All Questions
6,057 questions
4
votes
0
answers
135
views
$K$-group of category of bounded chain complexes of Projective modules with finite length homologies
For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
- 1,209
3
votes
1
answer
2k
views
Prime ideals of formal power series ring that are above the same prime ideal
Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the
ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$,
then $I[[X]]$, the set of power ...
- 938
2
votes
1
answer
158
views
Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?
Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
- 291
7
votes
2
answers
822
views
Determining the kernel of the localization map when defining the localization by generators and relations à la Serre
All rings considered will be commutative and unitary. Let $A$ be a ring, $S \subseteq A$ a multiplicatively closed subset. The localization $\lambda_S : A \longrightarrow A[S^{-1}]$ can be ...
- 7,893
9
votes
1
answer
714
views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
- 3,526
5
votes
1
answer
218
views
Closure of the product of subfunctors
Background:
Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every ...
- 30.3k
5
votes
0
answers
98
views
Rational functions with trivial Weil symbols at every point
Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
- 4,316
1
vote
1
answer
131
views
Question about Jacobian subalgebra
Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.
Is it true that $\mathbb{C}[x, ...
- 6,262
3
votes
0
answers
285
views
Infinite sum in power series ring
Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
- 31
132
votes
3
answers
21k
views
When is the tensor product of two fields a field?
Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often ...
CommunityBot
- 1
1
vote
1
answer
62
views
Continuations of derivations of Jacobian subring
Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...
- 6,262
5
votes
0
answers
166
views
When do the spectra of overrings glue to a proper morphism?
This question is motivated by the construction of blowups.
Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between.
Let $X ...
- 1,338
1
vote
1
answer
465
views
In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal?
I have asked the following question at MSE and got one answer. Any further ideas are welcome:
Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$.
Let $\lambda, \...
- 6,262
3
votes
0
answers
157
views
F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex
Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex?
Thank you
- 51
9
votes
2
answers
5k
views
Applications of algebraic geometry/commutative algebra to biology/pharmacology
Are there applications of algebraic geometry/commutative algebra to biology/pharmacology?
It might be that some Gröbner basis technique is used somewhere? I know there are some applications to ...
4
votes
0
answers
321
views
Polynomial objects in any concrete category
EDIT: The original question had a trivial answer: it's just a coproduct. New question below
New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
- 438
3
votes
0
answers
158
views
A characterization for a commutative ring with a special intersection property for prime ideals
Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
- 61
8
votes
4
answers
2k
views
Formally étale at all primes does not imply formally étale?
All rings are assumed to be commutative and unital, with all homomorphisms unital as well.
On last week's homework, there was a mistake in one of the questions:
(2.5) Let $R\to S$ be a ...
- 19.6k
4
votes
0
answers
244
views
Uniqueness of $\delta$-structure on a $p$-torsion ring
I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...
- 63.9k
1
vote
0
answers
90
views
Automorphisms of the completion of a strict henselian local ring $R$ which come from automorphisms of $R$
Let $A\rightarrow R$ be a local homomorphism of Noetherian strict henselian local rings with completions $\hat{A},\hat{R}$.
Let $u\in R^\times, x\in R$ be such that there is a unique $\hat{A}$-linear ...
- 7,527
2
votes
2
answers
265
views
Commuting nilpotent matrices and conjugation isomorphisms
Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices.
Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
- 385
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
3
votes
1
answer
224
views
Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map is surjective?
Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map $x\mapsto x^2$ is surjective?
This is certainly true for fields. For DVR's, you can ...
- 19.6k
4
votes
1
answer
367
views
Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite
I was told that if $A$ is the subring of $\mathbb{C}[x_1,\ldots, x_n]$ generated by the polynomials $p_1(x_1,\ldots, x_n),\ldots, p_1(x_1,\ldots, x_n)$, then the preimage $p^{-1}(c)$ via the map $p = (...
- 156k
1
vote
0
answers
243
views
Non-zero divisors on an $I$-completely flat module
Let $A$ be a commutative ring (not necessarily Noetherian), $I=(f_1, f_2, \dots, f_n) \subseteq A$ a finitely generated ideal that is generated by a regular sequence. Let $M$ be an $A$-module, and ...
- 1,052
33
votes
0
answers
2k
views
Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
5
votes
1
answer
178
views
A $q$-analogue of a characterization of polynomials by binomial coefficients
Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely ...
- 109k
2
votes
1
answer
168
views
For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$?
Suppose we have $a^2 \equiv b^2 \bmod 4R$ where $R$ is an integral domain. Under what conditions on $R$ can we conclude that $a \equiv b \bmod 2R$?
This would hold if $2 \in R$ is a prime or the ...
- 109k
3
votes
0
answers
115
views
Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?
Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...
- 980
0
votes
3
answers
1k
views
non-associative but commutative algebra [closed]
Is it possible(or may be easier) to give an example of non associative algebra but commutative?
At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...
- 73.2k
1
vote
0
answers
146
views
A series that is algebraic?
This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic ...
- 3,998
9
votes
1
answer
876
views
A series that is rational?
Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ belongs to $k(X,Y)$? At first, it looked like it was simple. But in fact, I have no ...
- 50.8k
3
votes
1
answer
246
views
Surjective monoid homomorphism $\text{End}(B)\to \text{End}(A)$ given surjection $g:B\to A$
For any set $A\neq\varnothing$ let $\text{End}(A)$ denote the endomorphism monoid, consisting of all functions $f:A\to A$, together with composition. If $A, B\neq \varnothing$ are sets and $g:B\to A$ ...
4
votes
1
answer
176
views
Certain endomorphisms of $\mathbb{C}(x,y)$
Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$
satisfying the following two conditions:
(i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(...
- 7,722
0
votes
0
answers
83
views
Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
- 1,703
8
votes
4
answers
573
views
On the fixed point of automorphism of $\mathbb F_3[[T]]$
Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq 2$...
CommunityBot
- 1
0
votes
0
answers
147
views
Is it possible to compute the basis of this module?
Let $A$ be a polynomial algebra in $n$ variables over field $\mathbb{F}$ of characteristic zero which is algebraically closed. Assume that $a_1,\ldots, a_n, b_1,\ldots, b_n\in A$ are such that $a_1b_1+...
- 291
10
votes
0
answers
474
views
Is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p$ coherent?
The question is as in the title:
Is the ring $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p = \mathbb{Z}_p \otimes_{\mathbb{Z}_{(p)}} \mathbb{Z}_p$ coherent?
As shown in the related question, the ...
- 2,976
2
votes
0
answers
113
views
Computing whether a set of polynomials cuts out a projective variety
I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....
- 980
2
votes
0
answers
147
views
Characterization of algebraic integers providing a prime ideal
Let $\alpha$ be an algebraic integer and let $\mathcal{O}_{\mathbb{Q}(\alpha)}$ be the ring of integers of $\mathbb{Q}(\alpha)$.
Question: How to characterize an algebraic integer $\alpha$ such that $\...
4
votes
0
answers
691
views
Why are there elementary equations that are not solvable in closed form?
Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.
$\log\colon x\mapsto\log(x)$; $x\...
- 63.9k
13
votes
1
answer
797
views
Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent?
Let $\mathbb{Q}_p$ denote the field of fractions of $\mathbb{Z}_p$. By the answers to this quesition the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ cannot be a Noetherian ring (...
- 28.7k
9
votes
4
answers
557
views
Depth of modules and regular sequences of endomorphisms
Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ is a finitely generated $R$-module of depth $t$. It is well known that every maximal regular sequence of $M$ has length $t$. Recall that $x_1,...
- 30.5k
1
vote
0
answers
164
views
When every localization of the polynomial ring over a ring has finitely many idempotents
Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have ...
- 11
15
votes
7
answers
4k
views
Different definitions of the dimension of an algebra
I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...
- 3,318
3
votes
1
answer
161
views
Behavior of invariants under reduction mod p
Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...
- 3,504
1
vote
0
answers
102
views
Notation question: bigraded direct sum of graded objects
In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...
- 5,429
9
votes
3
answers
490
views
Generalization of Weak Nullstellensatz?
I believe the following is standard, namely when $k = \bar{k}$ is algebraically closed there is a bijection between points and maximal ideals
\begin{eqnarray*}
k^n &\longrightarrow& \...
- 4,745
2
votes
0
answers
300
views
Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"
Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage ...
- 105k
10
votes
0
answers
562
views
Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...