_{n+1}^{a_{n+1}} \in k[x_1,\...
- 63.9k
3
votes
1
answer
149
views
Question on monoid algebras
Let $G$ be a finite monoid.
Question 1: In case the monoid algebra $A=kG$ is weakly symmetric (meaning soc(P)=top(P) for each indecomposable projective modules), is $kG$ even symmetric (meaning $A \...
CommunityBot
- 1
14
votes
2
answers
748
views
Does any derivation of commutative algebra preserve its nil-radical?
Given a commutative associative unital algebra over a field of characteristic zero.
Is it true that any derivation of it preseves its nil-radical?
More explicitly, let $D$ be a derivation of an ...
- 63.9k
0
votes
0
answers
413
views
When are the cotangent and tangent sheaves isomorphic?
Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
- 327
1
vote
1
answer
197
views
Chain of closed irreducible sets on Zariski Riemann spaces
Let $A$ be a domain and $K=\mathrm{Frac}(A)$.
The Zariski Riemann space $\mathrm{ZR}(K,A)$ is the set of all valuation rings of $K$ containing $A$. It comes with a natural center map
\begin{align}...
- 225
10
votes
2
answers
1k
views
Krull dimension of a local ring and completion
Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type.
Let $\hat{A}$ be its $\mathfrak{m}$-adic completion.
Do we have that $\...
- 131
11
votes
2
answers
950
views
Define Turing machine with algebraic concepts/structures
Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way) and then their definition is formalized (for example, in this way).
Is it ...
- 63.9k
3
votes
1
answer
271
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 26.5k
2
votes
0
answers
116
views
Bijection between monoid of nilpotent decreasing self-maps and local subsemigroup
Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
- 355
4
votes
1
answer
234
views
Detecting elements of nilpotent extensions via finitely generated ones
Let $K$ be a commutative unital ring field. Let $\pi:A \to K$ be a surjective homomorphism of commutative $K$-algebras with nilpotent kernel. (Recall that this means $\operatorname{Ker}(\pi)^n=0$ for ...
- 63.9k
7
votes
1
answer
499
views
Localization of symmetric monoidal categories and geometry
I have a series of vague questions, related to localization of symmetric monoidal categories.
Here is the context. Say we are working over a field of characteristic zero. Then the "one category ...
- 898
7
votes
0
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
9
votes
3
answers
2k
views
(Krull) dimension of any associated graded ring of a ring R equals the dimension of R
I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE.
For a local ring $(R,m)$, given any proper ideal $I$, the (Krull) dimension (from here on dimension means Krull ...
- 3,472
6
votes
3
answers
472
views
Spaces with unique endomorphism monoids
If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$.
We ...
CommunityBot
- 1
25
votes
3
answers
1k
views
What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?
This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
- 28.7k
2
votes
0
answers
359
views
Module structure for $\mathbb{Z}$
I am interested to know which module structures we can define in the additive group of integers $\mathbb{Z}$.
It is easy to prove that $\mathbb{Z}$ does not admit a vector space structure. (For more ...
- 63.9k
6
votes
0
answers
338
views
Are monomorphisms between algebraic spaces representable?
The question in the title can be reformulated as follows. Let $f : Y \to X$ be a monomorphism of algebraic spaces where $X$ is a scheme. Is it true that $Y$ is a scheme?
If $f$ is locally of finite ...
- 19.6k
5
votes
1
answer
388
views
Isomorphism of real closed fields
Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...
- 63.9k
3
votes
1
answer
155
views
Does ACC for principal ideal plus Krull dimension equal 0 imply DCC for principal ideals
Assume the ring is commutative and with 1.
We know that ACC + $\dim(R)=0$ imply DCC. However, if we only insist on the condition for principal ideals, can we conclude the same?
We know that having ...
- 63.9k