All Questions
6,055 questions
7
votes
3
answers
525
views
Is the class of inverse semigroups globally determined?
This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher ...
- 10.5k
17
votes
1
answer
687
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
- 7,893
1
vote
1
answer
100
views
Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
- 923
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 8,945
5
votes
0
answers
1k
views
The group of automorphisms of a polynomial ring in two variables over an integral domain
It is well-known that the group of automorphisms of a polynomial ring
$k[x,y]$, $k$ is any field, is a free product of $A(2)$ and $J(2)$
amalgamated along their intersection,
where $A(2)$ is its ...
0
votes
0
answers
90
views
Invariants of primary groups
In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
- 63.9k
7
votes
1
answer
415
views
An example of radical ideal which is irreducible but not prime
$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal.
In case $R$ is Noetherian, the radical of $I$ being ...
- 14.6k
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
1
vote
0
answers
132
views
Quotient of a polynomial ring with a prime ideal is Cohen$-$Macaulay
[Bruns-Herzog, Exercise 2.1.17] Let $k$ be a field and $R = k[x_1, . . . , x_n]$. Suppose $\mathfrak{p} \subset R$ is a prime ideal, $ht\mathfrak{p} \in \{0, 1, n − 1, n\}$. Show that $R/\mathfrak{p}$ ...
3
votes
0
answers
106
views
Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
- 412
0
votes
1
answer
172
views
Is the integral closure of a henselian local domain of dimension $1$ again local?
Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$.
If $R$ is henselian, then is $\overline R$ also a local ...
- 19.6k
0
votes
0
answers
178
views
Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced
I have a question about correctness of following statement claimed here in $\boxed{2} \ $:
Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...
- 6,038
19
votes
3
answers
3k
views
Total ring of fractions vs. Localization
Let $R$ be a commutative ring and denote by $K(R)$ its total ring of fractions, the localization of $R$ with respect to $R_{\mathrm{reg}}$. For every multiplicative subset $U \subseteq R$ there is a ...
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
- 16.2k
5
votes
3
answers
5k
views
Serre type vanishing theorem of coherent sheaves on quasi-projective variety?
For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheaf, $i\geq 1$ and sufficiently large $n$. I am wondering, is there a similar type of ...
- 6,367
2
votes
1
answer
194
views
Continuity of Moore-Penrose generalized inversion
Any matrix $A\in\mathbb{C}^{m\times n}$ has a unique generalized inverse $A^{\dagger}\in\mathbb{C}^{n\times m}$ with the properties $$AA^{\dagger}A=A,\qquad A^{\dagger}AA^{\dagger}=A^{\dagger},\qquad (...
- 63.9k
3
votes
1
answer
123
views
Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
- 12.9k
1
vote
1
answer
146
views
Cohen–Macaulayness of $k[[x^2, x^3, xy, y]]$ over $k[[x^2, y]]$
Let $k$ be a field and $R = k[[x^2, y]]$ and $S = k[[x^2, x^3, xy, y]]$. Since $R \subset S$, is $S$ Cohen–Macaulay as $R$-module?
To check this, what I have observed is that in $S$, the maximal ...
- 16.2k
0
votes
1
answer
153
views
Unitary representation of a group of automorphism on an abelian algebra
Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \...
- 42.8k
4
votes
1
answer
160
views
DG algebra structure on minimal free resolution of modules over regular local ring
Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
- 561
5
votes
2
answers
868
views
If every ascending chain of ideals leading up to an ideal stabilises, is the ideal finitely generated?
I'm a fourth-year undergraduate currently studying a master's degree. In the last couple of weeks, I have been wondering about the interaction of the Noetherian condition with the prime ideals of a ...
- 63.9k
2
votes
1
answer
223
views
Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 16.1k
5
votes
1
answer
357
views
Does the category of integral domains admit a symmetric monoidal structure?
Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...
- 63.1k
7
votes
1
answer
450
views
When is a module a filtered colimit of finitely presented submodules?
For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both:
a filtered colimit of finitely generated $R$-submodules (by considering all finite subsets of $M$ and ...
- 63.1k
0
votes
3
answers
2k
views
Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
5
votes
1
answer
210
views
Relation between row space and column space resp. null space and left null space over general rings
Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
- 156k
2
votes
1
answer
271
views
Apropos of two groups being globally isomorphic iff they are isomorphic
Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced ...
- 10.5k
4
votes
2
answers
193
views
Intersection of commutative factorial domains: completely integrally closed and Krull domain
Let $A=\bigcap_{t\in T}D_t$ be an integral domain such that $D_t$ is a commutative factorial domain for every $t\in T$. It is quite natural to see that $A$ is a completely integrally closed domain. ...
- 4,985
1
vote
2
answers
309
views
Ideal quotient and regular sequences
Let $(R,m)$ be a Noetherian local ring. $(x_1,...,x_d)\subset (y_1,...,y_d)$ are two $R$-regular sequences. If we write $x_i=\sum_{j=1}^{d}a_{ij}y_j$, then we have $(x_1,...,x_d):(y_1,...,y_d)=(x_1,......
- 878
5
votes
2
answers
788
views
When does End(M) consist entirely of zero, zero divisors, and units?
Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...
- 2,821
0
votes
0
answers
136
views
Understanding the relations without the knowledge of Plucker relations [duplicate]
Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so ...
- 839
2
votes
0
answers
60
views
Length of $\text{Tor}$ modules in complete intersection rings with characteristic $p$
Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$, it follows that $l(F(M))\geq p^nl(M)$, where $n=...
- 3,073
1
vote
1
answer
203
views
Degree three, codimension one subvarieties lying on a quadratic hypersurface
Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that
$V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...
- 39.3k
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 3,451
0
votes
1
answer
152
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 63.1k
0
votes
1
answer
327
views
Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
- 48.8k
6
votes
0
answers
178
views
Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
- 1,313
-1
votes
1
answer
294
views
Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?
The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
CommunityBot
- 1
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
3
votes
1
answer
263
views
(non)reduced stabilizer scheme
A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...
- 4,111
5
votes
1
answer
615
views
Is every character of the algebra of continuous functions on a locally compact space some evaluation?
Given any locally compact Hausdorff space $X$, let $C(X)$ denote the complex algebra of all complex-valued continuous functions on $X$.
Question. Given an arbitrary character (i.e. a non-zero ...
- 12.9k
5
votes
0
answers
285
views
Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
21
votes
1
answer
638
views
Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 28.7k
65
votes
4
answers
22k
views
When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
- 1,313
8
votes
2
answers
2k
views
Does every regular Noetherian domain have finite Krull dimension?
Does every regular Noetherian domain have finite Krull dimension?
Background: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. ...
- 4,713
1
vote
0
answers
106
views
Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
0
votes
0
answers
124
views
Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
- 2,907
0
votes
1
answer
213
views
number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
- 221
2
votes
1
answer
184
views
Gorenstein projective module over commutative local algebras
Let $A$ be a local commutative finite dimensional algebra over a field $K$.
An $A$-module $M$ is called Gorenstein projective if $M$ is reflexive, $Ext_A^i(M,A)=0=Ext_A^i(M^{*},A)$ for all $i>0$ ...
- 642
2
votes
1
answer
117
views
Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 12.9k