All Questions
6,056 questions
3
votes
0
answers
140
views
Dirichlet unit theorem for finite rings
Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
4
votes
0
answers
197
views
Quillen–Suslin theorem in a more general context
Let $A$ be a finite dimensional local Frobenius algebra that is Koszul.
Question: Is it true for the Koszul dual of $A$ that every finitely generated projective module is free? If not, is there a ...
- 26.5k
4
votes
1
answer
417
views
Is a solvable group satisfying a semigroup law?
Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We ...
- 883
39
votes
3
answers
8k
views
What is the "intuition" behind "brave new algebra"?
Y.I. Manin mentions in a recent interview
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
- 4,713
27
votes
3
answers
1k
views
Graded analogues of theorems in commutative algebra
Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...
- 30.3k
2
votes
0
answers
201
views
Automorphism group in finite dimensional case
Let $K$ be a field, $G_a := (K, +)$ be the additive group of $K$, and $X$ an affine variety.
I found the following claim: if $X$ admits a non-trivial $G_a$-action and $\dim(X) \ge 2$, then the group $\...
- 63.9k
3
votes
3
answers
486
views
On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map
\begin{align} \Phi_M: M \...
- 30.5k
2
votes
0
answers
92
views
Expressing elements in Verlinde ideal in terms of generators
It is known that the level $k$ Verlinde ring of $SU(n)$ is $R(SU(n))/I_k$, where $I_k$ is the Verlinde ideal. A set of generators of $I_k$ is given by $\{V_{(k+i)L_1}:=\text{Sym}^{k+i}V_{\text{std}}| ...
- 383
6
votes
1
answer
247
views
Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$
Let $p$ be a prime and let $n\geq 2$ be an integer.
The set $\mathbb{A}^n(\mathbb{F}_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}_p)$ ...
- 149k
3
votes
1
answer
260
views
K-projectivity for rings of finite homological dimension
Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
- 35.2k
2
votes
1
answer
167
views
Terminology for commutative ring whose Jacobson radical $J$ is nilpotent with semisimple quotient $R/J$
Is there a name for the following property of a commutative ring $R$:
its Jacobson radical $J$ is nilpotent, and $R/J$ is semi-simple?
(It is easily equivalent to: $R$ is a finite product of ...
- 1,507
3
votes
0
answers
69
views
Division algorithm for multivariable power series
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Consider the ring $R=\mathbb{Z}_p[[T]]$. Let $f,g \in R$ and assume that $f=a_0+a_1T+...$ with $a_i \in p\mathbb{Z}_p$ for $0\le i \le n-1$, but $...
- 429
8
votes
1
answer
645
views
Isomorphic morphisms. A 27-morphism category
Two morphisms of category $\ \mathbf C\ $ are isomorphic to one
another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that ...
- 4,786
6
votes
0
answers
369
views
Geometric meaning of localization at $(1+I)$?
Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
- 1,286
3
votes
0
answers
225
views
Complexity: Groebner bases method vs homotopy continuation method
Today, I came to know about homotopy continuation method to solve system of multivariate polynomials. This method finds its roots from the field of Numerical algebraic geometry.
I already know that ...
- 63.9k
0
votes
1
answer
139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 7,962
1
vote
1
answer
215
views
Using equational Jacobson condition to prove element lies in radical of ideal
Recall the Jacobson radical of a ring consists of elements $f\in A$ such that $1-gf\in A^\times$ for every $g\in A$. Say an ideal $I\vartriangleleft A$ is Jacobson if in the quotient $A/I$ the ...
CommunityBot
- 1
9
votes
0
answers
366
views
Proof of Artin–Rees / Krull intersection motivated by universal property of blowup
I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger.
Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ...
- 12.9k
1
vote
1
answer
400
views
Bound for multiplicities of closed points on scheme
Let $K$ be a perfect field, and let $f_1, \ldots, f_m \in K[X_1,\ldots,X_n]$ be polynomials. Consider the affine scheme
$$X = \mathrm{Spec}(K[X_1,\ldots, X_n]/(f_1,\ldots,f_m))$$
and let $N = \dim(X)$....
- 5,359
1
vote
1
answer
198
views
What are the properties of this set of infinite matrices and operations on them?
Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & . \\
0 & a_0 & a_1 & a_2 & . \\
0 & 0 & a_0 & a_1 & . \\
...
- 11.2k
3
votes
1
answer
102
views
Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)
Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $...
- 6,262
2
votes
1
answer
111
views
Uncountable integral domain such that every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra
Is there an uncountable integral domain such its every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra?
- 63.9k
1
vote
0
answers
47
views
Can the embedding dimension of a finite local algebra change after restricting to a finite subfield?
The embedding dimension of a commutative $k$-algebra is the minimum $n$ such that it is a quotient of the polynomial algebra in $n$-variables.
The embedding dimension of $\mathbb{F}_2\times \mathbb{F}...
- 325
2
votes
1
answer
150
views
Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as an $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$?
Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an extension of finite fields.
Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as an $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$ (i.e. ...
- 325
39
votes
5
answers
6k
views
Algebraic machinery for algebraic geometry
Hello everybody,
I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative ...
- 19.6k
2
votes
0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
4
votes
0
answers
135
views
Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain
I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...
- 259
1
vote
0
answers
99
views
Finding an injective envelope containing another injective envelope
Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...
- 12.9k
6
votes
1
answer
342
views
Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$
I am interested in the poset of all ideals of the local ring
$$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$
$n=1$ is trivial. $n=2$ takes little work and it is shown below....
- 575
14
votes
2
answers
1k
views
Invariants of matrices (by simultaneous $\mathrm{GL}_n$ conjugation) over arbitrary rings
$\DeclareMathOperator\GL{GL}$Let $R$ be a commutative ring, let $R[n] := R[M_d^{\oplus n}]$ be the polynomial ring on $nd^2$ variables corresponding to the coordinates of $n$-many $d\times d$ matrices....
- 7,527
2
votes
2
answers
621
views
Why is $M$ torsion-free?
I am studying the following article
https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf
The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof:
How does it help ...
- 30.5k
2
votes
0
answers
159
views
Mayer-Vietoris sequence from a bicartesian square of commutative rings
An article that I am reading quotes the following theorem (5.3 p.481, reformulated to focus on the commutative case) from Algebraic K-Theory by Hyman Bass:
Let $\require{AMScd}$
\begin{CD}
A @>p_2&...
0
votes
0
answers
95
views
On some loci of rings
Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set
$$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$
$$ nP(R) =\{\mathfrak p \in ...
- 89
3
votes
0
answers
132
views
When is the following a formula for local cohomology?
Suppose $R$ is a Noetherian local ring, and $\kappa$ its residue field. For $R$ module $M$, we can consider the module
$$N:=\kappa \otimes_S RHom(\kappa,M)$$ where $S$ is the derived ring of ...
- 2,356
1
vote
1
answer
248
views
A variation on $k(x^2,x^3)=k(x)$
Let $k$ be a field of characteristic zero, for example $k=\mathbb{R}$ or $k=\mathbb{C}$.
Of course, $k(x^2,x^3)=k(x)$, since $x=\frac{x^3}{x^2}$.
Let $f_1,\ldots,f_n,g_1,\ldots,g_m \in k[x]$, $n,m \...
- 149k
1
vote
0
answers
65
views
There is a ring with multiplication. Can we find a formula for division based on formula for multiplication?
Studying divergent integrals, I found a good formula for their multiplication:
$\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)...
- 10.1k
7
votes
1
answer
1k
views
Polynomials which are functionally equivalent over finite fields
Recall that two polynomials over a finite field are not necessarily considered equal, even if they evaluate to the same value at every point. For example, suppose $f(x) = x^2 + x + 1$ and $g(x) = 1$. ...
- 149k
2
votes
0
answers
174
views
de Rham cohomology of a specific ring
I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of:
$$
\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
- 63.9k
7
votes
1
answer
444
views
Does the strict henselization satisfy Going-Up?
$\DeclareMathOperator\sh{sh}$This question is cross-posted from Math.SE where it has gone unanswered for a week -- perhaps it is harder than I guessed. My question is this:
Let $A$ be a local ...
- 2,851
2
votes
1
answer
186
views
$k[X_1,\ldots,X_n]/Q$ is UFD for non-singular quadratic form $Q$ and $n\ge 5$
I am looking for a reference for the following result. Thanks in advance.
Let $k$ be a field of any characteristic other than $2$.
Klein and Nagata showed that the ring $R:=k[X_1,\ldots,X_n]/Q$ is a ...
- 26.5k
1
vote
1
answer
230
views
Properties of the generic matrix - struggles with constructive proofs
Write $A=(x_{ij})$ for the generic matrix (comprised of indeterminates) defined over $\mathbb Z[x_{11},\dots,x_{nn}]$. In their constructive commutative algebra book, Lombardi and Quitte write that ...
- 33.8k
3
votes
0
answers
161
views
A Nakayama type of claim for countably generated modules on complex affine varieties
Let $U\subset \mathbb{A}^n_{\mathbb{C}}$ be any Zariski open affine subvarity. Let $M$ be an $\mathcal{O}(U)$-module. Suppose $M$ satisfies $M\overset{L}{\otimes}\mathbb{C}_{\mathfrak{M}}\cong 0$ for ...
- 163
2
votes
1
answer
166
views
Submonoid of free monoid with certain properties
Let $N$ be a submonoid of a free monoid $M$ such that
$m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C)
Do such submonoids ...
- 491
4
votes
1
answer
502
views
Infinite linearly independent set in finitely generated module
Let $R$ be a (commutative, otherwise the answer is easy, see the comment below) ring and let $M$ be a finitely generated $R$-module. Is it possible that $M$ admits an infinite linearly independent set?...
- 63.9k
9
votes
0
answers
347
views
What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
19
votes
1
answer
2k
views
Examples of solid abelian groups
I am reading through Clausen's and Scholze's Lectures on condensed mathematics. I am struggling to understand the concept of solid abelian groups so I am looking for some examples.
Is the underlying ...
- 21.3k
4
votes
0
answers
236
views
Is this property of polynomials generic?
Let $n \geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure ...
- 26.9k
1
vote
0
answers
169
views
Choosing generators of a submodule with divisibility properties
Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$.
Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the ...
- 6,237
1
vote
0
answers
119
views
Commutative monoid gradings via group scheme actions
$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and ...
- 11.8k
35
votes
3
answers
5k
views
Matrix factorizations and physics
I have heard during some seminar talks that there are applications of the theory of
matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
- 5,711